Estimates for a class of slowly non-dissipative reaction-diffusion equations

Abstract

In this paper, we consider slowly nondissipative reaction-diffusion equations and establish several estimates. In particular, we manage to control Lp norms of the solution in terms of W1,2 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 Lp 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 L2 regularity implies Lp 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.