Nonexistence and Longtime Behaviors of Solutions to a Class of Nonlinear Degenerate Parabolic Equations Not in Divergence Form

In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation σtun=umdiv（｜▽um｜p-2▽um）＋γ｜▽um｜p＋βun with zero boundary condition. Blow-up time is derived when the blow-up does occur.

Nonlinear Degenerate Parabolic Equations with Time-dependent Singular Potentials for Baouendi–Grushin Vector Fields

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials： uq/ t=▽α·（‖z‖^-pγ｜▽αu｜^p-2▽αu）＋V（z, t）u^p-1, uq/ t=▽α·（‖z‖^-2γ▽αu^m）＋V（z, t）u^m, uq/ t=u^μ▽α·（u^τ｜▽αu｜^p-2▽αu）＋V（z, t）u^p-1＋μ＋τin a cylinder Ω×（0, T） with initial condition u（z, 0）=u0（z） ≥ 0 and vanishing on the boundary Ω×（0, T）, where Ω is a Carnot-Carathéodory metric ball in Rd＋k and the time-dependent singular potential function is V（z, t） ∈ L^1loc （Ω×（0, T））. We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.

Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields

The purpose of this paper is to investigate the nonexistence of positive so lutions of the following doubly nonlinear degenerate parabolic equations:{∂u=▽k·(u^(m-1)|▽ku|^(p-2)▽ku)+(w)u^(m+p-2),u(w,0)=u0(w)≥0,u(w,t)=0,inΩ×(0,T),inΩ,on∂Ω×(0,T),where Q is a Carnot-Carathéodory metric bal in IR^(2n+1)generated by Greiner vector fields,V∈L_(loc)(Ω),k∈N,m∈R,1<p<2n+2k and m+p-2>0.The better lower bound p*for m+p is found and the nonexistence results are proved for p*≤m+p<3.

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.

Homogenization of a Nonlinear Degenerate Parabolic Differential Equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problemwith Dirichlet boundary condition. Here the operator D（y, s, △s） is periodic in y and degenerated in △s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈ → 0 and also prove the corrector results of △u^∈ to strong convergence.

Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive （weak） solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.