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

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(Ω_T),\\ u(0,x)=u_0(x)\in L^1(Ω). \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-Hölder 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.

The manuscript upgrades arXiv:1707.06097 to the time-dependent setting and essentially uses the framework developed in arXiv:1806.06711

References in corpus (3)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Renormalized solutions of nonlinear parabolic equations with general measure data
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

Cited by in corpus (4)

• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces
• Bloch estimates in non-doubling generalized Orlicz spaces