BV SOLUTIONS TO A DEGENERATE PARABOLIC EQUATION FOR IMAGE DENOISING
当表面活性剂遇到大环分子
114Univ. Chem. 2023, 38 (12), 114–119收稿：2023-06-27；录用：2023-08-01；网络发表：2023-08-11*通讯作者，Email:*****************.cn基金资助：2021年基础学科拔尖学生培养计划2.0研究课题(20211014)；天津市首批虚拟教研室试点建设项目(化学类交叉人才培养课程建设虚拟教研室)•专题• doi: 10.3866/PKU.DXHX202306051 当表面活性剂遇到大环分子阮文娟，李悦，耿文超，郭东升*南开大学化学学院，天津 300071摘要：近年来，表面和胶体化学与大环化学的结合引起了科学家的普遍关注。
将多样的大环结构引入表面活性剂分子，不仅极大地丰富了表面活性剂分子的种类，还可以赋予其大环的主客体识别功能。
由此所开发出的大环两亲和超两亲分子已在生物成像和药物递送中表现出很高的应用潜力。
从传统表面活性剂到大环两亲和超两亲分子的发展、应用表明，不同领域的交叉融合对科学研究的发展是非常重要的。
关键词：表面活性剂；胶束；大环结构；大环两亲分子；超两亲分子中图分类号：G64；O6Encountering of Surfactants with Macrocyclic MoleculesWen-Juan Ruan, Yue Li, Wen-Chao Geng, Dong-Sheng Guo *College of Chemistry, Nankai University, Tianjin 300071, China.Abstract: In recent years, the combination of surface and colloid chemistry with macrocyclic chemistry has garnered widespread attention among scientists. The integration of diverse macrocyclic structures into surfactant molecules not only greatly enriches the diversity of surfactants, but also imparts them with the host-guest recognition functionality of macrocycles. Macrocyclic amphiphiles and supra-amphiphiles, developed from this approach, have demonstrated high potential in applications such as bioimaging and drug delivery. The evolution from traditional surfactants to macrocyclic amphiphiles and supra-amphiphiles underscores the importance of interdisciplinary integration in advancing scientific research.Key Words: Surfactants; Micelles; Macrocycles; Macrocyclic amphiphiles; Supra-amphiphiles表面活性剂及其所构筑的胶束是表面和胶体化学中所涉及的一类非常重要的体系。
刘长春简历 刘长春档案 刘长春资料
刘长春简历刘长春档案刘长春资料各位读友大家好，此文档由网络收集而来，欢迎您下载，谢谢刘长春简历刘长春档案刘长春资料刘长春，人物名，著名人物有：第一位正式参加奥运会的中国运动员刘长春、中共河南省开封市委书记刘长春、吉林大学副教授刘长春、内正蓝旗哈毕日嘎乡乌兰村长刘长春、浙江省台州市国土资源局原局长刘长春、《语文知识报》特邀记者刘长春、吉林省德惠市委副书记刘长春、北京舞蹈学院附中教师刘长春、内蒙古太仆寺旗组织部长刘长春、山东省第三地质矿产勘查院院长刘长春等。
中国奥运第一人刘长春(照片1)刘长春，男，中国奥运第一人，第一位正式参加奥运会的中国运动员。
1909年生，辽宁省大连市北河口人，擅长短跑。
他的技术特点是步频快、步幅大、动作向前性好。
1927年底就读于东北大学体育系，1932年毕业。
同年7月8日，冒着生命危险从上海出发，冲破日本侵略者的封锁，成功参加了在美国洛杉矶举行的第10届奥运会，成为第一位正式参加奥运会的中国运动员。
1936年再次代表中国参加在柏林举行的第11届奥运会，因旅途颠簸、恢复不及时，两次参加奥运会均未能进入复赛。
自上世纪30年代起，历任东北大学、北京师范大学、东北中正大学体育助教、讲师、副教授。
新中国成立后，执教大连工学院(现大连理工学院)三十余载，历任第五届全国政协委员、中华全国体育总会常委、中国奥委会副主席、辽宁省体育协会副理事长、第四届辽宁省政协常委等职。
有《田径指导法》、《田径裁判法》、《短跑运动》等学术著作传世。
人物生平中国奥运第一人刘长春(照片2)1929年5月31日至6月2日，刘长春在沈阳举行的第14届华北运动会上，一举打破100米、200米和400米3个短跑项目的全国纪录，成绩分别是秒、秒和秒。
当时这些成绩非常令人鼓舞，因1928年阿姆斯特丹奥运会100米冠军的成绩也是10秒8。
1932年洛杉矶奥运会，当时原本我国有意参加足球赛，因为我国的足球实力在远东首屈一指。
但是美国人对足球没什么兴趣，取消了足球比赛，代之以举办美式足球做为表演赛。
210979659_双折线卷筒端板受力分析及应用
双折线卷筒端板受力分析及应用刘须军中船第九设计研究院工程有限公司 上海 200090摘 要：双折线卷筒很好地解决了多层卷绕钢丝绳排绳问题，提高了钢丝绳的使用寿命，在工程中得到广泛应用。
但目前国内相关技术文献对双折线卷筒端板设计基本上采用传统方法，端板受力情况与实际应用仍有一定差异，给设计工作带来诸多困扰。
文中对端板受力分析和计算进行了探讨，并对不同计算方法进行了比较，利用Matlab 和有限元技术计算了端板的应力和变形，为提高端板设计的安全性和可靠性提供了分析手段和参考依据。
关键词：双折线卷筒；多层卷绕；端板；受力分析；Matlab中图分类号：TH21 文献标识码：B 文章编号：1001-0785（2023）05-0044-05Abstract: Bilinear reel solves the rope arrangement problem of multi-layer winding wire rope well and improves the service life of wire rope, so it has been widely used in engineering. However, at present, according to the relevant domestic technical literature, the design of the end plate of the bilinear reel basically adopts the traditional method, and the stress situation of the end plate is different from the actual application, which brings many problems to the design. In this paper, the stress analysis and calculation of end plate are discussed, and different calculation methods are compared. The stress and deformation of end plate are calculated by Matlab and finite element technology, which provides a reference for improving the safety and reliability of end plate.Keywords: bilinear reel; multi-layer winding; end plate; force analysis; Matlab0 引言随着各种大扬程起重设备需求的快速增加，多层缠绕卷筒的应用也越来越多。
castep最全的资料clark2
Band Structure Calculations;Electronic and OpticalPropertiesStewart ClarkUniversity of DurhamOutline•Introduction to band structures •Calculating band structures using Castep •Calculating optical properties•Examples results•Some applicationsBand Structure Stewart Clark -University of 2Band StructureStewart Clark -University of 3Band Structures•Bloch’s theorem introduces a wavevector k .•It can always be confined to the 1st BZ (any k outside the 1st BZ can be mapped back into it).•The band index appears in Bloch’s theorem; for each k there are many solutions.kn k n k n E H ,,.Ψ=ΨThis leads to a description of the energy levels of electrons in a periodic potential in terms of a family of continuous functions E n,k .This is the band structure of the solid.Band Structure Stewart Clark -University of4Simple MethodsFree electrons give parabolic bands Using a basic basis set for the wavefunctions improves resultsslightlyIn Castep we use an accurate plane wave basis set.Band Structure Stewart Clark -University ofAlgorithm•Calculate self-consistent density and potential.•For a chosen k-point in the band structure calculate it’s eigenvalues and eigenvectors (wavefunctions) as follows:–Start from random wavefunction.–For the first band, calculate the search direction using self-consistent potential.–Assuming parabolic energy curve, step wavefunction coefficients to bottom in given direction.–Repeat previous two steps (using orthogonal search direction) until no change occurs.–For subsequent bands at this k-point, perform similar process, ensuring the bands remain orthogonal to all lower bands.Band Structure Stewart Clark -University of 6Castep Input Files•Can control k-points used in band structures by several cell keywords-bs_kpoints_list-bs_kpoints_path-bs_kpoints_mp_grid•Controlled by param keyword task -task : BandStructureBand Structure Stewart Clark -University of 7Band StructureStewart Clark -University of 9Silicon Band StructureBand StructureStewart Clark -University of 10Conventional Cell?•Although the full set of levels can be described with restricted k , it is often useful to allow k to range over more of k -space.•The set of wavefunctions and energy levels for 2 values of k differing by a reciprocal lattice vector K are identical.kn K k n k n K k n E E r r ,,,,)()(=Ψ=Ψ++By not using a primitive cell, the band structure contains redundant information.For each of the k =k +K we can map them back into the 1st Brillouin zone of the primitive cell.Band StructureStewart Clark -University of 11Non-primitive CellDensity of States•Many electronic properties depend on the electronic structure throughout the whole Brillouin zone.• A band structure usually shows the electronic states along lines of high symmetry.•Instead, we need to sample the whole Brillouin zone in a method similar to the SCF.Band Structure Stewart Clark -University of 12Castep Input filesFor DOS calculations, get regular sampling grid using:bs_kpoints_mp_grid : i j k Plotting a DOS from k-points on high symmetry lines (from bs_kpoints_path) is not correctBand Structure Stewart Clark -University of 13Example of DOS CalculationBand Structure Stewart Clark -University of 14More interesting examples•Defect states-Local density of states (project onto atoms)•Magnetic Materials-Spins not degenerate-Spin initialisation (ferro vs antiferro, etc)•Optical properties-Dielectric functions, refractive index•Band offsets-Examining the electrostatic potential•Other examples-Surface states,Kohn-Sham orbitalsBand Structure Stewart Clark -University of 15H 0t ates Al bulkH +Magnetic Materials•By default CASTEP calculations are non-magnetic•All bands are doubly degenerate•To release this constraint use either: -spin_polarised : true-nspins : 2Band Structure Stewart Clark -University of 17Magnetic Materials•Can also set initial magnetic state (param file):–spin : 3•Can constrain spin (param file):–spin_fix : 6•Can initialise atom spin states (cell file): %block positions_fracFe 0.0 0.0 0.0 SPIN=2Fe 0.5 0.5 0.5 SPIN=-2%endblock positions_fracBand Structure Stewart Clark -University of 18Band StructureStewart Clark -University of 19Magnetic MaterialsCalculating Optical Properties•task keyword in param file-task : Optics•In the cell file use one of-optics_kpoints_list-optics_kpoints_path-optics_kpoints_mp_grid •Calculation similar to bandstructure, but optical matrix elements also calculated Band Structure Stewart Clark -University of 20Optical Properties •Properties include-Reflectivity-Absorption-Refractive Index-Dielectric Function-Conductivity•For-Polarised light-Unpolarised light-Polycrystalline materialsBand Structure Stewart Clark -University of 21Band StructureStewart Clark -University of22Example of Optical PropertiesBand StructureStewart Clark -University of 23BS Application –Band Offsets •Heterojunctions are formed when two different types of semiconductor are joined together.