Evolution equations on time-dependent Lebesgue spaces with variable exponents

We extend the results in Kloeden-Simsen [CPAA 2014] to \(p(x,t)\)-Laplacian problems on time-dependent Lebesgue spaces withvariable 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