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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 |
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.
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. |
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. |
[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. |
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