Evolution equations on time-dependent Lebesgue spaces with variable exponents
DOI:
https://doi.org/10.58997/ejde.2023.50Keywords:
Non-autonomous parabolic problems; variable exponents; p-Laplacian; pullback attractors; upper semicontinuityAbstract
We extend the results in Kloeden-Simsen [CPAA 2014] to \(p(x,t)\)-Laplacian problems on time-dependent Lebesgue spaces with
variable exponents. We study the equation $$\displaylines{ \frac{\partial u_\lambda}{\partial t}(t)-\operatorname{div}\big(D_\lambda(t,x)|\nabla u_\lambda(t)|^{p(x,t)-2}\nabla _\lambda(t)\big)
+|u_\lambda(t)|^{p(x,t)-2}u_\lambda(t) =B(t,u_\lambda(t)) }$$
on a bounded smooth domain \(\Omega\) in \(\mathbb{R}^n\),
\(n\geq 1\), with a homogeneous Neumann boundary condition, where the exponent \(p(\cdot)\in C(\bar{\Omega}\times [\tau,T],\mathbb{R}^+)\) satisfies \(\min p(x,t)>2\), and \(\lambda\in [0,\infty)\) is a parameter.
For more information see https://ejde.math.txstate.edu/Volumes/2023/50/abstr.html
References
C. O. Alves, S. Shmarev, J. Simsen, 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), no. 1, 265–294
S. Antontsev, S. Shmarev; Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up, Atlantis Studies in Differential Equations, 4. Atlantis Press, Paris, 2015.
S. Antonsev, S. Shmarev; A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), 515–545.
M. Avci; Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl., 2013: 55 (2013), 1–13.
M. Avci, B. Cekic, A. V. Kalinin, R. A. Mashiyev; Lp(x)(Ω)-estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl., 389 (2012), No. 2, 838–851.
V. Barbu; Nonlinear Semigroups and Differential Equations in Banach Space, Noordhoff International, 1976.
A. N. Carvalho, J. A. Langa, J. C. Robinson; Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, 2012.
Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (4) (2006) 1383–1406.
L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka; Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-laplacian Dirichlet problems, Nonlinear Anal., 52 (2003) 1843–1852.
X. L. Fan, D. Zhao; On the spaces Lp(x)(Ω) and W m,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424–446.
Z. Guo, Q. Liu, J. Sun, B. Wu; Reaction-diffusion systems with p(x)−growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904–2918.
P. E. Kloeden, P. Mar ́ın-Rubio, J. Real; Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062–2090.
P. E. Kloeden, J. Simsen; Pullback attractors for non-autonomous evolution equation with spatially variable exponents, Commun. Pure & Appl. Analysis, 13 , no. 6, (2014), 2543–2557.
T. F. Ma, P. Marın-Rubio, C. M. S. Chu ̃no; Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317–3342.
P. Mar ́ın-Rubio, J. Real; On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis, 71 (2009), 3956–3963.
K. Rajagopal, M. Ruzicka; Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001), 59–78.
M. R ̊uˇziˇcka; Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer-Verlag, Berlin, 2000.
J. Simsen; A global attractor for a p(x)-Laplacian parabolic problem, Nonlinear Anal., 73 (10) (2010), 3278–3283.
J. Simsen, M. J. D. Nascimento, 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.
S. Yotsutani; Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623–646.
Downloads
Published
License
Copyright (c) 2023 Jacson Simsen
This work is licensed under a Creative Commons Attribution 4.0 International License.
