Variable exponent perturbation of a parabolic equation with $p(x)$-Laplacian - Aldo Trajano Lourêdo; Manuel Milla Miranda; Marcondes Rodrigues Clark (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
This paper is concerned with the study of the global existence and the decay of solutions of an evolution problem driven by an anisotropic operator and a nonlinear perturbation, both of them having a variable exponent. Because the nonlinear perturbation leads to difficulties in obtaining a priori estimates in the energy method, we had to significantly modify the Tartar method. As a result, we could prove the existence of global solutions at least for small initial data. The decay of the energy is derived by using a differential inequality and applying a non-standard approach.