Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

We consider, in this paper, the following nonlinear equation with variable exponents: u(tt) - div (vertical bar del u vertical bar r((.)-2) del u) + a vertical bar u(t)vertical bar(m(.)-2)u(t) = b vertical bar u vertical bar(p(.)-2)u, where a, b > 0 are constants and the exponents of nonlinearity m, p, and r are given functions. We prove a finite-time blow-up result for the solutions with negative initial energy and for certain solutions with positive energy.