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

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 of R-n we establish the global existence of weak solutions u is an element of L-2(0, T; H-0(1) (Omega)) with u(t) is an element of 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) is an element of H-0(1) (Omega).