Singular solutions of parabolic $p$-Laplacian with absorption

SINGULAR SOLUTIONS OF PARABOLIC p-LAPLACIAN WITH ABSORPTION

We consider, for p ∈ (1, 2) and q > 1, the p-Laplacian evolution equation with absorption ut = div (|∇u|p−2∇u)− u in R × (0,∞). We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0,∞) \ {(0, 0)}, and satisfy u(x, 0) = 0 for all x = 0. We prove the following: (i) When q ≥ p− 1 + p/n, there does not exist any such singul...

Solutions with Dead Cores for a Parabolic P-Laplacian Equation

We study the solutions with dead cores and the decay estimates for a parabolic p-Laplacian equation with absorption by suband supersolution method. Special attention is given to the case where the solution of the steady-state problem vanishes in an interior region.

REGULARITY FOR ENTROPY SOLUTIONS OF PARABOLIC p-LAPLACIAN TYPE EQUATIONS

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut − div ap(x,∇u) = f in ]0, T [×Ω with initial datum in L1(Ω) and assuming Dirichlet’s boundary condition, where ap(., .) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1(]0, T [×Ω) and Ω is a domain in RN . We find spaces of type Lr(0, T ;Mq(Ω)) containing ...

Solutions for singular p-Laplacian equation in Rn

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of ∂tu−∆u+h(|x|) |u| p−1 u = 0 in R × (0, T ) with initial data kδ0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r, β > N(p−1)−2, we prove that the limit function u∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N(p−1)−2. If lim infr→0 r 2 ln(1/h(r)) > 0, u∞ has a persistent singulari...