Evolution equations on time-dependent Lebesgue spaces with variable exponents

Pub Date : 2023-07-24 DOI:10.58997/ejde.2023.50

J. Simsen

引用次数: 0

Abstract

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

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变指数时变Lebesgue空间上的演化方程

我们将Kloeden-Simsen [CPAA 2014]中的结果推广到$p(x,t)$ -拉普拉斯问题上的变指数时变Lebesgue空间。研究了在$\mathbb{R}^n$, $n\geq 1$中有界光滑域$\Omega$上的方程$$\displaylines{ \frac{\partial u_\lambda}{\partial t}(t)-\operatorname{div}\big(D_\lambda(t,x)|\nabla u_\lambda(t)|^{p(x,t)-2}\nabla _\lambda(t)\big)+|u_\lambda(t)|^{p(x,t)-2}u_\lambda(t) =B(t,u_\lambda(t)) }$$，该方程具有齐次Neumann边界条件，其中指数$p(\cdot)\in C(\bar{\Omega}\times [\tau,T],\mathbb{R}^+)$满足$\min p(x,t)>2$, $\lambda\in [0,\infty)$是一个参数。欲了解更多信息，请参阅https://ejde.math.txstate.edu/Volumes/2023/50/abstr.html

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