-
Previous Article
Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission
- DCDS-B Home
- This Issue
-
Next Article
Evolutionary de Rham-Hodge method
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. |
| [1] |
Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure and Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495 |
| [2] |
Antonio Carlos Fernandes, Marcela Carvalho Gonçcalves, Jacson Simsen. Non-autonomous reaction-diffusion equations with variable exponents and large diffusion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1485-1510. doi: 10.3934/dcdsb.2018217 |
| [3] |
Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653 |
| [4] |
Peter E. Kloeden, Meihua Yang. Forward attracting sets of reaction-diffusion equations on variable domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1259-1271. doi: 10.3934/dcdsb.2019015 |
| [5] |
Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036 |
| [6] |
Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054 |
| [7] |
Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101 |
| [8] |
Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1 |
| [9] |
Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079 |
| [10] |
Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543 |
| [11] |
Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399 |
| [12] |
Dingshi Li, Xuemin Wang. Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Electronic Research Archive, 2021, 29 (2) : 1969-1990. doi: 10.3934/era.2020100 |
| [13] |
Xiangming Zhu, Chengkui Zhong. Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3933-3945. doi: 10.3934/dcdsb.2021212 |
| [14] |
Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201 |
| [15] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 |
| [16] |
Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189 |
| [17] |
Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252 |
| [18] |
Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 |
| [19] |
Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891 |
| [20] |
Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]