## Estimates for a class of slowly non-dissipative reaction-diffusion equations.(English)Zbl 1358.35057

Authors’ abstract: 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.

### MSC:

 35K57 Reaction-diffusion equations 35B65 Smoothness and regularity of solutions to PDEs 58J35 Heat and other parabolic equation methods for PDEs on manifolds
Full Text:

### References:

 [1] G. Acosta and R.G. Durán, An optimal poincaré inequality in L1 for convex domains, Proc. Amer. Math. Soc. 132 (2004), 195-202. · Zbl 1057.26010 [2] H. Amann, Global existence for semilinear parabolic systems , J. reine angew. Math. 360 (1985), 47-83. · Zbl 0564.35060 [3] A.V. Babin and M.I. Vishik, Attractors of evolution equations , North-Holland, Amsterdam, 1992. · Zbl 0778.58002 [4] N. Ben-Gal, Non-compact global attractors for slowly non-dissipative pdes II: The connecting orbit structure , J. Dyn. Diff. Equat., [5] —-, Grow-up solutions and heteroclinics to infinity for scalar parabolic PDEs, Ph.D. thesis, Brown University, Providence, RI, 2010. [6] —-, Non-compact global attractors for slowly non-dissipative pdes I, The asymptotics of bounded and grow-up heteroclinics , preprint, 2011. [7] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations , Springer, New York, 2011. · Zbl 1220.46002 [8] P. Brunovskỳ and B. Fiedler, Connecting orbits in scalar reaction diffusion equations , Dynam. Rep. 1 (1988), 57-89. · Zbl 0679.35047 [9] —-, Connecting orbits in scalar reaction diffusion equations II, The complete solution , J. Diff. Equat. 81 (1989), 106-135. · Zbl 0699.35144 [10] L.C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, Arch. Rat. Mech. Anal. 197 (2010), 1053-1088. · Zbl 1273.70030 [11] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations , J. Diff. Equat. 125 (1996), 239-281. · Zbl 0849.35056 [12] 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. · Zbl 1322.35053 [13] A.Y. Goritskii and V.V. Chepyzhov, Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds , Sbor. Math. 196 (2005), 485. [14] J K. Hale, Asymptotic behavior of dissipative systems , Math. Surv. Mono. 25 , American Mathematical Society, Providence, RI, 1988. · Zbl 0642.58013 [15] J. Hell, Conley index at infinity , Ph.D. thesis, Freie Universität, Berlin, 2009. [16] D. Henry, Geometric theory of semilinear parabolic equations , volume 840, Springer-Verlag, Berlin, 1981. · Zbl 0456.35001 [17] O. Ladyzhenskaya, Attractors for semi-groups and evolution equations , Lincei Lectures, Cambridge University Press, Cambridge, 1991. [18] J. Pimentel, Asymptotic behavior of slowly non-dissipative systems , Ph.D. thesis, Universidade de Lisboa, Lisbon, Portugal, 2014. [19] 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. · Zbl 1364.37157 [20] C. Rocha, Properties of the attractor of a scalar parabolic pde, J. Dyn. Diff. Equat. 3 (1991), 575-591. · Zbl 0769.35032 [21] F. Rothe, A priori estimates for reaction-diffusion systems , in Nonlinear diffusion equations and their equilibrium states II, Springer, New York, 1988. · Zbl 0664.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.