# American Institute of Mathematical Sciences

March  2016, 15(2): 495-506. doi: 10.3934/cpaa.2016.15.495

## Reaction-Diffusion equations with spatially variable exponents and large diffusion

 1 Instituto de Matemática e Computaçã, Universidade Federal de Itajubá, 37500-903 Itajubá MG 2 Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 - Itajubá - Minas Gerais, Brazil 3 Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil

Received  April 2015 Revised  October 2015 Published  January 2016

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.
Citation: 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
##### References:
 [1] H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Company, Amsterdam, 1973. [2] H. Brézis, Analyse fonctionnelle:Théorie et applications, Masson, Paris, 1983. [3] A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equation, 116 (1995), 338-404. doi: 10.1006/jdeq.1995.1039. [4] A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q. [5] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522. [6] F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Comput. Math. Appl., 53 (2007), 595-604. doi: 10.1016/j.camwa.2006.02.032. [7] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of non-linear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. [8] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8. [9] X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5. [10] Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)-$growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918. doi: 10.1016/j.nonrwa.2011.04.015. [11] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8. [12] K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78. [13] M. Růžička, Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris S\'er. I., 329 (1999), 393-398. doi: 10.1016/S0764-4442(00)88612-7. [14] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in Lectures Notes in Mathematics (vol. 1748), Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. [15] J. Simsen, A global attractor for a $p(x)$-Laplacian inclusion, C. R. Acad. Sci. Paris Sér. I., 351 (2013), 87-90. doi: 10.1016/j.crma.2013.02.009. [16] J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617. doi: 10.1016/j.na.2009.03.041. [17] J. Simsen and M. S. Simsen, PDE and ODE limit problems for $p(x)$-Laplacian parabolic equations, J. Math. Anal. Appl., 383 (2011), 71-81. doi: 10.1016/j.jmaa.2011.05.003. [18] J. Simsen, M. S. Simsen and M. R. T. Primo, Continuity of the flows and upper semicontinuity of global attractors for $p_s(x)$-Laplacian parabolic problems, J. Math. Anal. Appl., 398 (2013), 138-150. doi: 10.1016/j.jmaa.2012.08.047. [19] J. Simsen, M. S. Simsen and M. R. T. Primo, On $p_s(x)$-Laplacian parabolic problems with non-globally Lipschitz forcing term, Z. Anal. Anwend., 33 (2014), 447-462. doi: 10.4171/ZAA/1522. [20] 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. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

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##### References:
 [1] H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Company, Amsterdam, 1973. [2] H. Brézis, Analyse fonctionnelle:Théorie et applications, Masson, Paris, 1983. [3] A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equation, 116 (1995), 338-404. doi: 10.1006/jdeq.1995.1039. [4] A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q. [5] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522. [6] F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Comput. Math. Appl., 53 (2007), 595-604. doi: 10.1016/j.camwa.2006.02.032. [7] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of non-linear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. [8] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8. [9] X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5. [10] Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)-$growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918. doi: 10.1016/j.nonrwa.2011.04.015. [11] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8. [12] K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78. [13] M. Růžička, Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris S\'er. I., 329 (1999), 393-398. doi: 10.1016/S0764-4442(00)88612-7. [14] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in Lectures Notes in Mathematics (vol. 1748), Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. [15] J. Simsen, A global attractor for a $p(x)$-Laplacian inclusion, C. R. Acad. Sci. Paris Sér. I., 351 (2013), 87-90. doi: 10.1016/j.crma.2013.02.009. [16] J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617. doi: 10.1016/j.na.2009.03.041. [17] J. Simsen and M. S. Simsen, PDE and ODE limit problems for $p(x)$-Laplacian parabolic equations, J. Math. Anal. Appl., 383 (2011), 71-81. doi: 10.1016/j.jmaa.2011.05.003. [18] J. Simsen, M. S. Simsen and M. R. T. Primo, Continuity of the flows and upper semicontinuity of global attractors for $p_s(x)$-Laplacian parabolic problems, J. Math. Anal. Appl., 398 (2013), 138-150. doi: 10.1016/j.jmaa.2012.08.047. [19] J. Simsen, M. S. Simsen and M. R. T. Primo, On $p_s(x)$-Laplacian parabolic problems with non-globally Lipschitz forcing term, Z. Anal. Anwend., 33 (2014), 447-462. doi: 10.4171/ZAA/1522. [20] 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. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
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