A doubly nonlinear parabolic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source

The aim of this work is to develop a self-contained existence and uniqueness theory for a doubly nonlinear parabolic problem with variable exponents and homogeneous Neumann boundary conditions. Our approach is based on Rothe's time-discretization method and builds on the results established by the author in a previous paper. We also investigate the asymptotic behavior of the solution. All arguments are presented in full detail, so that the paper is self-contained.