•Heterostructures are used extensively in the electronic (e.g. transistors) and optoelectronic (e.g.LED’s) industry.Calculating Band Offsets •Knowledge of the band structures of the two individual materials is not sufficient to determine the band offset.•Band energies are determined with respect to the average potential in the solid.•It is also necessary to perform a supercell calculation to determine how the potentials are lined up withrespect to each other.Band Structure Stewart Clark -University of 242useful for LED’s and near-IR detectorsBand StructureStewart Clark -University of 28Applications -Surfaces •We can also use the supercell method to calculate the band structures of surfaces.•The wavefunctions at surfaces decay exponentiallyinto the vacuum region.Band StructureStewart Clark -University of 29Details of GaAs SurfaceSurface supercell There is reconstruction of the surface –this is calculated first.A band structure of the supercell is then used for the surface band structure.Band StructureStewart Clark -University of 30The GaAs Band Structure •K-points are chosenusing the surfaceBrillouin zone.•Many regions have acontinuous energyspectrum, while gapsstill exist.•The details of thesurface band structuredepend on the details ofthe surfacereconstruction.Applications -Orbitals•In addition to the energy eigenvalues for a given material, a band structure calculation also gives the eigenvectors (wavefunctions) for any point in theBrillouin zone.•It should be noted that no proof exists which confirms that the DFT single particle wavefunctions generated here correspond to the many particle wavefunction.•However, the orbitals generated can lead to useful physics and give further insight in the the nature ofthe bonding in materials.Band Structure Stewart Clark -University of 31Example of DFT Orbitals •Instead of summing up the squares of all the wavefunctions from each electron, we can look ateach one individually.•Each one will be the charge density for a Kohn-Sham orbital.•This gives a chemistry point of view!•We can examine the electronic structure electron by electron.•Note: Kohn-Sham orbitals/one electron orbitals: meaning is not necessarily well defined!Band Structure Stewart Clark -University of 32Band StructureStewart Clark -University of 33Example of OrbitalsExample shown is for LaMnO 3–a magnetic material.Castep Band Structure KeywordsType the following:castep -help search bsBS_NEXTRA_BANDS *! number of extra bandstructure bandsBS_PERC_EXTRA_BANDS *! percentage of extra bandstructure bandsBS_NBANDS *! number of bands/k-point in band structure calculationBS_EIGENVALUE_TOL *! band convergence toleranceBS_XC_FUNCTIONAL *! BS exchange-correlation functionalBS_KPOINT_PATH *! Band structure pathBS_KPOINT_PATH_SPACING *! Band structure path spacingBS_KPOINT_LIST *! Band structure k-point listBS_KPOINT_MP_GRID *! Band structure Monkhorst-Pack grid Band Structure Stewart Clark -University of 34。
植物乳杆菌的生理特性及影响其增殖的因素
特产研究169Special Wild Economic Animal and Plant ResearchDOI：10.16720/ki.tcyj.2022.083植物乳杆菌的生理特性及影响其增殖的因素马翠柳1，王金铭1，袁伟涛1，赵德辉2，刘晗璐3※（1.中国农业科学院特产研究所，吉林长春130112；2.河北科技师范学院，河北秦皇岛066000；3.赤峰学院，内蒙古赤峰024000）摘要：植物乳杆菌作为一类有益菌，在人类生产生活中应用广泛。
在其生长繁殖过程中会产生具有抑菌效果、调节免疫功能的代谢产物，在一定程度上可起到类抗生素的作用。
碳源、氮源是植物乳杆菌生长过程中的必需营养物质，主导菌体细胞壁的形成和代谢物质的合成。
适宜的温度、pH及接种量等发酵条件对菌体繁殖速度和生产效益发挥关键作用。
本文对影响植物乳杆菌增殖的底物和条件进行综述，以期为增加生产效率、提高生产效益奠定基础。
关键词：植物乳杆菌；生理特性；增殖中图分类号：TS201.3文献标识码：A文章编号：1001-4721（2023）03-0169-05Physiological Characteristics and Factors of Proliferationon MA Cuiliu1,WANG Jinming1,YUAN Weitao1,ZHAO Dehui2,LIU Hanlu3※(1.Institute of Special Animal and Plant Sciences of Chinese Academy of Agricultural Sciences,Changchun130112, China;2.Hebei scientific and technical teachers'college,Qinhuangdao066000,China;3.Chifeng University,Chifeng024000,China)Absrtact:As a sort of probiotic，Lactobacillus plantarum was widely used in human production activity，metabolites with the antibacterial effects and immunomodulatory produced in the growth works as well as antibiotics.Carbon source and nitrogen source are essential culture medium，which dominate the generation of cell and synthesis of metabolic substance.Proper temperature，pH，bacterial load and so on play a key role in multiplicative process and performance.The medium and fermentation conditions of cultivate Lactobacillus plantarum are re-viewed，which would lay a foundation for increasing production efficiency and enhancing benefit.Keywords:Lactobacillus plantarum;physiological characteristics;proliferation乳酸菌为当今益生菌生产应用中最广泛的菌种，乳酸杆菌属包含50多个不同的物种，在乳酸菌中占据主要地位。
β-氨基酸聚合物用于协同增效逆转白色念珠菌对伊曲康唑的耐药性
功 能 高 分 子 学 报Vol. 35 No. 6 532Journal of Functional Polymers2022 年 12 月文章编号： 1008-9357(2022)06-0532-08DOI: 10.14133/ki.1008-9357.20220403001β-氨基酸聚合物用于协同增效逆转白色念珠菌对伊曲康唑的耐药性马凯茜， 张东辉， 施 超， 顾佳蔚， 刘润辉（华东理工大学生物反应器工程国家重点实验室, 超细材料制备与应用教育部重点实验室,教育部医用生物材料工程研究中心, 材料科学与工程学院, 上海 200237）摘 要： 设计合成了与伊曲康唑具有协同活性的系列β-氨基酸聚合物。
通过β-氨基酸N-硫代羧基酸酐(β-NTA)开环聚合的方法，将不同比例疏水性单体DL-β-正亮氨酸N-羧基硫代羰基环内酸酐(简称Bu)和阳离子单体N(α)-Z-DL-2,3-二氨基丙酸N-羧基硫代羰基环内酸酐(简称DAP)进行共聚，得到了系列β-氨基酸聚合物(DAP x Bu y)n。
抗菌测试表明，制备的(DAP x Bu y)n聚合物可通过协同增效，有效逆转白色念珠菌(C. albicans)对伊曲康唑的耐药性，使伊曲康唑的抗真菌最低抑制质量浓度从单药的大于200 μg/mL降低至协同后的3.1 μg/mL，即从无效逆转为高效抗真菌活性。
此外，(DAP x Bu y)n聚合物在400 μg/mL的高浓度下基本没有造成明显的人血红细胞溶血和细胞毒性。
(DAP x Bu y)n聚合物能实现高效协同增效和逆转真菌对伊曲康唑的耐药性。
关键词： β-氨基酸聚合物；伊曲康唑；协同增效；逆转真菌耐药性；抗真菌中图分类号： R318.08 文献标志码： ASynergistic Effect of β-Amino Acid Polymers and Itraconazole onReversing Drug Resistance in C. albicansMA Kaiqian, ZHANG Donghui, SHI Chao, GU Jiawei, LIU Runhui（State Key Laboratory of Bioreactor Engineering, Key Laboratory for Ultrafine Materials of Ministry of Education, Research Center for Biomedical Materials of Ministry of Education, School of Materials Science and Engineering, East ChinaUniversity of Science and Technology, Shanghai 200237, China）Abstract: In this study, a series of β-amino acid polymers which have synergistic antifungal activity with itraconazole were designed and synthesized. The random copolymers (DAP x Bu y)n were obtained by ring-opening polymerization of β-amino acid N-thiocarboxyanhydrides (β-NTA) under room temperature using 4-tert-Butylbenzylamine (t BuBz-NH2) as an initiator, with DL-β-norleucine N-thiocarboxyanhydrides as hydrophobic monomer and N(α)-Z-DL-2,3-diaminopropionic acid N-thiocarboxyanhydrides as cationic monomer. The effect of (DAP x Bu y)n combined with itraconazole on C. albicans was evaluated by checkerboard antifungal test. The test showed that (DAP x Bu y)n copolymers could effectively reverse itraconazole resistance in C. albicans through synergistic effect, while the minimum inhibitory concentration (MIC) of antifungal of itraconazole was reduced from more than 200 μg/mL to 3.1 μg/mL after exposure to (DAP x Bu y)n, indicating that the收稿日期： 2022-04-03基金项目： 国家自然科学基金(22075078, 21861162010)；上海市优秀学术带头人(20XD1421400)作者简介： 马凯茜（1996—），女，硕士生，研究方向为生物医用高分子材料。
SDSecondDerivativeTest-MIT：SD二阶导数测试-麻省理工学院
SD.Second Derivative Test1.The Second Derivative TestWe begin by recalling the situation for twice differentiable functions f(x)of one variable. Tofind their local(or“relative”)maxima and minima,we1.find the critical points,i.e.,the solutions of f′(x)=0;2.apply the second derivative test to each critical point x0:f′′(x0)>0⇒x0is a local minimum point;f′′(x0)<0⇒x0is a local maximum point.The idea behind it is:at x0the slope f′(x0)=0;if f′′(x0)>0,then f′(x)is strictly increasing for x near x0,so that the slope is negative to the left of x0and positive to the right,which shows that x0is a minimum point.The reasoning for the maximum point is similar.If f′′(x0)=0,the test fails and one has to investigate further,by taking more derivatives, or getting more information about the graph.Besides being a maximum or minimum,such a point could also be a horizontal point of inflection.The analogous test for maxima and minima of functions of two variables f(x,y)is a little more complicated,since there are several equations to satisfy,several derivatives to be taken into account,and another important geometric possibility for a critical point,namely a saddle point.This is a local minimax point;around such a point the graph of f(x,y) looks like the central part of a saddle,or the region around the highest point of a mountain pass.In the neighborhood of a saddle point,the graph of the function lies both above and below its horizontal tangent plane at the point.(Your textbook has illustrations.) The second-derivative test for maxima,minima,and saddle points has two steps.1.Find the critical points by solving the simultaneous equations f x(x,y)=0,f y(x,y)=0. Since a critical point(x0,y0)is a solution to both equations,both partial derivatives are zero there,so that the tangent plane to the graph of f(x,y)is horizontal.2.To test such a point to see if it is a local maximum or minimum point,we calculate the three second derivatives at the point(we use subscript0to denote evaluation at(x0,y0), so for example(f)0=f(x0,y0)),and denote the values by A,B,and C:(1)A=(f xx)0,B=(f xy)0=(f yx)0,C=(f yy)0,(we are assuming the derivatives exist and are continuous).Second-derivative test.Let(x0,y0)be a critical point of f(x,y),and A,B,and C be as in(1).ThenAC−B2>0,A>0or C>0⇒(x0,y0)is a minimum point;AC−B2>0,A<0or C<0⇒(x0,y0)is a maximum point;AC−B2<0⇒(x0,y0)is a saddle point.SD.SECOND DERIVATIVE TEST 1If AC −B 2=0,the test fails and more investigation is needed.Note that if AC −B 2>0,then AC >0,so that A and C must have the same sign.Example 1.Find the critical points of w =12x 2+y 3−12xy and determine their type.Solution.We calculate the partial derivatives easily:(2)w x =24x −12y w y =3y 2−12x A =w xx =24B =w xy =−12C =w yy =6yTo find the critical points we solve simultaneously the equations w x =0and w y =0;we get w x =0w y =0⇒y =2x y 2=4x ⇒4x 2=4x ⇒x =0,1⇒(x,y )=(0,0)(x,y )=(1,2).Thus there are two critical points:(0,0)and (1,2).To determine their type,we use the second derivative test:we have AC −B 2=144y −144,so thatat (0,0),we have AC −B 2=−144,so it is a saddleat (1,2),we have AC −B 2=144and A >0,so it is minimum point.A plot of the level curves is given at the right,which firms the above.Note that the behavior of the level near the origin can be determined by using the w ≈12x 2−12xy ;this shows the level curves near (0,0)like those of the function x (x −y ):the family of x (x −y )=c ,with asymptotes given by the degenerate bola x (x −y )=0,i.e.,the pair of lines x =0(the y -axis)x −y =0(the diagonal line y =x ).2.Justification for the Second-derivative The test involves the quantity AC −B 2.In general,B 2−4AC or B 2−AC or their negatives,it means the quadratic formula is involved,in one of its two forms (the second is often used to get rid of the excess two’s):Ax 2+Bx +C =0⇒x =−B ±√B 2−4AC 2A(3)Ax 2+2Bx +C =0⇒x =−B ±√B 2−AC A(4)This is what is happening here.We want to know whether,near a critical point P 0,the graph of our function w =f (x,y )always stays on one side of its horizontal tangent plane (P 0is then a maximum or minimum point),or whether it lies partly above and partly below the tangent plane (P 0is then a saddle point).As we will see,this is determined by how the graph of a quadratic function f (x )lies with respect to the x -axis.Here is the basic lemma.Lemma.For the quadratic function Ax 2+2Bx +C ,AC −B 2>0,A >0or C >0⇒Ax 2+2Bx +C >0for all x ;(5)AC −B 2>0,A <0or C <0⇒Ax 2+2Bx +C <0for all x ;(6)AC −B 2<0⇒ Ax 2+2Bx +C >0,for some x ;Ax 2+2Bx +C <0,for some x.(7)218.02NOTESProof of the Lemma.To prove(5),we note that the quadratic formula in the form(4) shows that the zeros of Ax2+2Bx+C are imaginary,i.e.,it has no real zeros.Therefore its graph must lie entirely on one side of the x-axis;which side can be determined from either A or C,sinceA>0⇒limx→∞Ax2+2Bx+C=∞;C>0⇒Ax2+2Bx+C>0when x=0.If A<0or C<0,the reasoning is analogous and proves(6).If on the other hand AC−B2<0,formula(4)shows the quadratic function has two real roots,so that its parabolic graph crosses the x-axis twice,and hence lies partly above and partly below it.This proves(7).Proof of the Second-derivative Test in a special case.The simplest function is a linear function,w=w0+ax+by,but it does not in general have maximum or minimum points and its second derivatives are all zero.The simplest functions to have interesting critical points are the quadratic functions,which we write in the form(the2’s will be explained momentarily):(8)w=w0+ax+by+12(Ax2+2Bxy+Cy2).Such a function has in general a unique critical point,which we will assume is(0,0);this gives the function a special form,which we can determine by evaluating its partial derivatives at(0,0):(9)(w x)0=a(w y)0=bw xx=Aw xy=Bw yy=C(The neat look of the above explains the12and2B in(8).)Since(0,0)is a critical point,(9)shows that a=0and b=0,so our quadratic function has the form(10)w−w0=12(Ax2+2Bxy+Cy2).We moved w0to the left side since the tangent plane at(0,0)is the horizontal plane w=w0, and we are interested in whether the graph of the quadratic function lies above or below this tangent plane,i.e.,whether w−w0>0or w−w0<0at points other than the origin.If(x,y)=(0,0),then either x=0or y=0;say y=0.Then we write(10)as(11)w−w0=y22 Ax y 2+2B x y +CWe know that y2>0if y=0;applying our previous lemma to the factor on the right of (11),(or if y=0,switching the roles of x and y in(11)and applying the lemma),we getAC−B2>0,A>0or C>0⇒w−w0>0for all(x,y)=(0,0);⇒(0,0)is a minimum point;AC−B2>0,A<0or C<0⇒w−w0<0for all(x,y)=(0,0);⇒(0,0)is a maximum point;AC−B2<0⇒ w−w0>0,for some(x,y);w−w0<0,for some(x,y);⇒(0,0)is a saddle point.SD.SECOND DERIVATIVE TEST3Argument for the Second-derivative Test for a general function.This part won’t be rigorous,only suggestive,but it will give the right idea.We consider a general function w=f(x,y),and assume it has a critical point at(x0,y0), and continuous second derivatives in the neighborhood of the critical point.Then by a generalization of Taylor’s formula to functions of several variables,the function has a best quadratic approximation at the critical point.To simplify the notation,we will move the critical point to the origin by making the change of variablesu=x−x0,v=y−y0.Then the best quadratic approximation is(if the x,y on the left and u,v on the right is upsetting,just imagine u and v replaced everywhere by x−x0and y−y0):(13)w=f(x,y)≈w0+12 Au2+2Buv+Cv2 ;here the coefficients A,B,C are given as in(1)by the second partial derivatives with respect to u and v at(0,0),or what is the same(according to the chain rule—see the footnote below), by the second partial derivatives with respect to x and y at(x0,y0).(Intuitively,one can see the coefficients have these values by differentiating both sides of(13)and pretending the approximation is an equality.There are nolinear terms in u and v on the right since(0,0)is a critical point.)Since the quadratic function on the right of(13)is the best approximation to w=f(x,y) for(x,y)close to(x0,y0),it is reasonable to suppose that their graphs are essentially the same near(x0,y0),so that if the quadratic function has a maximum,minimum or saddle point there,so will f(x,y).Thus our results for the special case of a quadratic function having the origin as critical point carry over to the general function f(x,y)at a critical point(x0,y0),if we interpret A,B,C as the second partial derivatives at(x0,y0).This is what the second derivative test says.Exercises:Section2HFootnote:Using u=x−x0and v=y−y0,we can apply the chain rule for partial derivatives,which tells us that for all x,y and the corresponding u,v,we havew x=w u ∂u∂x+w v∂v∂x=w u,since u x=1and v x=0,and similarly,w y=w v.Therefore at the corresponding points,(w x)(x0,y0)=(w u)(0,0),(w y)(x0,y0)=(w v)(0,0),and differentiating once more and using the same reasoning,(w xx)(x0,y0)=(w uu)(0,0),(w xy)(x0,y0)=(w uv)(0,0),(w yy)(x0,y0)=(w vv)(0,0).18.02Notes and Exercises by A.Mattuck,with the assistance of T.Shifrin and S.LeDucc M.I.T.20101。
