ESTIMATES FOR A CLASS OF SLOWLY NON-DISSIPATIVE REACTION-DIFFUSION EQUATIONS

In this paper, 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,W-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.