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

In this paper, we first prove the well-posedness for thenon-autonomous reaction-diffusion equations on the entire space $\R^N$ in thesetting of locally uniform spaces with singular initial data. Thenwe study the asymptotic behavior of solutions of such equation andshow 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 externalforces being translation uniform bounded but not translation compactin $L_b^p(\R;L^q_U(\R^N))$. We also obtain the uniform attracting propertyin the stronger topology