Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity

The authors of this article study the existence and uniqueness of weak solutions of the initial-boundary value problem for ut=div((|u|σ+d0)|∇u|p(x,t)−2∇u)+f(x,t)(0<σ<2).. They apply the method of parabolic regularization and Galerkin's method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L2(Ω) norm as t → ∞.