Blow-up for a pseudo-parabolic equation with variable nonlinearity depending on [formula omitted] and negative initial energy

Abstract

We study the Dirichlet problem for the pseudo-parabolic equation ut−diva(x,t)|∇u|p(x,t)−2∇u−Δut=b(x,t)|u|q(x,t)−2uin the cylinder QT=Ω×(0,T), where Ω⊂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(|Ω|,sup(x,t)∈QTp(x,t),sup(x,t)∈QTq(x,t)), such if the initial energy is negative, E(0)=∫Ωa(x,0)p(x,0)|∇u0(x)|p(x,0)−b(x,0)q(x,0)|u0(x)|q(x,0)dx<−M,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.