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Non-autonomous reaction-diffusion equations with variable exponents and large diffusion
Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 - Itajubá - Minas Gerais, Brazil |
$\left\{ {\begin{array}{*{20}{l}}{\frac{{\partial {u_s}}}{{\partial t}}(t) - {D_s}{\rm{div}}(|\nabla {u_s}{|^{{p_s}(x) - 2}}\nabla {u_s}) + C(t)|{u_s}{|^{{p_s}(x) - 2}}{u_s} = B({u_s}(t)),\;\;t > \tau ,}\\{{u_s}(\tau ) = {u_{\tau s}},}\end{array}} \right.$ |
$u_{τ s}∈ H: = L^2(Ω),$ |
$Ω\subset\mathbb{R}^n$ |
$n≥ 1$ |
$B:H \to H$ |
$L≥ 0$ |
$D_s∈[1,∞)$ |
$C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$ |
$p_s(·)∈ C(\bar{Ω})$ |
$p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$ |
$p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$ |
$s∈ \mathbb{N},$ |
$p_s(·) \to p$ |
$L^∞(Ω)$ |
$D_s \to ∞$ |
$s \to∞,$ |
$a,p>2$ |
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. |

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