A nonlinear viscoelastic plate equation with p⃗(x,t)-Laplace operator: Blow up of solutions with negative initial energy

In this paper we consider a nonlinear class viscoelastic plate equation with a lower order by perturbation of p⃗(x,t)-Laplace operator of the form utt+Δ2u−Δp⃗(x,t)u+∫0tg(t−s)Δu(s)ds−ϵΔut+f(u)=0,(x,t)∈QT=Ω×(0,T), associated with initial and Dirichlet–Neumann boundary conditions. Under suitable conditions on g,f and the variable exponent of the p⃗(x,t)-Laplace operator, we prove a blow up in finite time with negative initial energy in the presence of a strong damping ϵΔut(ϵ>0) acting in the domain. This equation corresponds to a viscoelastic version arising in dynamics of elastoplastic flows and plate vibrations.