Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces

Abstract

In this paper, we first prove the well-posedness for the non-autonomous reaction-diffusion equations on the entire space $\R^N$ in the setting of locally uniform spaces with singular initial data. Then we study the asymptotic behavior of solutions of such equation and show the existence of $(H^1,q_U(\R^N),H^1,q_\phi(\R^N))$-uniform(w.r.t. $g\in\mcH_L^q_U(\R^N)(g_0)$) attractor $\mcA_\mcH_L^q_U(\R^N)(g_0)$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^p(\R;L^q_U(\R^N))$. We also obtain the uniform attracting property in the stronger topology.