Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

TL;DR

This paper establishes the existence and uniqueness of renormalized solutions for nonlinear parabolic equations in complex anisotropic Musielak-Orlicz spaces without Lavrentiev's phenomenon, addressing challenges in non-reflexive, inhomogeneous, and time-dependent settings.

Contribution

It introduces new methods for handling fully anisotropic, non-reflexive Musielak-Orlicz spaces in time-dependent PDEs, ensuring approximation properties and solution existence.

Findings

Proved existence and uniqueness of solutions in complex spaces.

Developed new approximation-in-time techniques.

Addressed inhomogeneous and anisotropic space challenges.

Abstract

We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in $\mathbb{R}^n$. Namely we study \begin{equation*} \left\{\begin{array}{l } \partial_t u-{\rm div} A(t,x,\nabla u)= f(t,x) \in L^1(\Omega_T),\\ u(0,x)=u_0(x)\in L^1(\Omega). \end{array}\right. \end{equation*} The growth of the monotone vector field $A$ is assumed to be controlled by a generalized nonhomogeneous and anisotropic $N$-function $M:[0,T)\times \Omega \times\mathbb{R}^n \to[0,\infty)$. Existence and uniqueness of renormalized solutions are proven in absence of~Lavrentiev's phenomenon. The condition we impose to ensure approximation properties of the space is a certain type of balance of interplay between the behaviour of $M$ for large $|\xi|$ and small changes of time and space variables. Its instances are log-H\"older continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (changing in time). The noticeable challenge of this paper is considering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time. New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions.