A nonlinear viscoelastic plate equation with p⃗(x, t)-Laplace operator: Blow up of solutions with negative initial energy. (June 2021) - Antontsev, S.; Ferreira, J.
Abstract: In this paper we consider a nonlinear class viscoelastic plate equation with a lower order by perturbation of p ⃗ ( x, t ) -Laplace operator of the form u t t + Δ 2 u − Δ p ⃗ ( x, t ) u + ∫ 0 t g ( t − s ) Δ u ( s ) d s − ϵ Δ u t + f ( u ) = 0, ( x, t ) ∈ Q T = Ω × ( 0, T ), associated with initial and Dirichlet–Neumann boundary conditions. Under suitable conditions on g, f and the variable exponent of the p ⃗ ( x, t ) -Laplace operator, we prove a blow up in finite time with negative initial energy in the presence of a strong damping ϵ Δ u t ( ϵ > 0 ) acting in the domain. This equation corresponds to a viscoelastic version arising in dynamics of elastoplastic flows and plate vibrations.