Global existence of weak solutions for the nonlocal energy-weighted reaction-diffusion equations. (English) Zbl 1407.35111

The authors introduce and study the non-local energy-weighted reaction-diffusion equation \[ u_t = -E_\varepsilon^M(u) \nabla_X E_\varepsilon(u) \] where \[ E_\varepsilon(u) =\int_\Omega \left[\frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon}\right]\,dx \] for some double-well potential \(W\), \[ E_\varepsilon^M(u) = \min(E_\varepsilon(u), M) \] and \(\nabla_X\) is the gradient in \(X=L^2(\Omega)\).
By intricately adapting the Galerkin method to this modified reaction-diffusion setting, the authors show that there exists a weak solution \(u\in L^2(0,T,H_0^1(\Omega))\) for any initial conditions \(u_0\in H_0^1(\Omega)\) if Neumann boundary conditions are imposed at \(\partial \Omega\) and if \(\|f\|_{C^0}<\infty\) and \(\|f\|_{C^1}<\infty\) for \(f(u)=W'(u)\).


35K57 Reaction-diffusion equations
35D30 Weak solutions to PDEs
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35A35 Theoretical approximation in context of PDEs
Full Text: DOI Euclid


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