## Rocky Mountain Journal of Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 7 September 2016

https://projecteuclid.org/euclid.rmjm/1473275770

Digital Object Identifier
doi:10.1216/RMJ-2016-46-3-1011

Mathematical Reviews number (MathSciNet)
MR3544843

Zentralblatt MATH identifier
1358.35057

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1473275770

#### References

• G. Acosta and R.G. Durán, An optimal poincaré inequality in L1 for convex domains, Proc. Amer. Math. Soc. 132 (2004), 195–202.
• H. Amann, Global existence for semilinear parabolic systems, J. reine angew. Math. 360 (1985), 47–83.
• A.V. Babin and M.I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, 1992.
• N. Ben-Gal, Non-compact global attractors for slowly non-dissipative pdes II: The connecting orbit structure, J. Dyn. Diff. Equat., to appear.
• ––––, Grow-up solutions and heteroclinics to infinity for scalar parabolic PDEs, Ph.D. thesis, Brown University, Providence, RI, 2010.
• ––––, Non-compact global attractors for slowly non-dissipative pdes I, The asymptotics of bounded and grow-up heteroclinics, preprint, 2011.
• H. Brezis,Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
• P. Brunovsk\`y and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynam. Rep. 1 (1988), 57–89.
• ––––, Connecting orbits in scalar reaction diffusion equations II, The complete solution, J. Diff. Equat. 81 (1989), 106–135.
• L.C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, Arch. Rat. Mech. Anal. 197 (2010), 1053–1088.
• B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Equat. 125 (1996), 239–281.
• D.A. Gomes, E. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Diff. Equat. 40 (2015), 40–76.
• A.Y. Goritskii and V.V. Chepyzhov, Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sbor. Math. 196 (2005), 485.
• J K. Hale, Asymptotic behavior of dissipative systems, Math. Surv. Mono. 25, American Mathematical Society, Providence, RI, 1988.
• J. Hell, Conley index at infinity, Ph.D. thesis, Freie Universität, Berlin, 2009.
• D. Henry, Geometric theory of semilinear parabolic equations, volume 840, Springer-Verlag, Berlin, 1981.
• O. Ladyzhenskaya, Attractors for semi-groups and evolution equations, Lincei Lectures, Cambridge University Press, Cambridge, 1991.
• J. Pimentel, Asymptotic behavior of slowly non-dissipative systems, Ph.D. thesis, Universidade de Lisboa, Lisbon, Portugal, 2014.
• J. Pimentel and C. Rocha, A permutation related to non-compact global attractors for slowly non-dissipative equations, J. Dyn. Diff. Equat. 28 (2016), 1–28.
• C. Rocha, Properties of the attractor of a scalar parabolic pde, J. Dyn. Diff. Equat. 3 (1991), 575–591.
• F. Rothe, A priori estimates for reaction-diffusion systems, in Nonlinear diffusion equations and their equilibrium states II, Springer, New York, 1988.