Global well-posedness of solutions to a class of double phase parabolic equation with variable exponents

Abstract

The main objective of this paper is to study a class of parabolic equation driven by double phase operator with initial-boundary value conditions. As is well known, subcritical hypotheses play an important role in investigating well-posedness result to parabolic and elliptic equations. The highlight of this paper is to overcome the difficulties without sub-critical assumption creates by restricting the domain. We firstly obtain the local solution separately on the radial and the nonradial cases by the appropriate approach of subdiffer-ential, Palais principle of symmetric criticality and variational methods. Later, using the potential well method, the results of global solution and decay estimates of energy functional are proved when the initial energy is subcritical. Finally, we derived the blow-up in finite time of solutions when the initial data satisfies different conditions. The present work extends and complements some of earlier contributions related parabolic equations involving p(x)-Laplacian.