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

Convergence of quasilinear parabolic equations to semilinear equations

Flank D. M. Bezerra 1, , Jacson Simsen 2,, and  Mariza Stefanello Simsen 2,

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.

Keywords: Reaction-diffusion equations, parabolic problems, variable exponents, attractors, upper semicontinuity.
Mathematics Subject Classification: Primary:35B40, 35B41, 35K57;Secondary:35K59.
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:
[1]

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

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[3]

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

[4]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[5]

J. Simsen and M. S. Simsen, On $p(x)$-Laplacian parabolic problems, Nonlinear Stud., 18 (2011), 393-403.   Google Scholar

[6]

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

[7]

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

[8]

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

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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

show all references

References:
[1]

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

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[3]

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

[4]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[5]

J. Simsen and M. S. Simsen, On $p(x)$-Laplacian parabolic problems, Nonlinear Stud., 18 (2011), 393-403.   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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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