Convergence of quasilinear parabolic equations to semilinear equations

Flank D. M. Bezerra; Jacson Simsen; Mariza Stefanello Simsen

doi:10.3934/dcdsb.2020258

Article Contents

2021, Volume 26, Issue 7: 3823-3834. Doi: 10.3934/dcdsb.2020258

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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
^* Corresponding author: Jacson Simsen

Received: April 2020

Revised: July 2020

Early access: August 2020

Published: July 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary:35B40, 35B41, 35K57;Secondary:35K59.

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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.
[2]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976.
[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.
[4]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, 1981.
[5]	J. Simsen and M. S. Simsen, On $p(x)$-Laplacian parabolic problems, Nonlinear Stud., 18 (2011), 393-403.
[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.
[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.
[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.
[9]	A. S. Tersenov, The one dimensional parabolic $p(x)-$Laplace equation, NoDEA, 23 (2016), 1-11. doi: 10.1007/s00030-016-0377-y.