英语数学词汇D
数学专业词汇对照以字母D开头d integrable d 可积d integral d 积分d'alembert principle 达朗贝尔原理d'alembert ratio test 达朗贝尔比例试验法d'alembert solution 达朗贝尔解d'alembertian 达朗伯符;达郎贝尔算子damped harmonic oscillation 阻尼谐振动damped oscillation 阻尼振动damped vibration 阻尼振动damping 阻尼damping factor 阻尼因子dantzig van de panne method 但泽范德潘方法darboux tangent 达布切线darboux theorem 达布定理data 数据data processing 数据处理data storage 数据存储器data storage register 数据存储寄存器death process 死亡过程death rate 死亡率debugging 堤序deca 十decade 十个decade scaler 十进制计数器decagon 十边形decahedron 十面体decameter 十米decay curve 衰变曲线deci 分decidability 可判定性decile 十分位数decimal 十进位的decimal arithmetic 十进算术decimal binary conversion 十二进制变换decimal digit 十进制数字decimal expansion 十进制展开decimal fraction 十进小数decimal notation 十进制记数法decimal number 十进小数decimal number system 十进制decimal of many places 多位十进小数decimal part 小数部分decimal place 小数位decimal point 小数点decimal representation 十进制记数法decimal system 十进制decimal to binary conversion 十二进制变换decimetre 分米decision 判定decision domain 决策域decision function 判定函数decision problem 判定问题decision procedure 判定过程decision space 判定空间decision theory 决策论decision variable 决策变量decision vector 决策向量decisive 决定的declination 倾斜decoder 译码器decomposability 可分解性decomposable form 可分解形式decomposable matrix 可分解矩阵decomposable operator 可分解算子decompose 分解decomposition 分解decomposition field 分解域decomposition formula 分解公式decomposition group 分解群decomposition in a direct sum 直和分解decomposition into linear factors 线性因子分解decomposition into partial fractions 部分分数分解decomposition operator 分解算子decomposition principle 分解原理decomposition theorem 分解定理decrease 减少decreasing function 递减函数decrement 减量dedekind axiom 绰金公理dedekind completion 绰金完备化dedekind cut 绰金切断dedekind domain 绰金环dedekind ring 绰金环dedekind set 绰金集dedekind sum 绰金和deduce 演绎deducibility 可推断deduction 演绎法deductive method 演绎法deductive proof 演绎证明defect *defect indices 扛数defect of operators 算子的*defect of spline 样条的筐defect relation 控系defect subspaces 坑空间defective number *defective value 康deferent 圆心轨迹deficiency *deficiency index 扛标deficient number *definability 可定义性definable 可定义的define 定义definiendum 被定义者definiens 定义者defining contrast 定义对比defining equation 定义方程defining field 定义域defining relations 定义关系definite 定的definite divergence 定发散definite integral 定积分definiteness 梅性definition by induction 用归纳法定义definition by transfinite induction 依超限归纳法的定义deflation 降阶deform 使变形deformable 可变形的deformation 变形deformation ratio 形变比率deformation retract 形变收缩核deformation retraction 形变收缩degeneracy 退化degeneracy operator 退化算子degenerate 退化degenerate case 退化情况degenerate core 简并核degenerate distribution 退化分布degenerate eigenvalue 退化本盏degenerate extreme point 退化极值点degenerate kernel 退化核degenerate parabolic equation 退化抛物型方程degenerate polyhedron 退化多面体degenerate set 退化集degenerate simplex 退化单形degeneration 退化degree 次数degree of a polynomial 多项式的次数degree of a representation 表示度degree of accuracy 精确度degree of an equation 方程式的次数degree of approximation 近似度degree of freedom 自由度degree of inseparability 不可分次数degree of mapping 映射度degree of stability 稳定度degree of symmetry 对称度del 倒三角形del operator 倒三角形delay 延迟delay equation 延滞方程delay line store 延迟线存储器delay time 延迟时间delete 删去deleted neighborhood 去心邻域deletion 删除delocalization 非局部化delta function 狄垃克函数deltoid 形曲线demarcation 划分界线demi continuous 半连续的demonstrate 证明论证demonstration 证明denominate number 庚denomination 名称denominator 分母denote 指示dense 稠密的dense in itself 自密的dense in itself set 自密集dense set 稠集dense subset 稠子集denseness 稠密性denseness of set 集的密度densimetry 密度测定density 密度density distribution 密度分布density function 密度函数density matrix 密度矩阵density of distribution 分布密度density of simultaneous distribution 联合分布密度density theorem 密度定理denumerability 可数性denumerable 可数的denumerable set 可数集denumeration 计算depend 依赖dependence 相关dependent 相关的dependent equations 相关方程组dependent variable 应变数dependent variate 应变量depression 降低depth line 深度线derivability 可微性derivable 可微的derivate 导出数derivation 微分derivative 导数derivative of a distribution 分布导数derivative of a vector 向量导数derivative of higher order 高阶导数derivative of n th order n 阶导数derive 导出derived algebra 导出代数derived equation 导出方程derived function 导数derived functor 导函子derived graph 导出图derived rule of inference 推理的导出规则derived series 导出列derived set 推导集derived unit 导出单位derogatory matrix 减次阵descartes rule of signs 笛卡儿正负号规则descending central series 降中心列descending chain 降链descending chain condition 降链条件descending difference 前向差分descending induction 递减归纳descending order 递减次序descending power series 递减幂级数descent 下降descent method 下降法description 描述description operator 摹状算子descriptive form 描述形式descriptive function 描述形式descriptive geometry 画法几何descriptive set theory 描述集论descriptive statistics 描述统计学design 计划design of experiments 实验设计detached coefficients 分离系数determinant 行列式determinant of infinite order 无限行列式determinant of the coefficients 系数行列式determinant of the coefficients of a linear form 线性形式的系数行列式determinantal divisor 行列式因子determinantal equation 行列式方程determinate 一定的determinate automaton 确定性自动机determinate system 确定组determine 决定出determined system 确定组determining equation 决定方程determining factor 决定因素deterministic digital system 确定性数字系统deterministic optimization 确定性最优化deterministic process 确定过程deterministic programming 确定性最优化develop 展开developability 可展性developable 可展的developable function 可展函数developable surface 可展曲面development 展开development in power series 幂级数展开deviate 偏离deviation 偏差deviation from the mean 平均偏差diadic system 二进制数系diagnostic routine 诊断程序diagonal 对角线diagonal continued fraction 对角连分数diagonal dominancy 对角优势diagonal element 对角元素diagonal form 对角型diagonal map 对角映射diagonal matrix 对角阵diagonal method 对角线法diagonal morphism 对角射diagonal of a determinant 行列式的对角线diagonal of the face 面对角线diagonal point 对边点diagonal procedure 对角线法diagonal process 对角线法diagonal sequence 对角序列diagonal sum 矩阵的迹diagonal sum rule 对角求和规则diagonalizable matrix 可对角化矩阵diagonalization 对角线化diagonalize 对角化diagonally dominant matrix 对角占优矩阵diagram 图表diagram scheme 图解概型diameter 直径diameter of a circle 圆的直径diametric plane 径面diamond shaped 菱形的dichotomy 二分法diffeomorphic mapping 微分同胚映射diffeomorphism 微分同胚映射difference 差difference boundary value problem 差分边值问题difference differential equation 差分微分方程difference equation 差分方程difference group 差群difference method 差分法difference operator 差分算子difference product 差积difference quotient 均差difference schema 差分格式difference sequence 差数序列difference set 差集difference table 差分表different 共轭差积differentiability 可微性differentiable 可微的differentiable function 可微函数differentiable manifold of class c c 类微分廖differential 微分differential algebra 微分代数differential analyzer 微分分析仪differential and integral calculus 微积分differential calculus 微分学differential circuit 微分电路differential coefficient 微分系数differential cross section 微分截面differential curve 微分曲线differential difference equation 差分微分方程differential equation 微分方程differential equation with delayed argument 延滞方程differential equation with deviating argument 偏差自变数微分方程differential equation with lag 滞后微分方程differential equation with separated variables 分离变数型微分方程differential expression 微分式differential form 微分形式differential form of the first kind 第一种微分形式differential game 微分对策differential geometry 微分几何学differential ideal 微分理想differential method 微分法differential of arc 微弧differential operator 微分算子differential parameter 微分参数differential quotient 微分系数differential ring 微分环differential scattering 微分散射截面differential topology 