Blow-up for a pseudo-parabolic equation with variable nonlinearity depending on (x, t) and negative initial energy. (June 2023) - Antontsev, Stanislav; Kuznetsov, Ivan; Shmarev, Sergey
Abstract: We study the Dirichlet problem for the pseudo-parabolic equation u t − div a ( x, t ) | ∇ u | p ( x, t ) − 2 ∇ u − Δ u t = b ( x, t ) | u | q ( x, t ) − 2 u in the cylinder Q T = Ω × ( 0, T ), where Ω ⊂ R d 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 ) ∈ Q T p ( x, t ), sup ( x, t ) ∈ Q T q ( x, t ) ), such if the initial energy is negative, E ( 0 ) = ∫ Ω a ( x, 0 ) p ( x, 0 ) | ∇ u 0 ( x ) | p ( x, 0 ) − b ( x, 0 ) q ( x, 0 ) | u 0 ( x ) | q ( x, 0 ) d x < − 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.