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June, 2018 Global Existence of Weak Solutions for the Nonlocal Energy-weighted Reaction-diffusion Equations
Mao-Sheng Chang, Hsi-Chun Wu
Taiwanese J. Math. 22(3): 695-723 (June, 2018). DOI: 10.11650/tjm/8167

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 $u_0 \in H^1_0(\Omega)$.

Citation

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Mao-Sheng Chang. Hsi-Chun Wu. "Global Existence of Weak Solutions for the Nonlocal Energy-weighted Reaction-diffusion Equations." Taiwanese J. Math. 22 (3) 695 - 723, June, 2018. https://doi.org/10.11650/tjm/8167

Information

Received: 14 February 2017; Revised: 20 June 2017; Accepted: 26 June 2017; Published: June, 2018
First available in Project Euclid: 8 September 2017

zbMATH: 06965393
MathSciNet: MR3807333
Digital Object Identifier: 10.11650/tjm/8167

Subjects:
Primary: 35K57
Secondary: 35K55 , 35K61

Keywords: Allen-Cahn energy , Allen-Cahn equation , Galerkin method , global existence , Gradient flow , nonlocal , reaction-diffusion equation

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 3 • June, 2018
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