Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function

Abstract

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems.

References in corpus (1)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces

Cited by in corpus (2)

• 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