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A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. (English) Zbl 1112.35108
Summary: We study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose $L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation, and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.

MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 35B35 Stability of solutions of PDE 35B40 Asymptotic behavior of solutions of PDE 35K55 Nonlinear parabolic equations 35K57 Reaction-diffusion equations 45K05 Integro-partial differential equations
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References:
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