Global existence and regularity for a pseudo-parabolic equation with p(x, t)-Laplacian

We study the Dirichlet problem for the pseudo-parabolic equation ut = div(|Vu|p(x,t)-2Vu) + Delta ut + f (x, t, u, Vu) in the cylinder (x, t) E QT = omega x (0, T), omega c Rd, d > 2. It is shown that under appropriate conditions on the regularity of the data and the growth of the source f with respect to the second and third arguments, the problem has a global in time solution with the properties u E L infinity(0, T; H02(omega)), ut, |Vut|E L2(QT), |Vu| E L infinity(0,T; Lp(center dot)(omega)) n Lp(center dot,center dot)+delta(QT) with some delta > 0. For special choices of the source f, sufficient conditions of uniqueness are derived, stability of solutions with respect to perturbations of the nonlinear structure of the equation is proven, and the rate of vanishing of llullW1,2(omega) is found.(c) 2023 Elsevier Inc. All rights reserved.