Open Access
2016 Estimates for a class of slowly non-dissipative reaction-diffusion equations
Edgard A. Pimentel, Juliana F.S. Pimentel
Rocky Mountain J. Math. 46(3): 1011-1028 (2016). DOI: 10.1216/RMJ-2016-46-3-1011


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 $W^{1,2}$ 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.


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Edgard A. Pimentel. Juliana F.S. Pimentel. "Estimates for a class of slowly non-dissipative reaction-diffusion equations." Rocky Mountain J. Math. 46 (3) 1011 - 1028, 2016.


Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 1358.35057
MathSciNet: MR3544843
Digital Object Identifier: 10.1216/RMJ-2016-46-3-1011

Primary: 35B65 , 35K57 , 58J35

Keywords: Gagliardo-Nirenberg inequality , Reaction-diffusion equations , Slowly non-dissipative systems , Sobolev regularity

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 3 • 2016
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