Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 50, pp. 1-13.

Title: Evolution equations on time-dependent Lebesgue spaces with variable exponents
       
Author: Jacson Simsen (Univ. Federal de Itajuba, Minas Gerais, Brazil)

Abstract: 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
$$\displaylines{ 
\frac{\partial u_\lambda}{\partial t}(t)-\operatorname{div}\big(D_\lambda(t,x)|\nabla
u_\lambda(t)|^{p(x,t)-2}\nabla u_\lambda(t)\big)
+|u_\lambda(t)|^{p(x,t)-2}u_\lambda(t) 
=B(t,u_\lambda(t))
}$$
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.

Submitted February 18, 2023. Published July 24, 2023.
Math Subject Classifications: 35K55, 35K92, 35A16, 35B40, 35B41, 37B55.
Key Words: Non-autonomous parabolic problems;  variable exponents;  p-Laplacian; pullback attractors; upper semicontinuity.