Optimal reaction exponent for some qualitative properties  of solutions to the $p$-heat equation

Susana Merchán; Luigi Montoro; I. Peral

doi:10.3934/cpaa.2015.14.245

Article Contents

2015, Volume 14, Issue 1: 245-268. Doi: 10.3934/cpaa.2015.14.245

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Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation

1.
Dipartimento di Matematica e Informatica, UNICAL, Ponte Pietro Bucci, 31 B, 87036, Arcavacata di Rende, Cosenza
2.
Dipartimento di Matematica, UNICAL, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende, Cosenza
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid

Received: January 2014

Revised: February 2014

Published: January 2015

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Abstract

This article deals with the following quasilinear parabolic problem \begin{eqnarray} u_t-\Delta_p u=h(x)u^{q}, u\geq 0 & \text{in} \,\, \Omega\times (0,\infty),\\ u(x,t)=0 & \text{on}\,\, \partial \Omega\times (0,\infty), \\ u(x,0)=f(x), \,\, f\geq 0 & \text{in} \,\, \Omega, \end{eqnarray} where $-\Delta_p u=-div(|\nabla u|^{p-2}\nabla u)$, $p>1$, $q>0$, $h(x)>0$ and $f(x)\geq 0$ are non negative functions satisfying suitable hypotheses. We assume the domain $\Omega$ is either a bounded regular domain or the whole $\mathbb{R}^N$. The main contribution of this work is to prove that the optimal exponent in the reaction term in order to prove existence of a global positive solution is $q_0=\min\{1,(p-1)\}$. More precisely, we obtain the following conclusions
If $1 < p < 2$ and $0 < q < p-1$, there is no finite extinction time.

If $p > 2$ and $0 < q< 1$, there is no finite speed of propagation.
In both cases the result is optimal.

Keywords:

Mathematics Subject Classification: 35K59, 35K65, 35K67, 35K92, 35B09.

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