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September/October 2014 Degenerate parabolic equations with singular lower order terms
Ida de Bonis, Linda Maria De Cave
Differential Integral Equations 27(9/10): 949-976 (September/October 2014).

Abstract

In this paper, we give existence and regularity results for nonlinear parabolic problems with degenerate coercivity and singular lower order terms, whose simplest example is \begin{eqnarray*} \begin{cases} u_t-\Delta_p u= {\frac{f(x,t)}{u^\gamma}} & \mbox{in}\;\Omega\times (0,T)\\ u(x,t)=0 & \mbox{on}\;\partial\Omega\times(0,T)\\ u(x,0)=u_0 (x) & \mbox{in}\;\Omega\; \end{cases} \end{eqnarray*} with $\gamma>0$, $p\geq 2$, $\Omega$ a bounded open set of $\mathbb{R}^{\mathrm{N}}$ ($N\geq 2$), $0 < T < +\infty$, $f\geq 0$, $f\in L^m(Q_T)$, $m\geq 1$ and $u_0\in L^\infty(\Omega)$ such that $$ \forall \, \omega\subset\subset\Omega\; \exists\;d_{\omega} > 0\,:\,u_{0}\geq d_{\omega}\;\mbox{in}\,\;\omega\,. $$ The aim of the paper is to extend the existence and regularity results recently obtained for the associated singular stationary problem. One of the main difficulties that arises in the parabolic case is the proof of the strict positivity of the solution in the interior of the parabolic cylinder, in order to give sense to the weak formulation of the problem. The proof of this property uses Harnack's inequality.

Citation

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Ida de Bonis. Linda Maria De Cave. "Degenerate parabolic equations with singular lower order terms." Differential Integral Equations 27 (9/10) 949 - 976, September/October 2014.

Information

Published: September/October 2014
First available in Project Euclid: 1 July 2014

zbMATH: 1340.35175
MathSciNet: MR3229098

Subjects:
Primary: 35K55, 35K65, 35K67

Rights: Copyright © 2014 Khayyam Publishing, Inc.

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Vol.27 • No. 9/10 • September/October 2014
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