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

Abstract

The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in RN. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic N-function M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large |ξ| 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 (possibly changing in time). New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions.