Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions

Abstract

We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \[\partial_t u-\mathrm{div} A(x,\nabla u)= f\in L^1(Ω_T),\] on a Lipschitz bounded domain in ${\mathbb{R}}^n$. The growth of the weakly monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M$. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^{\ast }$ (neither ${\Delta }_2$, nor ${\nabla }_2$). The condition we impose on $M$ is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.

arXiv admin note: text overlap with arXiv:1701.08970

References in corpus (5)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces
• Renormalized solutions of nonlinear parabolic equations with general measure data
• Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space
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Cited by in corpus (8)

• Maximal regularity for local minimizers of non-autonomous functionals
• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces
• Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function
• An extended variational theory for nonlinear evolution equations via modular spaces