July  2021, 26(7): 3823-3834. doi: 10.3934/dcdsb.2020258

## Convergence of quasilinear parabolic equations to semilinear equations

 1 Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa - PB, Brazil 2 Instituto de Matemática e Computação, Universidade Federal de Itajubá, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá - MG, Brazil

* Corresponding author: Jacson Simsen

Received  April 2020 Revised  July 2020 Published  July 2021 Early access  August 2020

Fund Project: J. Simsen was partially supported by the Brazilian research agency FAPEMIG - Process PPM 00329-16

In this work we consider a family of reaction-diffusion equations with variable exponents reaching as a limit problem a semilinear equation. We provide uniform estimates for the solutions and we prove that the solutions of the family of quasilinear equations with variable exponents converge to the solution of a limit semilinear equation when the exponents go to 2. Moreover, the robustness of the global attractors is also studied.

Citation: Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258
##### References:
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show all references

##### References:
  C. Alves, S. Shmarev, J. Simsen and M. S. Simsen, The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior, J. Math. Anal. Appl., 443 (2016), 265-294.  doi: 10.1016/j.jmaa.2016.05.024. Google Scholar  V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976. Google Scholar  F. Bezerra, J. Simsen and M. S. Simsen, Semilinear limit problems for reaction-diffusion equations with variable exponents, J. Differential Equations, 266 (2019), 3906-3924.  doi: 10.1016/j.jde.2018.09.021. Google Scholar  D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, 1981. Google Scholar  J. Simsen and M. S. Simsen, On $p(x)$-Laplacian parabolic problems, Nonlinear Stud., 18 (2011), 393-403. Google Scholar  J. Simsen, M. S. Simsen and F. B. Rocha, Existence of solutions for some classes of parabolic problems involving variable exponents, Nonlinear Stud., 21 (2014), 113-128. Google Scholar  J. Simsen, M. S. Simsen and M. R. T. Primo, Reaction-diffusion equations with spatially variable exponents and large diffusion, Commun. Pure Appl. Anal., 15 (2016), 495-506.  doi: 10.3934/cpaa.2016.15.495. Google Scholar  J. Simsen, M. S. Simsen and A. Zimmermann, Study of ODE limit problems for reaction-diffusion equations, Opuscula Math., 38 (2018), 117-131.  doi: 10.7494/OpMath.2018.38.1.117. Google Scholar  A. S. Tersenov, The one dimensional parabolic $p(x)-$Laplace equation, NoDEA, 23 (2016), 1-11.  doi: 10.1007/s00030-016-0377-y. Google Scholar
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