Asymptotic behavior for doubly degenerate parabolic equations

We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form

(1)$\text{?ρ}\text{?t}\text{=}\text{div}\text{ρ?c}{}^1\text{?}\text{F′(ρ)+V}\text{in}\text{(0,∞)×Ω,}\text{and}\text{ρ(t=0)=ρ}{}_0\text{in}\text{{0}×Ω,}$

where $\text{Ω}$ is $\text{R}{}^{\mathrm{n}}$, or a bounded domain of $\text{R}{}^{\mathrm{n}}$ in which case $\text{ρ?c}{}^1\text{[?(F′(ρ)+V)]·ν=0}$ on $\text{(0,∞)×?Ω}$. We investigate the case where the potential V is uniformly c-convex, and the degenerate case where V=0. In both cases, we establish an exponential decay in relative entropy and in the c-Wasserstein distance of solutions – or self-similar solutions – of (1) to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all p>1, the HWI inequalities obtained by Otto and Villani (J. Funct. Anal. 173 (2) (2000) 361–400) when p=2. This class of PDEs includes the Fokker–Planck, the porous medium, fast diffusion and the parabolic p-Laplacian equations. To cite this article: M. Agueh, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Asymptotic expansions for parabolic systems

We consider a parabolic system in a half space. A theorem, similar to one proved by Meyers and Pazy for elliptic equations outside the unit ball is proved, namely, if the coefficients, the right side, and the initial conditions of the parabolic system have asymptotic expansions at infinity with respect to the space variable, then so does the solution of the corresponding Cauchy problem. Some generalizations and examples are given.     

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

This paper is concerned with the Cauchy problem for the nonlinear parabolic equation $${\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},$$ , where $$\begin{gathered} N \geqslant 1, \hfill \\ F \in C(R^N \times (0,\infty ) \times R \times R^N ), \hfill \\ \phi \in L^\infty (R^N ) \cap L^1 (R^N ,(1 + |x|^K )dx)forsomeK \geqslant 0 \hfill \\ \end{gathered} $$ . We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of the solution in W ^1,q (R ^N ) with 1 ≤ q ≤ ∞.     

Asymptotic finite solutions of degenerate quasilinear parabolic equations with small diffusion

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 77–88, August, 1991.     

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.     

Blow-up for doubly degenerate nonlinear parabolic equations

In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations. The project is supported by the Natural Science Foundation of Fujian Province of China (No. Z0511048)     

Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity $u_t?{△}_{X}u=u\log ?|u|$, where $X=(X_1,X_2,?,X_m)$ is an infinitely degenerate system of vector fields, and ${△}_X:={\sum }_{j=1}^m{X}_j^2$ is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions.     

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

A Harnack inequality for weighted degenerate parabolic equations

In this paper we generalize the recent result of DiBenedetto, Gianazza, Vespri on the Harnack inequality for degenerate parabolic equations to the case of a weighted p-Laplacian type operator in the spatial part. The weight is assumed to belong to the suitable Muckenhoupt class.