Continuity of the flow and robustness for evolution equations with non globally Lipschitz forcing term
| Autor: | Marcos Roberto Teixeira Primo, Mariza Stefanello Simsen, Jacson Simsen |
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| Rok vydání: | 2019 |
| Předmět: |
General Mathematics
010102 general mathematics Lambda Lipschitz continuity 01 natural sciences 010101 applied mathematics Combinatorics symbols.namesake Computational Theory and Mathematics Flow (mathematics) Dirichlet boundary condition Domain (ring theory) symbols Exponent Nabla symbol Sensitivity (control systems) 0101 mathematics Statistics Probability and Uncertainty Mathematics |
| Zdroj: | São Paulo Journal of Mathematical Sciences. 14:223-241 |
| ISSN: | 2316-9028 1982-6907 |
| DOI: | 10.1007/s40863-019-00133-8 |
| Popis: | We study the sensitivity with respect to exponent and diffusion parameters for the problem $$\begin{aligned} \left\{ \begin{array}{l}\frac{\partial u_{\lambda }}{\partial t}-\text {div}(D_{\lambda }(x)|\nabla u_{\lambda }|^{p_{\lambda }(x)-2}\nabla u_{\lambda })+f(x,u_{\lambda })=g(x),\;\; t>0,\\ u_{\lambda }(0)=u_{0\lambda },\end{array}\right. \end{aligned}$$ under homogeneous Dirichlet boundary conditions, where \(\lambda \in [0,\lambda _0],\)\(\varOmega \subset {\mathbb {R}}^n\) (\(n\ge 1\)) is a smooth bounded domain, \(u_{0\lambda }\in H:=L^2(\varOmega ),\)\(g\in L^2(\varOmega ),\)\(p_\lambda (\cdot ) \rightarrow p(\cdot )\), \(D_\lambda (\cdot ) \rightarrow D(\cdot )\) in \(L^\infty (\varOmega )\) as \(\lambda \rightarrow 0,\) and \(f:\varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a non globally Lipschitz Caratheodory mapping. We prove continuity of the flow and upper semicontinuity of the family of global attractors. |
| Databáze: | OpenAIRE |
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