微分拓扑differentiate 微分differentiating circuit 微分电路differentiation 微分differentiation of a function 函数的微分法differentiation of implicit function 隐函数微分法differentiation operator 微分算子differentiation symbol 微分记号differentiation term by term 逐项微分differentiation theorem 微分定理differentiator 微分器diffraction 衍射diffraction angle 衍射角diffraction curve 衍射曲线diffraction disc 绕射盘diffusion 扩散diffusion coefficient 扩散系数diffusion constant 扩散常数diffusion equation 扩散方程diffusion process 扩散过程digamma function 双函数digit 数字digital 数字的digital computer 数字计算机digital control 数字控制digital differential analyzer 数字微分分析仪digital recorder 数字式自动记录器digital simulation 数据模拟digitize 计数化dihedral angle 二面角dihedral group 二面体群dihedron 二面体dilatation 单项变换dilated maximum principle 扩张极大值原理dilemma 二难推论dimension 量纲dimension theorem 维数定理dimension theory 维数论dimensional 量纲的dimensional analysis 维量分析dimensional equation 量纲方程dimensionality 量纲dimensionless 无量纲的dimensionless quantity 无因次量dimer 二聚物dimetric 二维的diophantine analysis 丢番图分析diophantine equation 丢番图方程diplohedron 扁方二十四面体dirac delta distribution 狄垃克函数dirac equation 狄拉克方程dirac measure 狄拉克测度direct 直接的direct analytic continuation 直接解析开拓direct correspondence 直接对应direct decomposition 直分解direct factor 直积因子direct image 直接象direct limit 归纳极限direct method 直接法direct numerical method 直接数值法direct predecessor 直前仟direct product 直积direct successor 紧接后元direct sum 直和direct system 归纳系direct union 直并directed circuit 有向回路directed distance 有向距离directed edge sequence 有向棱序列directed graph 有向图directed group 有向群directed line 有向元directed line segment 有向线段directed path 有向通路directed quantity 有向量directed set 有向集directed system 有向系directing curve 有向曲线direction 方向direction angle 方向角direction cosine 方向余弦direction field 方向场direction of principal axis 轴方向direction of principal curvature 助率方向direction parameter 方向参数directional 定向的directional derivative 方向导数directional differentiation 方向微分法directional field 方向场directivity 方向性directly proportional 直接比例的directoin search program 方向检颂序director circle 准圆director cone 准锥面director plane 准平面directrix 准线directrix of a conic 二次曲线的准线dirichlet boundary condition 狄利克雷边界条件dirichlet conditions 狄利克雷条件dirichlet distribution 狄利克雷分布dirichlet domain 狄利克雷域dirichlet drawer principle 狄利克雷抽屉原理dirichlet function 狄利克雷函数dirichlet integral 狄利克雷积分dirichlet principle 狄利克雷原理dirichlet problem 狄利克雷问题dirichlet product 狄利克雷乘积dirichlet series 狄利克雷级数dirichlet space 狄利克雷空间dirichlet theorem 狄利克雷定理disagreement 不符合disappearance 消失disassembly 拆卸disc 圆盘disconnected space 不连通空间discontinuity 不连续discontinuity interval 不连续区间discontinuity on the left 左方不连续性discontinuity on the right 右方不连续性discontinuous function 不连续函数discontinuous group 不连续群discontinuous random variable 不连续变量discontinuous set 不连续集discontinuous term 不连续项discontinuous variate 不连续变量discontinuum 密断统discount 折扣discount factor 折扣因子discrete 分立的discrete category 离散范畴discrete continuous system 离散连续系统discrete distribution 离散分布discrete distribution function 离散分布函数discrete flow 离散流discrete fourier transform 离散傅里叶变换discrete group 离散群discrete mathematics 离散数学discrete optimization 离散最佳化discrete optimization problem 离散最优化问题discrete problem 离散问题discrete process 离散随机过程discrete programming 离散规划discrete random variable 离散随机变量discrete series 离散序列discrete set 离散集discrete spectrum 离散谱discrete state 离散状态discrete system 离散系统discrete time 离散时间discrete topological space 离散拓扑空间discrete topology 离散拓扑discrete uniform distribution 离散均匀分布discrete valuation 离散赋值discreteness 离散性discretization 离散化discretization error 离散化误差discrimator 判别式函数discriminant 判别式discriminant analysis 判别分析discriminant function 判别式函数discriminant of a polynomial 多项式的判别式discriminatory analysis 判别分析disjoint elements 不相交元素disjoint relations 不相交关系disjoint sets 不相交集disjoint sum 不相交并集disjoint union 不相交并集disjointed set 不相交集disjunction 析取disjunction sign 析取记号disjunction symbol 析取记号disjunctive normal form 析取范式disjunctive proposition 选言命题disk 圆盘disorder 无秩序disorder order transformation 无序有序变化dispersion 方差dispersion matrix 方差矩阵dispersion relations 分散关系dispersive 扩散的displacement 位移displacement operator 位移算符display statusconcomitant 相伴式disposition 配置disproportion 不相称disproportionate 不成比例的dissection 剖分dissimilar terms 不同类项dissipation 散逸dissipation of energy 消能dissipative function 散逸函数dissipative measurable transformation 散逸可测变换dissipative system 耗散系dissociation 解离dissociation constant 分离常数distance axioms 距离公理distance between two points 两点间距distance circle 距离圆distance function 距离函数distance matrix 距离矩阵distance meter 测距仪distance point 距离点distinction 差别distinguish 辨别distinguished polynomial 特异多项式distortion 畸变distortion angle 畸变角distortion theorem 畸变定理distortionless 无畸变的distributed constant 分布常数distributed parameter 分布参数distribution 分布distribution coefficient 分布系数distribution curve 分布曲线distribution family 分布族distribution function 分布函数distribution law 分布律distribution of prime numbers 素数分布distribution parameter 分布参数distribution ratio 分布系数distribution rule 分布规则distribution space 广义函数空间distribution with negative skewness 负偏斜分布distribution with positive skewness 正偏斜分布distributionfree test 无分布检验distributive 分配的distributive lattice 分配格distributive law 分配律distributivity 分配性disturbance 扰动disturbing function 扰动函数diverge 发散divergence 发散divergence of a series 级数发散divergence of tensor field 张量场的散度divergence of vector field 向量场的散度divergent sequence 发散序列divergent series 发散级数divide 除divided difference 均差dividend 被除数divider compasses 除法器两脚规dividers 除法器两脚规divisibility 可除性divisible 可除的divisible element 可除元素division 除法;划分division algebra 可除代数division algorithm 辗转相除法division of a line segment 线段的分割division ring 可除环division transformation 有剩余的除法division with remainder 有剩余的除法divisor 因divisor class 除子类divisor function 除数函数divisor problem 除数问题documentation 文件编制documentation of program 程序文档dodecagon 十二边形dodecagonal 十二边形的dodecahedral number 十二面体数dodecahedron 十二面体dog curve 追踪曲线domain 定义域domain of attraction 吸引范围domain of convergence 收敛域domain of definition 定义域domain of dependence 依赖域domain of existence 存在域domain of integration 积分区域domain of integrity 整环domain of meromorphy 亚纯域domain of regularity 正则域domain of transitivity 可递域domain of unsolvability 不可解域domain of variability 定义域dominant 帜dominant strategy 优策略dominant weight 最高权dominate 支配dominated convergence 控制收敛dominating set 控制集domination 支配domination principle 优势原理domino problem 多米诺问题dot 点dot chart 点图表dot product 纯量积dotted 点线的dotted line 点线dotted spinor 有点旋量double 双的double angle formulas 倍角公式double chain complex 双链复形double complex 二重复形double cone 对顶锥double coset 重倍集double cusp 双尖点double element 二重元素double exponential distribution 二重指数分布double folium 双叶线double fourier series 二重傅里叶级数double integral 二重积分double laplace transformation 二重拉普拉斯变换double layer 双层double layer potential 双层位势double limit 二重极限double line 二重线double loop 双环路double negation 双重否定double orthogonal system 二重正交系double periodicity 双周期性double plane 二重面double point 重点double point of curve 曲线的二重点double poisson distribution 二重泊松分布double product 二重积double ratio 交比double root 重根double sequence 二重数列double series 二重级数double subscript 双下标double sum 二重和double tangent 二重切线double valued function 双值函数double vector product 二重向量积doubly periodic function 双周期函数dozen 一打draw 拉drum 磁鼓dual abelian variety 对偶阿贝耳簇dual automorphism 逆自同构dual base 对偶基dual basis 对偶基dual category 对偶范畴dual cell 对偶胞腔dual complex 对偶复形dual cone 对偶锥dual curve 对偶曲线dual figure 对偶图dual form 对偶形式dual formula 对偶公式dual graph 对偶图dual group 特贞群dual ideal 对偶理想dual isomorphism 对偶同构dual lattice 对偶格dual mapping 对偶映射dual module 对偶模dual number 对偶数dual operation 对偶运算dual operator 对偶算子dual problem 对偶问题dual relation 对偶关系dual representation 对偶表示dual simplex method 对偶单形法dual spaces 对偶空间dual system 对偶系统dual theorem 对偶定理dual vector space 对偶向量空间duality 对偶性duality principle 对偶原理duality relation 对偶关系duality theorem 对偶定理duel 竞赛dummy index 哑指标duodecimal notation 十二进记数法duodecimal system 十二进制duodecimal system of numbers 十二进数系duplication formula 倍角公式duplication of the cube 倍立方duration 持久时间dyad 并向量dyadic expansion 二进展开dyadic product 并向量积dyadic rational 二进有理数dynamic optimization 动态最优化dynamic programming 动态规划dynamic store 动态存储器dynamic system 动力系统dynamical variables 动态变数dynamics 力学dynkin diagram 丹金图形。
