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 $\mathbb{R}^N$ 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 $(H^{1,q}_U(\mathbb R^N),H^{1,q}_\phi(\mathbb R^N))$-uniform (w.r.t. $g\in\mathcal{H}_{L^q_U(\mathbb R^N)}(g_0)$) attractor $\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb R^N)}(g_0)}$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^p(\mathbb R;L^q_U(\mathbb R^N))$. We also obtain the uniform attracting property in the stronger topology.