Rocky Mountain Journal of Mathematics

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

Edgard A. Pimentel and Juliana F.S. Pimentel

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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.

Article information

Rocky Mountain J. Math., Volume 46, Number 3 (2016), 1011-1028.

First available in Project Euclid: 7 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 35K57: Reaction-diffusion equations 58J35: Heat and other parabolic equation methods

Slowly non-dissipative systems reaction-diffusion equations Gagliardo-Nirenberg inequality Sobolev regularity


Pimentel, Edgard A.; Pimentel, Juliana F.S. Estimates for a class of slowly non-dissipative reaction-diffusion equations. Rocky Mountain J. Math. 46 (2016), no. 3, 1011--1028. doi:10.1216/RMJ-2016-46-3-1011.

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