On nonlinear parabolic equations with singular lower order term

Autor: Mounim El ouardy, Youssef El Hadfi, Abdelaaziz Sbai, Aziz Ifzarne
Rok vydání: 2021
Předmět:
Zdroj: Journal of Elliptic and Parabolic Equations. 8:49-75
ISSN: 2296-9039
2296-9020
DOI: 10.1007/s41808-021-00138-5
Popis: In this paper we study existence and regularity results for solution to a nonlinear and singular parabolic problem. The model is $$\begin{aligned} \left\{ \begin{array}{lll} \dfrac{\partial u}{\partial t}-\text{ div }((a(x,t)+|u|^{q})\nabla u)=\frac{f}{u^{\gamma }} &{} \text{ in } &{} Q,\\ u(x,t)=0 &{} \text{ on } &{} \Gamma ,\\ u(x,0)=u_{0}(x) &{} \text{ in } &{} \varOmega , \end{array} \right. \end{aligned}$$ where $$\varOmega $$ is a bounded open subset of $$\mathbb {R}^{N},$$ $$N\ge 2,$$ Q is the cylinder $$\varOmega \times (0,T),$$ $$T>0,$$ $$\Gamma $$ the lateral surface $$\partial \varOmega \times (0,T),$$ $$q>0,$$ $$\gamma >0,$$ and f is non-negative function belonging to some Lebesgue space $$L^{m}(Q),$$ $$m\ge 1$$ and $$u_{0}\in L^{\infty }(\varOmega )$$ such that $$\begin{aligned} \forall \; \omega \subset \subset \varOmega ,\; \exists \; D_{\omega }>0: \; u_{0}\ge D_{\omega }\;\; \text{ in }\; \omega . \end{aligned}$$
Databáze: OpenAIRE
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