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April  2019, 24(4): 1485-1510. doi: 10.3934/dcdsb.2018217

Non-autonomous reaction-diffusion equations with variable exponents and large diffusion

Antonio Carlos Fernandes , , Marcela Carvalho Gonçcalves and  Jacson Simsen

Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 - Itajubá - Minas Gerais, Brazil

Corresponding author: jacson@unifei.edu.br

Received  November 2017 Revised  March 2018 Published  April 2019 Early access  June 2018

Fund Project: J. Simsen has been partially supported by FAPEMIG - processes PPM 00329-16 (Brazil) and CEX-APQ-00814-16. M.C. Gonçalves was supported with CAPES scholarship (Brazil).

In this work we prove continuity of solutions with respect to initial conditions and a couple of parameters and we prove upper semicontinuity of a family of pullback attractors for the problem
$\left\{ {\begin{array}{*{20}{l}}{\frac{{\partial {u_s}}}{{\partial t}}(t) - {D_s}{\rm{div}}(|\nabla {u_s}{|^{{p_s}(x) - 2}}\nabla {u_s}) + C(t)|{u_s}{|^{{p_s}(x) - 2}}{u_s} = B({u_s}(t)),\;\;t > \tau ,}\\{{u_s}(\tau ) = {u_{\tau s}},}\end{array}} \right.$
under homogeneous Neumann boundary conditions,
$u_{τ s}∈ H: = L^2(Ω),$
$Ω\subset\mathbb{R}^n$
(
$n≥ 1$
) is a smooth bounded domain,
$B:H \to H$
 is a globally Lipschitz map with Lipschitz constant
$L≥ 0$
,
$D_s∈[1,∞)$
,
$C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$
 is bounded from above and below and is monotonically nonincreasing in time,
$p_s(·)∈ C(\bar{Ω})$
,
$p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$
$p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$
 for all
$s∈ \mathbb{N},$
 when
$p_s(·) \to p$
 in
$L^∞(Ω)$
 and
$D_s \to ∞$
 as
$s \to∞,$
 with
$a,p>2$
 positive constants.
Keywords: Reaction-diffusion equations, parabolic problems, variable exponents, pullback attractors, upper semicontinuity.
Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B41.
Citation: 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
References:
[1]

C. O. Alves, S. Shmarev, J. Simsen and M. 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 Space, Noordhoff International Publishing, 1976.

[3]

H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equation, 116 (1995), 338-404.  doi: 10.1006/jdeq.1995.1039.

[6]

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.

[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.  doi: 10.1137/0135001.

[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]

X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(Ω)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[11]

X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m, p(x)}(Ω)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[12]

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.

[13]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.

[14]

S. Kondo and M. Mimura, A reaction-diffusion system and its shadow system describing harmful algal blooms, Tamkang Journal of Mathematics, 47 (2016), 71-92.  doi: 10.5556/j.tkjm.47.2016.1916.

[15]

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.

[16]

J. Simsen, M. J. D. Nascimento and M. S. Simsen, Existence and upper semicontinuity of pullback attractors for non-autonomous p−Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.  doi: 10.1016/j.jmaa.2013.12.019.

[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, 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.

[19]

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

[20]

S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.  doi: 10.2969/jmsj/03140623.

show all references

References:
[1]

C. O. Alves, S. Shmarev, J. Simsen and M. 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 Space, Noordhoff International Publishing, 1976.

[3]

H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equation, 116 (1995), 338-404.  doi: 10.1006/jdeq.1995.1039.

[6]

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.

[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.  doi: 10.1137/0135001.

[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]

X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(Ω)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[11]

X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m, p(x)}(Ω)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[12]

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.

[13]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.

[14]

S. Kondo and M. Mimura, A reaction-diffusion system and its shadow system describing harmful algal blooms, Tamkang Journal of Mathematics, 47 (2016), 71-92.  doi: 10.5556/j.tkjm.47.2016.1916.

[15]

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.

[16]

J. Simsen, M. J. D. Nascimento and M. S. Simsen, Existence and upper semicontinuity of pullback attractors for non-autonomous p−Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.  doi: 10.1016/j.jmaa.2013.12.019.

[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, 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.

[19]

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

[20]

S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.  doi: 10.2969/jmsj/03140623.

Figure 4.  Figure with $x'+(\sin(t)+1.1)|x|^2x = x$, with $p = 4$
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Figure 5.  Figure with $x'+(\sin(t)+1.1)|x|^7x = x$, with $p = 9$
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Figure 1.  Figure with $x'+C(t)|x|^2x = x$, with $p = 4$ and $C(t) = 1.1$ if $t\leq 0$ and $C(t) = e^{-t}+0.1$ if $t>0$
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Figure 2.  Figure with $x'+(e^{-t^2}+0.1)|x|^2x = x$, with $p = 4$
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Figure 3.  Figure with $x'+(\cos(t)+1.1)|x|^2x = x$, with $p = 4$
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