In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], 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 L1 to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global L∞ bound via interpolation of a polynomially growing H1 bound with the almost exponential L1 convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.
Citation
Desvillettes L, Fellner K (2008) Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds. Rev Mat Iberoamericana: 407–431. Available: http://dx.doi.org/10.4171/rmi/541.
Acknowledgements
This work has been supported by the European IHP network “HYKE-HYperbolic andKinetic Equations: Asymptotics, Numerics, Analysis”, Contract Number: HPRN-CT-2002-00282. K.F. has also been supported by the Austrian Science Fund FWF projectP16174-N05, by theWittgenstein Award 2000 of Peter A. Markowich, and by the KAUSTinvestigator award 2008 of Peter A. Markowich.
