Differential and Integral Equations

Degenerate parabolic equations with singular lower order terms

Ida de Bonis and Linda Maria De Cave

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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.

Article information

Source
Differential Integral Equations, Volume 27, Number 9/10 (2014), 949-976.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.die/1404230052

Mathematical Reviews number (MathSciNet)
MR3229098

Zentralblatt MATH identifier
1340.35175

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 35K67: Singular parabolic equations

Citation

de Bonis, Ida; De Cave, Linda Maria. Degenerate parabolic equations with singular lower order terms. Differential Integral Equations 27 (2014), no. 9/10, 949--976. https://projecteuclid.org/euclid.die/1404230052


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