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SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH A GRADIENT

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Acta Mathematica Scientia 2015年第5期35卷 1023-1036页

作者：王影王明新 School of Mathematics and Information Science North China University of Water Resources and Electric Power Natural Science Research Center Harbin Institute of Technology

In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.

关键词： quasilinear elliptic equations existence and nonexistence gradient terms singular weights

Source-type solutions of heat equations with convection in several variables space

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Science China Mathematics 2011年第6期54卷 1145-1173页

作者： LU GuoFu YIN HongMin Institute of Applied Mathematics Putian University Putian 351100 China Department of Mathematics Washington State University Pullman WA 99164 USA

In this paper we study the source-type solution for the heat equation with convection： ut = △u ＋ b·▽un for （x,t） ∈ ST→ RN × （0,T] and u（x,0） = δ（x） for x ∈ RN, where δ（x） denotes Dirac measure in = RN,N 2,n 0 and b = （b1,...,bN） ∈ RN is a vector. It is shown that there exists a critical number pc = N＋2 such that the source-type solution to the above problem exists and is unique if 0 N n 〈 pc and there exists a unique similarity source-type solution in the case n = N＋1 , while such a solution does not exist N if n 〉 pc. Moreover, the asymptotic behavior of the solution near the origin is studied. It is shown that when 0 〈 n 〈 N＋1 the convection is too weak and the short time behavior of the source-type solution near the origin N is the same as that for the heat equation without convection.

关键词： source-type solution heat equation with convection existence and nonexistence

SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION

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Acta Mathematica Scientia 1995年第4期15卷 431-441页

作者：赵俊宁 JILIN UNIV DEPT MATHCHANGCHUN 130023PEOPLES R CHINA

In this paper we discuss the existence and nonexistence of singular solutions for a porous medium equations with convection and absorption terms.

关键词： convection diffusion equation singular solution existence and nonexistence