Initial trace of positive solutions of a class of  degenerate heatequation with absorption

Tai Nguyen Phuoc; Laurent Véron

doi:10.3934/dcds.2013.33.2033

Article Contents

2013, Volume 33, Issue 5: 2033-2063. Doi: 10.3934/dcds.2013.33.2033

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Initial trace of positive solutions of a class of degenerate heat equation with absorption

Tai Nguyen Phuoc^1, and
Laurent Véron^2,

1.
Department of Mathematics, Technion, 32000 Haifa
2.
Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Université François Rabelais, Tours

Received: March 31, 2012

Revised: August 31, 2012

Published: April 2013

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Abstract

We study the initial value problem with unbounded nonnegative functions or measures for the equation $ ∂_t u-Δ_p u+f(u)=0$ in $\mathbb{R}^ × (0,\infty)$ where $p>1$, $Δ_p u = div(|∇ u|^{p-2} ∇ u )$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $kΔ_0$ and we study their limit when $k → ∞$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)= ∫_0^s f(σ)dσ$. We also give new results dealing with uniqueness and non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case $f(u)=u^α ln^β (u+1)$, where $α>0$ and $β ≥ 0$.

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Mathematics Subject Classification: Primary: 35K92, 35K61; Secondary: 35A02.

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