边界退化的半线性抛物方程解的渐近行为
第61卷 第4期吉林大学学报(理学版)V o l .61 N o .4 2023年7月J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n )J u l y2023d o i :10.13413/j .c n k i .jd x b l x b .2022424边界退化的半线性抛物方程解的渐近行为郭 微1,靳曼莉1,井鑫鑫2(1.北华大学数学与统计学院,吉林吉林132013;2.山东科技大学数学与系统科学学院,山东青岛266590)摘要:用加权能量估计和构造自相似上解的方法,研究一类在边界退化的半线性抛物方程初边值问题解的渐近行为,得到了问题解的整体存在性和爆破性,建立了F u j i t a 型定理,并刻画了临界F u j i t a 指标与退化扩散项和非线性源项之间的定量关系.关键词:边界退化;临界F u ji t a 指标;半线性抛物方程中图分类号:O 175.23 文献标志码:A 文章编号:1671-5489(2023)04-0801-07A s y m pt o t i cB e h a v i o r o f S o l u t i o n s t o S e m i l i n e a r P a r a b o l i cE q u a t i o n sw i t hB o u n d a r y D e g e n e r a c yG U O W e i 1,J I N M a n l i 1,J I N G X i n x i n2(1.S c h o o l o f M a t h e m a t i c s a n dS t a t i s t i c s ,B e i h u aU n i v e r s i t y ,J i l i n 132013,J i l i nP r o v i n c e ,C h i n a ;2.C o l l e g e o f M a t h e m a t i c s a n dS y s t e m sS c i e n c e ,S h a n d o n g U n i v e r s i t y o f S c i e n c e a n dT e c h n o l o g y ,Q i n g d a o 266590,S h a n d o n g Pr o v i n c e ,C h i n a )A b s t r a c t :W es t u d i e dt h ea s y m p t o t i c b e h a v i o ro fs o l u t i o n sf o rac l a s s o fi n i t i a l -b o u n d a r y v a l u e p r o b l e m so fas e m i l i n e a r p a r a b o l i ce q u a t i o n w i t h b o u n d a r y d e g e n e r ac y b y u s i n g th e m e t h o d so f w e i g h t e de n e r g y e s t i m a t e sa n dc o n s t r u c t i n g s e l f -s i m i l a ru p p e rs o l u t i o n s .W eo b t a i n e dt h e g l o b a l e x i s t e n c e a n db l o w i n g -u pp r o p e r t i e s o f s o l u t i o n s t o t h e p r o b l e m ,e s t a b l i s h e dF u j i t a t y p e t h e o r e m ,a n d c h a r a c t e r i z e dt h e q u a n t i t a t i v er e l a t i o n s h i p b e t w e e nt h ec r i t i c a lF u j i t ae x p o n e n ta n dt h ed e g e n e r a t e d i f f u s i o n t e r ma n d t h en o n l i n e a r s o u r c e t e r m.K e yw o r d s :b o u n d a r y d e g e n e r a c y ;c r i t i c a l F u j i t a e x p o n e n t ;s e m i l i n e a r p a r a b o l i c e q u a t i o n 收稿日期:2022-10-24.第一作者简介:郭 微(1976 ),女,汉族,博士,教授,从事非线性扩散方程理论的研究,E -m a i l :g u o w e i ji l i n @163.c o m.通信作者简介:靳曼莉(1980 ),女,汉族,博士,副教授,从事微分方程定性理论的研究,E -m a i l :64278144@q q.c o m.基金项目:吉林省教育厅科学技术研究项目(批准号:J J K H 20210031K J ;J J K H 20210029K J ).0 引 言考虑如下在边界退化的半线性抛物方程初边值问题解的渐近行为:∂u ∂t =∂∂x x λ∂u ∂æèçöø÷x +b (x )u p ,x >0, t >0, (1) l i m x ң0+xλ∂u ∂x(x ,t )=0,t >0,(2) u (x ,0)=u 0(x ),x >0,(3)其中:λ>0;p >1;b ɪC 1[0,+ɕ)满足Copyright ©博看网. All Rights Reserved.l 1(x γ+1)ɤb (x )ɤl 2(x γ+1), x >0,(4)这里0<l 1ɤl 2,γȡ0.方程(1)在边界x =0上是退化的.对于具边界退化的半线性抛物方程的初边值问题(1)-(3),其解的渐近行为由方程(1)在边界x =0上的退化强度和非线性源项决定,本文建立问题(1)-(3)的F u ji t a 型定理.F u ji t a [1]研究了如下含内部源热传导方程∂u ∂t=Δu +u p , x ɪℝn , t >0的C a u c h y 问题,证明了当1<p <p c =1+2/n 时该问题不存在非负非平凡的整体解,而当p >p c =1+2/n 时既存在非负非平凡整体解(小初值)又存在非负爆破解(大初值).指标p c 称为临界F u j i t a 指标.文献[2-3]证明了临界情形p =p c 也属于爆破情形.F u j i t a 的工作揭示了非线性偏微分方程的一类重要性质,目前已有很多研究拓展了F u ji t a 的结果,包括各种非线性发展方程和方程组㊁不同类型的非线性源㊁不同类型的区域以及不同类型的边值条件等[4-17],其中,文献[10]首先研究了如下两个在边界退化的半线性抛物方程的初边值问题:∂u ∂t =∂∂x x λ∂u ∂æèçöø÷x +up ,0<x <1, t >0,l i m x ң0+xλ∂u ∂x(x ,t )=0, u (1,t )=0,t >0,u (x ,0)=u 0(x ),0<x <1和∂u ∂t =∂∂x x λ∂u ∂æèçöø÷x +up ,x >0, t >0,l i m x ң0+xλ∂u ∂x(x ,t )=0,t >0,u (x ,0)=u 0(x ),x >0,式中λ>0,p >1.对于有界区间上的问题,文献[10]证明了如果退化较强即λȡ2,则任意非平凡解都在有限时刻内爆破,如果退化不强即0<λ<2,则该问题既存在非平凡整体解(小初值)又存在爆破解(大初值).对于无界区间上的问题,文献[10]建立了F u j i t a 定理,证明了临界F u ji t a 指标为pc =3-λ,0<λ<2,+ɕ,λȡ2{.本文建立问题(1)-(3)的F u ji t a 型定理.与文献[10]所考虑的问题相比,方程(1)中的非线性源项具有更一般的系数,该系数会影响问题(1)-(3)的临界F u j i t a 指标.问题(1)-(3)的临界F u j i t a 指标是pc =3-λ+γ,0<λ<2,+ɕ,λȡ2{.(5)即当λȡ2时,对p >1,问题(1)-(3)的任意非平凡解都在有限时刻爆破;当0<λ<2时,对1<p ɤ3-λ+γ,问题(1)-(3)的任意非平凡解都在有限时刻爆破,而对p >3-λ+γ,问题(1)-(3)既存在非平凡整体解(小初值)又存在爆破解(大初值).受文献[10-14]中的方法启发,为证明问题(1)-(3)解的爆破性质,采用加权能量估计的方法,通过选取恰当的权函数,比较退化扩散项与非线性源项之间的相互作用,建立能量估计进而得到解的爆破性质;为证明问题(1)-(3)存在非平凡的整体解,构造自相似形式的上解.由于方程(1)的非线性源具有一般的系数,因此本文需要一些更细致复杂的运算和估计.特别地,对于临界情形需要建立一系列精确的估计.1 预备知识定义1 设0<T ɤ+ɕ.如果非负函数u 满足下列条件,则u 称为问题(1)-(3)在(0,T )上的下解(上解):208 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.1)对任意的0<췍T <T ,有u ɪL ɕ((0,+ɕ)ˑ(0,췍T )),且对任意的R >0有∂u ∂t ,x λ/2∂u ∂xɪL 2((0,R )ˑ(0,췍T));2)对任意的0<췍T <T 及任意满足在t =췍T 或者当x 充分大时为零的非负函数φɪC 1([0,+ɕ)ˑ[0,췍T ]),积分不等式ʏ췍T 0ʏ+ɕ∂u ∂t (x ,t )φ(x ,t )+x λ∂u ∂x (x ,t )∂φ∂x (x ,t æèçöø÷)d x d t ɤ(ȡ)ʏ췍Tʏ+ɕb (x )u p (x ,t )φ(x ,t )d x d t 成立;3)式(3)在迹的意义下成立.如果u 既是问题(1)-(3)在(0,T )上的上解,又是问题(1)-(3)在(0,T )上的下解,则称u 是问题(1)-(3)在(0,T )上的解.下面不加证明地给出问题(1)-(3)的适定性定理和比较原理,类似证明可参考文献[10].引理1 设^u 和㊅u 分别是问题(1)-(3)在(0,T )上的上解和下解,则㊅u ɤ^u 于(0,+ɕ)ˑ(0,T ).引理2 设0ɤu 0ɪL ɕ((0,+ɕ)),且对任意的R >0,有x λ/2u ᶄ0ɪL 2((0,R )),则问题(1)-(3)至少存在一个局部解.如果u 是问题(1)-(3)在(0,+ɕ)上的解,则称u 是全局解.否则,存在T >0,使得u 是问题(1)-(3)在(0,T )上的解,并满足l i m t ңT-s u p (0,+ɕ)u (㊃,t )=+ɕ,则称u 在有限时刻T 爆破,T 称为爆破时间.2 F u ji t a 型定理的证明定理1 假设λȡ2,b ɪC 1([0,+ɕ))满足式(4),对任意非平凡的初值0ɤu 0ɪL ɕ((0,+ɕ)),且对任意的R >0有x λ/2u ᶄ0ɪL 2((0,R )),则问题(1)-(3)的解在有限时刻爆破.证明:当λȡ2时,类似文献[10]中定理2.5的证明可得结论.定理2 假设0<λ<2,b ɪC 1([0,+ɕ))满足式(4).1)如果1<p <p c =3-λ+γ,对任意非平凡的初值0ɤu 0ɪL ɕ((0,+ɕ)),且对任意R >0有x λ/2u ᶄ0ɪL 2((0,R )),则问题(1)-(3)的解在有限时刻爆破;2)如果p >p c =3-λ+γ,对任意非平凡的初值0ɤu 0ɪL ɕ((0,+ɕ)),且对任意R >0有x λ/2u ᶄ0ɪL 2((0,R )),则当初值u 0充分小时,问题(1)-(3)的解全局存在,当初值u 0充分大时,问题(1)-(3)的解在有限时刻爆破.证明:仿文献[10]中定理2.4选取ψR :ψR (x )=1,0<x ɤR ,121+c o s (x -R )πæèçöø÷R ,R <x <2R ,0,x ȡ2R ìîíïïïï.(6)易得ψR ɪC 1([0,+ɕ)),ψR 分片光滑且满足(x λψᶄR (x ))ᶄ=-λπ2R x λ-1s i n (x -R )πR -π22R 2x λc o s (x -R )πR ȡ-λπ2R 2x λ-π22R2x λc o s (x -R )πR ȡ-2λπ2R λ-2ψR (x ), R <x <2R .由定义1可得u 满足d d t ʏ0+ɕu (x ,t )ψR (x )dx =ʏ+ɕu (x ,t )(x λψᶄR (x ))ᶄd x +ʏ+ɕb (x )u p (x ,t )ψR (x )d x ȡ-2λπ2Rλ-2ʏ+ɕu (x ,t )ψR(x )d x +ʏ+ɕb (x )u p(x ,t )ψR(x )d x , t >0.(7)由H öl d e r 不等式可得308 第4期 郭 微,等:边界退化的半线性抛物方程解的渐近行为 Copyright ©博看网. All Rights Reserved.ʏ+ɕ0u (x ,t )ψR(x )d x ɤʏ+ɕ(x γ+1)-1/(p-1)ψR (x )d ()x (p -1)/pʏ+ɕ(x γ+1)u p(x ,t )ψR(x )d ()x 1/pɤC(p -1)/p1R(p -γ-1)/pʏ+ɕ(x γ+1)u p(x ,t )ψR(x )d ()x 1/p, t >0,(8)其中C 1是与R 无关的正常数.将式(8)代入式(7)得d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ-2λπ2R λ-2ʏ+ɕ0u (x ,t )ψR(x )d x +C 1-p 1Rγ-p+1ʏ+ɕ0u (x ,t )ψR(x )d ()x p, t >0.(9)由于p <p c 且u 0是非负的,故存在充分大的R >0,使得2λ+1π2Rλ-2<C 1-p1Rγ-p+1ʏ+ɕu 0(x )ψR (x )d ()x p-1.因此,由式(9)可得d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ12C 1-p 1R γ-p +1ʏ+ɕu (x ,t )ψR (x )d ()x p, t >0.故存在T >0,使得l i m t ңT-ʏ+ɕ0u (x ,t )ψR(x )d x =+ɕ,从而l i m t ңT-s u p (0,+ɕ)u (㊃,t )=+ɕ,表明u 在有限时刻爆破.下面讨论情形2)即p >p c .当初值充分大时,考虑问题(1)-(3)的爆破解.设u 是问题(1)-(3)的一个非负非平凡解,由定义1和式(4)可得u 满足d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ-2λπ2R λ-2ʏ+ɕu (x ,t )ψR (x )d x +ʏ+ɕb (x )u p (x ,t )ψR (x )d x ȡ-2λπ2Rλ-2ʏ+ɕu (x ,t )ψR (x )d x +l 1ʏ+ɕ(x γ+1)u p (x ,t )ψR (x )d x , t >0,(10)其中ψR (x )是式(6)定义的函数.由H öl d e r 不等式可得ʏ+ɕu (x ,t )ψR(x )d x ɤʏ+ɕ(x γ+1)-1/(p-1)ψR(x )d ()x (p -1)/p㊃ʏ+ɕ(x γ+1)u p(x ,t )ψR(x )d ()x 1/p, t >0.(11)将式(11)代入式(10)可得d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ-2λπ2R λ-2ʏ+ɕ0u (x ,t )ψR(x )d x +l 1ʏ+ɕ(x γ+1)-1/(p -1)ψR(x )d ()x 1-p㊃ʏ+ɕ0u (x ,t )ψR(x )d ()x p, t >0.(12)选取u 0充分大,使得2λ+1π2Rλ-2ɤl 1ʏ+ɕ(x γ+1)-1/(p-1)ψR (x )d ()x 1-pʏ+ɕ0u 0(x )ψR(x )d ()x p-1.类似于情形1)的讨论可得u 在有限时刻爆破.当初值充分小时证明方程(1)具有如下形式的自相似上解:u (x ,t )=(t +1)-αw ((t +1)-βx ), x >0, t >0,(13)其中α=2-λ+γ(2-λ)(p -1),β=1(2-λ).当0<x <1时,由式(4)可知b (x )有界;当x ȡ1时,由式(4)可得12l 1x γɤb (x )ɤ2l 2x γ.将式(13)代入式(1),经过运算可得(r λw ᶄ(r ))ᶄ+12-λr w ᶄ(r )+2-λ+γ(2-λ)(p -1)w (r )+2(l 2r γ+1)w p (r )ɤ0, r >0.(14)令w (r )=εe -A r 2-λ, r >0,(15)408吉林大学学报(理学版)第61卷Copyright ©博看网. All Rights Reserved.其中ε>0为待定常数,A 满足2-λ+γ(2-λ)2(p -1)<A <1(2-λ)2.将式(15)代入式(14)可得(r λw ᶄ(r ))ᶄ+12-λr w ᶄ(r )+2-λ+γ(2-λ)(p -1)w (r )+2(l 2r γ+1)w p (r )=εe-A r 2-λA (2-λ)A (2-λ)-12-æèçöø÷λr 2-λéëêê+2-λ+γ(2-λ)(p -1)-A (2-λæèçöø÷)+2(l 2r γ+1)εp -1e -A (p -1)r 2-ùûúúλ, r >0.根据A 的定义可知A (2-λ)<12-λ, 2-λ+γ(2-λ)(p -1)<A (2-λ),从而可知式(14)成立.注意到x λ∂^u ∂æèçöø÷x (0,t )=0,t >0,因此,存在充分小的ε>0,使得u 0(x )ɤ^u (x ,0), x >0.(16)由引理1可知,如果u 0满足式(16),则问题(1)-(3)的解全局存在.3 临界情形下面考虑当0<λ<2时的临界情形p =p c =3-λ+γ.