Fundamental solutions of degenerate or singular elliptic equations

Exitence of fundamental solutions is established for a class of degenerate or singular, second-order, linear elliptic partial differential equations. This class contains, for example, Tricomi's equation in the upper half-plane which arises in the study of aerodynamics; the equation of Weinstein's generalized axially symmetric potential theory which arises in the study of fluid dynamics and elasticity; and Schrödinger's equation with a singular potential which arises in quantum mechanics.     

Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations

We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: \((i)\) the concentrated part of the solution with respect to the Newtonian capacity is constant; \((ii)\) the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem \((P)\) are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given.     

Global existence and decay estimates for solutions of degenerate parabolic equations

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionInthispaper,weconsiderthefollowinginitial--boundaryvalueproblemwhereQ~fix(o,co),aQ=aflx(o,co),fiisaboundeddomaininEuclideanspaceR"(n22)withsmoothboundaryonandac=(u.,,'Iu..)denotesthegradientoffunctionu(x).Weassumethefunctionsal(x,t,u,p)(i=1,2,',n)anda(x,t,u,p)arelocallyH5ldercontinuousonfix(0,co)suchthatwherealtuandparepositiveconstants,m,aZIa3.hi,b2,alIadZ20,or321areconstants,m*E[0,m 2),hi16z/0,afl m*/…     

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

The initial and boundary value problem for the degenerate parabolic equation v_t = Δ(?(v)) + F(v) in the cylinder $\text{Ω × ¦0, ∞), Ω ? R}{}^{\mathrm{n}}$ bounded, for a certain class of point functions ? satisfying ?′(v) ? 0 (e.g., $\text{?(v) = ¦v¦}{}^{\mathrm{m}}\text{sign}\text{v}$) is considered. In the case that F(v) sign $\text{v ? C(1 + ¦?(v)¦}{}^{\mathrm{\alpha }}\text{), α < 1}$, the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that $\text{F(v)}\text{?(v)}$ is nondecreasing a condition is given on the initial state v(x, 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.     

Classical solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence and comparison principle of classical solutions for a class of fully nonlinear degenerate parabolic equations.     

Existence of solutions for degenerate quasilinear parabolic equations of higher order

In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory. Supported by the funds of the State Educational Commission of China for returned scholars from abroad.     

Travelling wave solutions for some degenerate parabolic equations (II)

In this paper, we study the existence and regularity of travelling wave front solutions for some degenerate parabolic equations

Gaussian bounds for degenerate parabolic equations

Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A₂ or QC class. We show that there is a heat kernel Wt(x,y) associated to the parabolic equation wut=divA∇u, and Wt satisfies classic Gaussian bounds:

     

Identification problems for degenerate parabolic equations

This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different degenerate identification problems. Applications to identification problems for the Stokes system, Poisson-heat equation, and Maxwell system are given to illustrate the theory.