American Institute of Mathematical Sciences

Advanced
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  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, 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

[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

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

[1]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[3]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[5]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[6]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[7]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[8]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[9]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[10]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[11]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[12]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[13]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385
[14]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[15]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446
[16]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435
[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[18]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[19]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452
[20]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences