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Estimates for a class of slowly non-dissipative reaction-diffusion equations. (English) Zbl 1358.35057

Authors’ abstract: We consider slowly non-dissipative reaction-diffusion equations and establish several estimates. In particular, we manage to control \(L^p\) norms of the solution in terms of \(W^{1,2}\) norms of the initial conditions, for every \(p>2\). This is done by carefully combining preliminary estimates with Gronwall’s inequality and the Gagliardo-Nirenberg interpolation theorem. By considering only positive solutions, we obtain upper bounds for the \(L^p\) norms, for every \(p>1\), in terms of the initial data. In addition, explicit estimates concerning perturbations of the initial conditions are established. The stationary problem is also investigated. We prove that \(L^2\) regularity implies \(L^p\) regularity in this setting, while further hypotheses yield additional estimates for the bounded equilibria. We close the paper with a discussion of the connection between our results and some related problems in the theory of slowly non-dissipative equations and attracting inertial manifolds.

MSC:

35K57 Reaction-diffusion equations
35B65 Smoothness and regularity of solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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