Reaction-Diffusion equations with spatially variable exponents and large diffusion

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

Reaction-Diffusion equations with spatially variable exponents and large diffusion

Jacson Simsen ^1, , Mariza Stefanello Simsen ^2, and Marcos Roberto Teixeira Primo ^3,

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

Abstract
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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.

Keywords: parabolic problems, variable exponents, Reaction-Diffusion equations, attractors, upper semicontinuity..

Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B4.

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.

show all references

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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