Existence of solutions for some quasilinear parabolic systems with weight and weak monotonicity general data

Autor: Abdelkrim Barbara, Elhoussine Azroul, El Houcine Rami
Rok vydání: 2020
Předmět:
Zdroj: SeMA Journal. 77:507-529
ISSN: 2281-7875
2254-3902
DOI: 10.1007/s40324-020-00226-x
Popis: We prove the existence of weak solution u for the nonlinear parabolic systems: $$\begin{aligned} {(QPS){\omega }} \left\{ \begin{array}{rcl} \partial _{t} u -div\sigma (x,t,u,Du) &{} = &{} v(x,t) + f(x,t,u,Du) + divg(x,t,u)\; \text{ in } \;\; \Omega _{T}\\ u(x,t)&{} =&{} 0\;\; \text{ on } \;\;\partial \Omega \times (0,T)\\ u(x,0)&{} =&{}u_{0}(x)\;\; \text{ on } \;\;\Omega \end{array} \right. \end{aligned}$$ which is a Dirichlet Problem. In this system, v belongs to $$L^{p'}(0,T,W^{-1,p'}(\Omega ,\omega ^{*},\mathbb {R}^{m}))$$ and $$u_{0} \in L^{2}(\Omega ,\omega _{0}, \mathbb {R}^{m})$$ , f and g satisfy some standards continuity and growth conditions. We prove existence of a weak solution of different variants of this system under classical regularity for some $$p_{s}\in ]\frac{2n}{n+2}\; ;\;\infty [,$$ growth and coercivity for $$\sigma $$ but with only very mild monotonicity assumptions.
Databáze: OpenAIRE
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