Blow-up for a pseudo-parabolic equation with variable nonlinearity depending on (x, t) and negative initial energy

We study the Dirichlet problem for the pseudo-parabolic equation ut -div (a(x, t)IVuIP(x,t) 2Vu) -& UDelta;ut = b(x, t)IuI9(x,t) 2u in the cylinder QT = Q x (0, T), where Q C Rd is a sufficiently smooth domain. The positive coefficients a, b and the exponents p > 2, q > 2 are given Lipschitz-continuous functions. The functions a, p are monotone decreasing, and b, q are monotone increasing in t. It is shown that there exists a positive constant M = M(IQI, sup(x,t)& epsilon;QT p(x, t), sup(x,t)& epsilon;QT q(x, t)), such if the initial energy is negative, & int; (a(x, 0) E(0) = p(x, 0) IVu0(x)IP(x,0) -b(x, 0) q(x, 0)Iu0(x)I9(x,0)) dx < -M, & OHM; then the problem admits a local in time solution with negative energy E(t). If p and q are independent of t, then M = 0. For the solutions from this class, sufficient conditions for the finite time blow-up are derived.