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Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds. (English) Zbl 1171.35330
Authors’ abstract: In the continuation of [J. Math. Anal. Appl. 319, No. 1, 157–176 (2006; Zbl 1096.35018)], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in \(L^1\) to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global \(L^\infty\) bound via interpolation of a polynomially growing \(H^1\) bound with the almost exponential \(L^1\) convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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