Global Existence of Weak Solutions for the Nonlocal Energy-weighted Reaction-diffusion Equations

Abstract

The reaction-diffusion equations provide a predictable mechanism for pattern formation. These equations have a limited applicability. Refining the reaction-diffusion equations must be a good way for supplying the gap between the mathematical simplicity of the model and the complexity of the real world. In this manuscript, we introduce a modified version of reaction-diffusion equation, which we have named ‘‘nonlocal energy-weighted reaction-diffusion equation’’. For any bounded smooth domain $\Omega \subset \mathbb{R}^n$, we establish the global existence of weak solutions $u \in L^2(0,T;H^1_0(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ to the initial boundary value problem of the nonlocal energy-weighted reaction-diffusion equation for any initial data $u_0 \in H^1_0(\Omega)$.