引理3 假设0<λ<2,b ɪC 1([0,+ɕ))满足式(4),p =pc ,u 是问题(1)-(3)在(0,+ɕ)上的解.则对任意R >0,有ʏ+ɕu (x ,t )ψR (x )d x ɤ2(λ+1)/(p -1)π2/(p -1)C 1, t >0,(17)d d t ʏ+ɕu (x ,t )ψR(x )d x ȡ-p -1p2λp /(p -1)π2p /(p-1)C 1R λ-2, t >0(18)d d tʏ+ɕu (x ,t )ψR(x )d x ȡR λ-2ʏ+ɕu (x ,t )ψR(x )d ()x 1/2-2λπ2ʏ2RRu (x ,t )ψR(x )d ()x 1/2[+C 1-p1ʏ+ɕu (x ,t )ψR(x )d ()x (2p-1)/]2, t >0,(19)其中ψR (x )是式(6)定义的函数,C 1是定理2中给出的常数.证明:首先,由式(9)可得d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ-2λπ2R λ-2ʏ+ɕu (x ,t )ψR(x )d x +C 1-p1R λ-2ʏ+ɕu (x ,t )ψR(x )d ()x p, t >0.(20)下面利用反证法证明式(17)成立.否则,存在t 0>0,使得2λ+1π2ɤC 1-p1ʏ+ɕu (x ,t )ψR(x )d ()x p-1.则根据式(20)知d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡ12C 1-p 1R λ-2ʏ+ɕ0u (x ,t )ψR (x )d()x p, t >t 0,从而u 在有限时刻爆破,与假设矛盾.因此式(17)成立.下面证明式(18)成立.由式(20)和Y o u n g 不等式,有d d t ʏ+ɕ0u (x ,t )ψR (x )d x ȡC 1-p 1R λ-2-2λπ2C p -11ʏ+ɕu (x ,t )ψR (x )d x +ʏ+ɕu (x ,t )ψR (x )d ()x []pȡC 1-p1Rλ-2-1pʏ+ɕu (x ,t )ψR (x )d ()x éëêp-p -1p2λp /(p -1)π2p /(p -1)C p 1+ʏ+ɕu (x ,t )ψR (x )d ()x ùûúúpȡ-p -1p2λp /(p -1)π2p /(p -1)C 1R λ-2, t >0,508 第4期 郭 微,等:边界退化的半线性抛物方程解的渐近行为 Copyright ©博看网. All Rights Reserved.式(18)得证.最后证明式(19)成立.根据式(7)可得d d t ʏ+ɕ0u (x ,t )ψR (x )d x =ʏ2RRu (x ,t )(x λψᶄR (x ))ᶄd x +ʏ+ɕb (x )u p (x ,t )ψR (x )d x ȡ-2λπ2Rλ-2ʏ2RRu (x ,t )ψR (x )d x +C 1-p 1R λ-2ʏ+ɕu (x ,t )ψR (x )d ()x pȡRλ-2ʏ+ɕ0u (x ,t )ψR (x )d ()x 1/2-2λπ2ʏ2RRu (x ,t )ψR (x )d ()x 1/[2+C 1-p1ʏ+ɕu (x ,t )ψR(x )d ()x (2p-1)/]2, t >0,即式(19)成立,证毕.定理3 假设0<λ<2,b ɪC 1([0,+ɕ))满足式(4).如果p =p c =3-λ+γ,对任意非平凡的初值0ɤu 0ɪL ɕ((0,+ɕ)),且对任意R >0有x λ/2u ᶄ0ɪL 2((0,R )),则问题(1)-(3)的解在有限时刻爆破.证明:假设u 是问题(1)-(3)在(0,+ɕ)上的解.对任意R >0,令w R (t )=ʏ+ɕ0u (x ,t )ψR(x )d x , t ȡ0,其中ψR (x )是式(6)定义的函数.记Λ=s u p R >0,t >0w R (t )=s u p t >0ʏ+ɕ0u (x ,t )d x .(21)由式(17)和u 0的非平凡性可得0<Λ<+ɕ.固定ε0ɪ(0,Λ),M 0>0,使得2λπ2(ε0+M 0)1/2ɤ12C 1-p 1(Λ-ε0)(2p-1)/2.根据式(21)可知,存在t 1ȡ0和R 0>0,使得w R 0(t 1)ȡΛ-ε0.(22)对任意t ȡt 1,在式(18)中取R =R 0并由式(22)可得ʏ+ɕu (x ,t )ψR 0(x )d x ȡʏ+ɕ0u (x ,t 1)ψR 0(x )d x -p -1p2λp /(p -1)π2p /(p -1)C 1R λ-20(t -t 1)ȡΛ-ε0-p -1p2λp /(p -1)π2p /(p -1)C 1R λ-20(t -t 1),(23)由式(21),(23)可得ʏ4R 02R 0u (x ,t )ψ2R 0(x )d x ɤʏ+ɕu (x ,t )d x -ʏ+ɕu (x ,t )ψR 0(x )d x ɤε0+p -1p2λp /(p -1)π2p /(p-1)C 1R λ-20(t -t 1).(24)在式(19)中选取R =2R 0,有d d tw 2R 0(t )ȡ(2R 0)λ-2w 1/22R 0(t )-2λπ2ʏ4R 02R 0u (x ,t )ψ2R 0(x )d ()x1/2+C 1-p 1w (2p -1)/22R 0(t []), t >t 1,(25)由式(21),(22),(24),(25)可得d d tw 2R 0(t )ȡ2λ-3C 1-p 1R λ-20(Λ-ε0)p , t 1<t <t 2,(26)其中t 2=t 1+p (p -1)C 12-λp /(p -1)π-2p /(p -1)M 0R 2-λ0.根据式(26)和式(22)可得w 2R 0(t 2)ȡw 2R 0(t 1)+2λ-3C 1-p 1R 0λ-2(Λ-ε0)p (t 2-t 1)ȡΛ-ε0+γ0,其中γ0=p p -12(3-λ-3p )/(p -1)C -p 1π-2p /(p -1)M 0(Λ-ε0)p.同理可得w 4R 0(t 3)ȡw 4R 0(t 2)+γ0ȡw 2R 0(t 2)+γ0ȡΛ-ε0+2γ0,这里t 3=t 2+p (p -1)C12-λp /(p -1)π-2p /(p -1)M 0(2R 0)2-λ.依次重复上述步骤,对任意正整数i ,可得608 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.w 2i R 0(t i +1)ȡw 2i R 0(t i )+γ0ȡw 2i -1R 0(t i )+γ0ȡΛ-ε0+i γ0,其中t i +1=t i +p (p -1)C12-λp /(p -1)π-2p /(p-1)M 0(2i -1R 0)2-λ.因此s u pt >0ʏ+ɕ0u (x ,t )d x =+ɕ,与式(21)矛盾.证毕.参考文献[1] F U J I T A H.O n t h eB l o w i n g u p o f S o l u t i o n s o f t h eC a u c h y P r o b l e mf o r u t =Δu +u 1+α[J ].JF a cS c iU n i vT o k y o S e c tⅠ,1966,13:109-124.[2] HA Y A K AWA K.O nN o n e x i s t e n c e o fG l o b a l S o l u t i o n s o f S o m eS e m i l i n e a rP a r a b o l i cD i f f e r e n t i a lE q u a t i o n s [J ].P r o c J a pa nA c a d ,1973,49:503-505.[3] K O B A Y A S H IK ,S I A R O T ,T A N A K A H.O nt h eB l o w i n g u p P r ob l e mf o rS e m i l i n e a r H e a tE q u a t i o n s [J ].JM a t hS oc J a p a n ,1977,29(1):407-424.[4] A N D R E U C C ID ,C I R M IG R ,L E O N A R D I S ,e t a l .L a r geT i m eB e h a v i o r o f S o l u t i o n s t o t h eN e u m a n nP r o b l e m f o ra Q u a s i l i n e a rS e c o n d O r d e r D e g e n e r a t e P a r a b o l i c E q u a t i o ni n D o m a i n s w i t h N o n c o m p a c tB o u n d a r y [J ].JD i f f e r e n t i a l E q u a t i o n s ,2001,174(2):253-288.[5] D E N G K ,L E V I N E H A.T h eR o l eo fC r i t i c a lE x p o n e n t s i nB l o w -u p T h e o r e m s :T h eS e q u e l [J ].J M a t h A n a l A p pl ,2000,243(1):85-126.[6] 郭微,高云柱.一类非线性耦合对流扩散方程组的临界F u ji t a 曲线[J ].吉林大学学报(理学版),2020,58(2):271-276.(G U O W ,G A O Y Z .C r i t i c a lF u j i t aC u r v e sf o raC l a s so f N o n l i n e a rC o u p l e d C o n v e c t i o n -D i f f u s i o n S y s t e m s [J ].J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n ),2020,58(2):271-276.)[7] J I N G XX ,N I E Y Y ,WA N G CP .A s y m p t o t i cB e h a v i o ro fS o l u t i o n s t oC o u p l e dS e m i l i n e a rP a r a b o l i cS y s t e m s w i t hB o u n d a r y D e g e n e r a c y [J ].E l e c t r o n JD i f f e r e n t i a l E qu a t i o n s ,2021,2021:67-1-67-17.[8] J I N G X X ,N I E Y Y ,WA N GCP .A s y m p t o t i cB e h a v i o r o f S o l u t i o n s t oC o u p l e dS e m i l i n e a rP a r a b o l i cE q u a t i o n s w i t hG e n e r a lD e g e n e r a t eD i f f u s i o nC o e f f i c i e n t s [J ].D i s c r e t eC o n t i nD y nS y s t S e rB ,2023,28(2):959-983.[9] L E V I N E H A.T h eR o l e o fC r i t i c a l E x p o n e n t s i nB l o w -u p T h e o r e m s [J ].S I AM R e v ,1990,32(2):262-288.[10] WA N G C P .A s y m p t o t i c B e h a v i o ro fS o l u t i o n st oa C l a s so fS e m i l i n e a rP a r a b o l i c E q u a t i o n s w i t h B o u n d a r y D e g e n e r a c y [J ].P r o cA m e rM a t hS o c ,2013,141(9):3125-3140.[11] WA N GCP ,Z H E N GSN.C r i t i c a lF u j i t aE x p o n e n t so fD e g e n e r a t ea n dS i n g u l a rP a r a b o l i cE q u a t i o n s [J ].P r o c R o y S o cE d i n b u r ghS e c tA ,2006,136(2):415-430.[12] WA N GCP ,Z H E N G S N ,WA N G ZJ .C r i t i c a lF u j i t aE x p o n e n t sf o raC l a s so fQ u a s i l i n e a rE q u a t i o n s w i t h H o m o g e n e o u sN e u m a n nB o u n d a r y D a t a [J ].N o n l i n e a r i t y,2007,20(6):1343-1359.[13] Z H E N GSN ,WA N GCP .L a r g eT i m eB e h a v i o u r o f S o l u t i o n s t o aC l a s s o fQ u a s i l i n e a rP a r a b o l i cE q u a t i o n sw i t h C o n v e c t i o nT e r m s [J ].N o n l i n e a r i t y,2008,21(9):2179-2200.[14] Z HO U Q ,N I E Y Y ,HA N X Y.L a r g e T i m eB e h a v i o ro fS o l u t i o n st oS e m i l i n e a rP a r a b o l i c E q u a t i o n s w i t h G r a d i e n t [J ].JD y nC o n t r o l S y s t ,2016,22(1):191-205.[15] 周倩,那杨,高冬.一类耦合半线性扩散方程组的F u ji t a 临界曲线[J ].吉林大学学报(理学版),2020,58(4):733-743.(Z HO U Q ,N A Y ,G A O D.F u j i t aC r i t i c a l C u r v e s f o r aC l a s s o fC o u p l e dS e m i l i n e a rD i f f u s i o nS y s t e m s [J ].J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n ),2020,58(4):733-743.)[16] Z HO U M J ,L E N G Y.E x i s t e n c ea n d N o n e x i s t e n c eo ft h eS o l u t i o n st ot h e C a u c h y P r o b l e m o f Q u a s i l i n e a r P a r a b o l i cE qu a t i o nw i t haG r a d i e n tT e r m [J ].L i t h M a t hJ ,2021,61(1):123-142.[17] Z HA O X T ,Z HO U M J ,J I N G X X.A s y m p t o t i cB e h a v i o ro fS o l u t i o n st o P o r o u s M e d i u m E qu a t i o n s w i t h B o u n d a r y D e g e n e r a c y [J ].E l e c t r o n JD i f f e r e n t i a l E q u a t i o n s ,2021,2021:96-1-96-19.(责任编辑:赵立芹)708 第4期 郭 微,等:边界退化的半线性抛物方程解的渐近行为Copyright ©博看网. All Rights Reserved.。
主要学术贡献
主要学术贡献主要从事非线性偏微分方程的研究，着重探讨流体动力学等领域中的数学理论及其应用。
多次应邀到美国、香港和新加坡等地学术访问或演讲。
主持或参加了多个科研项目，在国内外学术刊物发表论文60余篇。
主讲高等数学，数学物理方程，偏微分方程泛函方法，流体动力学的数学理论，非线性偏微分方程的某些理论。
指导博士研究生11人，其中9人已经毕业并获得博士学位。
指导硕士研究生26人，其中20人已经毕业并获得硕士学位。
合作编写教材《数学物理方程》一部，已由高等教育出版社出版。
获奖励情况(1) 1999年获得教育部科学技术进步一等奖；(2) 2000年获得教育部首届青年教师奖；(3) 2002年获得吉林省第七届青年科技奖；(4) 2006年获得吉林省长春市政府特殊津贴。
科研项目1) “相变和图像处理等领域中的某些非线性扩散方程”，教育部优秀青年教师教学与科研奖励基金项目，项目负责人,2000—2004；2) “图像处理中的非线性扩散模型”，国家自然科学基金项目青年基金项目，项目负责人,2001—2003；3) “流体动力学等领域中的具有退化性或奇异性的某些数学模型”，国家自然科学基金面上项目，项目负责人,2006—2008；4）“数学与其它领域交叉的若干专题”，国家重点基础研究发展计划973计划，参加者，2006—2011；5）“带有奇异性的某些流体动力学模型”,国家自然科学基金面上项目，项目负责人，2010年～2012年。
