Initial trace of positive solutions of a class of degenerate heat equation with absorption
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Publication:6222844
DOI10.3934/DCDS.2013.33.2033zbMath1510.35168arXiv1101.1576MaRDI QIDQ6222844
Phuoc-Tai Nguyen, Laurent Véron
Publication date: 8 January 2011
Abstract: We study the initial value problem with unbounded nonnegative functions or measures for the equation $ prt_tu-Gd_p u+f(u)=0$ in $BBR^N i(0,infty)$ where $p>1$, $Gd_p u = ext{div}(abs {
abla u}^{p-2}
abla u)$ 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 $kgd_0$ and we study their limit when $k oinfty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)=int_0^s f(gs)dgs$. We also give new results dealing with 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 $f(u)=u^ga ln^gb(u+1)$, where $ga>0$ and $gbgeq 0$.
Initial value problems for second-order parabolic equations (35K15) Quasilinear parabolic equations with (p)-Laplacian (35K92) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02)
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