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 W1,2 norms of the initial conditions, for every p > 2. This is done by carefully combining preliminary estimate...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2016-01, Vol.46 (3), p.1011-1028
Main Authors: PIMENTEL, EDGARD A., PIMENTEL, JULIANA F.S.
Format: Article
Language:English
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Summary: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 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 L 2 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.
ISSN:0035-7596
1945-3795
DOI:10.1216/RMJ-2016-46-3-1011