发表论文目录1.Yuan, Hongjun; Xu, Xiaojing Existence and uniqueness of solutions for a classof non-Newtonian fluids with singularity and vacuum. J. Differential Equations 245 (2008), no. 10, 2871--2916.2.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie; Xu, Xiaojing On theCauchy problem for the evolution $p$-Laplacian equations with gradient term and source. J. Differential Equations 235 (2007), no. 2, 544--585.3.Lei, Yutian; Wu, Zhuoqun; Yuan, Hongjun Radial minimizers of a Ginzburg-Landaufunctional. Electron. J. Differential Equations1999, No. 30, 21 pp.4.Wu, Zhuoqun; Yuan, Hongjun; Yin, Jingxue Some properties of solutions for asystem of dynamics of biological groups. Comm. Partial Differential Equations22 (1997), no. 9-10, 1389--1403.5.Yuan, Hong Jun, Hölder continuity of interfaces for the porous medium equationwith absorption. Comm. Partial Differential Equations 18 (1993), no. 5-6, 965--976.6.Yuan, Hongjun; Wang, Changjia Unique solvability for a class of fullnon-Newtonian fluids of one dimension with vacuum. Z. Angew. Math. Phys. 60 (2009), no. 5, 868--898. 35Q357.Yin, Li; Xu, Xiaojing; Yuan, Hongjun Global existence and uniqueness ofsolution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum. Z. Angew. Math. Phys. 59 (2008), no. 3, 457--474.8.Xu, Xiaojing; Yuan, Hongjun Existence of the unique strong solution for a classof non-Newtonian fluids with vacuum. Quart. Appl. Math. 66 (2008), no. 2, 249--279.9.Wang, Changjia; Yuan, Hongjun Global strong solutions for a class ofheat-conducting non-Newtonian fluids with vacuum. Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3680–3703,10.Lining, Tong; Hongjun, Yuan Classical solutions to Navier-Stokes equationsfor nonhomogeneous incompressible fluids with non-negative densities. J. Math.Anal. Appl. 362 (2010), no. 2, 476–504.11.Lian, Songzhe; Gao, Wenjie; Cao, Chunling; Yuan, Hongjun Study of thesolutions to a model porous medium equation with variable exponent ofnonlinearity. J. Math. Anal. Appl. 342 (2008), no. 1, 27--38.12.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie The limiting problemof the drift-diffusion-Poisson model with discontinuous $p$-$n$-junctions.J. Math. Anal. Appl. 347 (2008), no. 1, 157--168.13.Yuan, Hongjun; Chen, Mingtao Positive solutions for a class of $p$-Laplaceproblems involving quasi-linear and semi-linear terms. J. Math. Anal. Appl.330 (2007), no. 2, 1179--1193.14.Xin, Zhouping; Yuan, Hongjun Vacuum state for spherically symmetric solutionsof the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ. 3 (2006), no. 3, 403--442.15.Yuan, Hongjun; Tong, Lining; Xu, Xiaojing BV solutions for the Cauchy problemof a quasilinear hyperbolic equation with $\sigma$-finite Borel measure and nonlinear source. J. Math. Anal. Appl. 311 (2005), no. 2, 715--735.16.Yuan, Hongjun; Xu, Xiaojing; Gao, Wenjie; Lian, Songzhe; Cao, ChunlingExtinction and positivity for the evolution $p$-Laplacian equation with $L^1$ initial value. J. Math. Anal. Appl. 310 (2005), no. 1, 328--337.17.Hongjun, Yuan; Songzhe, Lian; Wenjie, Gao; Xiaojing, Xu; Chunling, CaoExtinction and positivity for the evolution $p$-Laplacian equation in $R^n$.Nonlinear Anal. 60 (2005), no. 6, 1085--1091.18.Hongjun, Yuan; Xiaoyu, Zheng Existence and uniqueness for a quasilinearhyperbolic equation with $\sigma$-finite Borel measures as initial conditions.J. Math. Anal. Appl. 277 (2003), no. 1, 27--50.19.Yuan, Hongjun The Cauchy problem for a singular conservation law with measuresas initial conditions. J. Math. Anal. Appl. 225 (1998), no. 2, 427--439.20.Hongjun, Yuan Source-type solutions of a singular conservation law withabsorption. Nonlinear Anal. 32 (1998), no. 4, 467--492.21.Yuan, Hong Jun Extinction and positivity for the evolution $p$-Laplacianequation. J. Math. Anal. Appl. 196 (1995), no. 2, 754--763.22.Yuan, Hong Jun The Cauchy problem for a quasilinear degenerate parabolicsystem. Nonlinear Anal. 23 (1994), no. 2, 155--164.23.Yuan, Hong Jun Finite velocity of the propagation of perturbations for generalporous medium equations with strong degeneracy. Nonlinear Anal. 23 (1994), no. 6, 721--729.24.Yuan, Hongjun Continuity of weak solutions for quasilinear parabolicequations with strong degeneracy. Chin. Ann. Math. Ser. B 28 (2007), no. 4, 475--498.25.Yuan, Hong Jun; Lian, Song Zhe; Cao, Chun Ling; Gao, Wen Jie; Xu, Xiao JingExtinction and positivity for a doubly nonlinear degenerate parabolic equation.Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1751--1756.26.Yuan, Hong Jun; Tong, Li Ning BV solutions for a quasilinear hyperbolicequation with nonlinear source and finite Radon measures as initial conditions.(Chinese) Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 1, 54–70,27.Yuan, Hongjun; Wang, Shu The zero-Mach limit of a class of combustion flow.J. Partial Differ. Equ. 22 (2009), no. 4, 362–375,28.Ren, Chang Yu; Guan, Jin Rui; Yuan, Hong Jun A class of general-form parabolicMonge-Ampère equations. (Chinese) Chinese Ann. Math. Ser. A 30 (2009), no.3, 421--432. 35K9629.Yuan, Hongjun; Yan, Han Existence and uniqueness of BV solutions for a classof degenerate Boltzmann equations with measures as initial conditions. J.Partial Differ. Equ. 22 (2009), no. 2, 127--152. 35F2530.Yuan, Hongjun; Xu, Xiaojing Some entropy inequalities for a quasilineardegenerate hyperbolic equation. J. Partial Differential Equations18 (2005), no. 4, 289--303.31.Yuan, Hong Jun; Wu, Gang Quasilinear degenerate parabolic equation with Diracmeasure. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 4, 515--526;translation in Chinese J. Contemp. Math.26 (2005), no. 3, 291--30232.Yuan, Hongjun; Jin, Yang Existence and uniqueness of BV solutions for theporous medium equation with Dirac measure as sources. J. Partial Differential Equations 18 (2005), no. 1, 35--58.33.Yuan, Hong Jun; Xu, Xiao Jing Existence and uniqueness of BV solutions fora quasilinear degenerate hyperbolic equation with local finite measures asinitial conditions. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 1, 39--48; translation in Chinese J. Contemp. Math.26 (2005), no. 1, 43--5434.Yuan, Hongjun Instantaneous shrinking and localization of functions in$\roman Y_\lambda(m,p,q,N)$ and their applications. Chinese Ann. Math. Ser.B 22 (2001), no. 3, 361--380.35.Yuan, Hongjun Cauchy's problem for degenerate quasilinear hyperbolicequations with measures as initial values. J. Partial Differential Equations12 (1999), no. 2, 149--178.36.Yuan, Hongjun Localization condition for a nonlinear diffusion equation.Chinese J. Contemp. Math. 17 (1996), no. 1, 45--58.37.Yuan, Hongjun Existence and nonexistence of interfaces of weak solutions fornonlinear degenerate parabolic systems. J. Partial Differential Equations 9 (1996), no. 2, 177--185.38.Yuan, Hongjun Extinction and positivity for the non-Newtonian polytropicfiltration equation. J. Partial Differential Equations 9 (1996), no. 2, 169--176.39.Yuan, Hong Jun A localization condition for a class of nonlinear diffusionequations. (Chinese) Chinese Ann. Math. Ser. A 17 (1996), no. 1, 47--58.40.Zhao, Junning; Yuan, Hongjun The Cauchy problem of some nonlinear doublydegenerate parabolic equations. Chinese J. Contemp. Math. 16 (1995), no. 2, 173--192.41. Zhao, Jun Ning; Yuan, Hong Jun The Cauchy problem for a class of nonlinear doubly degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A 16 (1995), no. 2, 181--196.42. Wu, Zhuo Qun; Yuan, Hong Jun Uniqueness of generalized solutions for a quasilinear degenerate parabolic system. J. Partial Differential Equations 8 (1995), no. 1, 89--96.43.Yuan, Hong Jun Regularity of free boundary for certain degenerate parabolic equations. Chinese J. Contemp. Math. 15 (1994), no. 1, 77--86.44.Zhao, Jun Ning; Yuan, Hong Jun Uniqueness of the solutions of $u_t=\Delta u^m$ and $u_t=\Delta u^m-u^p$ with initial datum a measures: the fast diffusion case. J. Partial Differential Equations 7 (1994), no. 2, 143--159.45.Yuan, Hong Jun Regularity of the free boundary for a class of degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A15 (1994), no. 1, 89--97.。