Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

TL;DR

This paper establishes existence and uniqueness of solutions for a nonlinear parabolic PDE in complex anisotropic Musielak-Orlicz spaces that vary in time and space, without Lavrentiev's phenomenon, using advanced integration techniques.

Contribution

It introduces a framework for solving parabolic equations in non-reflexive, inhomogeneous Musielak-Orlicz spaces with anisotropic growth, extending prior results to more general settings.

Findings

Proved existence and uniqueness of solutions under specified conditions.

Developed advanced integration by parts formula for Musielak-Orlicz spaces.

Demonstrated approximation properties in non-reflexive, inhomogeneous spaces.

Abstract

We study a general nonlinear parabolic equation on a Lipschitz bounded domain in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ \Omega_T,\\ u(t,x)=0 &\ \mathrm{ on} \ (0,T)\times\partial\Omega,\\ u(0,x)=u_0(x)&\text{in}\ \Omega, \end{array}\right. \end{equation*} with $f\in L^\infty(\Omega_T)$ and $u_0\in L^\infty(\Omega)$. The growth of the monotone vector field $A$ is controlled by a generalized fully anisotropic $N$-function $M:[0,T)\times\Omega\times\mathbb{R}^N\to[0,\infty)$ inhomogeneous in time and space, and under no growth restrictions on the last variable. It results in the need of the integration by parts formula which has to be formulated in an advanced way. Existence and uniqueness of solutions are proven when the Musielak-Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To ensure approximation properties of the space we impose natural assumption that the asymptotic behaviour of the modular function is sufficiently balanced. Its instances are log-H\"older continuity of variable exponent or optimal closeness condition for powers in double phase spaces. The noticeable challenge of this paper is cosidering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time.