Robustness with respect to exponents for nonautonomous reaction–diffusion equations
Rodrigo Antonio Samprogna, Jacson Simsen · Electronic journal of qualitative theory of differential equations · 2018
In this work we consider a family of nonautonomous problems with homogeneous Neumann boundary conditions and spatially variable exponents with equation of the form \begin{equation*} \frac{\partial u_{\lambda}}{\partial_t}(t)-\operatorname{div}\left(D(t)| abla u_{\lambda}(t)|^{p_{\lambda}(x)-2} abla u_{\lambda}(t)\right)+|u_{\lambda}(t)|^{p_{\lambda}(x)-2}u_{\lambda}(t)=B(t,u_{\lambda}(t)). \end{equation*} We study the continuity of the flow and we study the behavior of attractors when $p_{\lambda}(\cdot)\to p(\cdot)$ in $L^{\infty}(\Omega)$ as $\lambda\to\infty$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$.