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Nonlinear degenerate parabolic equations with a singular nonlinearity. (English) Zbl 1534.35251

Summary: In this paper, we study the existence and regularity results for some parabolic equations with degenerate coercivity, and a singular right-hand side. The model problem is \[ \begin{cases} \frac{\partial u}{\partial t} - \operatorname{div} \left(\frac{\left (1+|\nabla u|^{-\Lambda}\right) |\nabla u|^{p-2} \nabla u}{(1+|u|)^{\theta}} \right)=\frac{f}{(e^u-1)^{\gamma}} & \text{in }Q_T, \\ u(x,0)=0 & \text{on } \Omega, \\ u =0 & \text{on } \partial Q_T, \end{cases} \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\) \(N \geq 2\), \(T>0\), \(\Lambda \in [0, p-1)\), \(f\) is a non-negative function belonging to \(L^m (Q_T)\), \(Q_T=\Omega \times (0,T)\), \(\partial Q_T=\partial \Omega \times (0,T)\), \(0 \leq \theta < p-1+\frac{p}{N}+\gamma (1+\frac{p}{N})\) and \(0 \leq \gamma < p-1\).

MSC:

35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K59 Quasilinear parabolic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
Full Text: DOI

References:

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