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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

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 & 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.  Google Scholar

[2]

H. Brézis, Analyse fonctionnelle:Théorie et applications, Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[12]

K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78. Google Scholar

[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.  Google Scholar

[14]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in Lectures Notes in Mathematics (vol. 1748), Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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.  Google Scholar

[2]

H. Brézis, Analyse fonctionnelle:Théorie et applications, Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78. Google Scholar

[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.  Google Scholar

[14]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in Lectures Notes in Mathematics (vol. 1748), Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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