Blow up of solutions for a r(x)-Laplacian Lam'{e} equation with variable-exponent nonlinearities and arbitrary initial energy level

Abstract

In this paper, we consider the nonlinear $r(x)-$Laplacian Lam'{e} equation $$ u_{tt}-Delta_{e}u-divbig(|nabla u|^{r(x)-2}nabla ubig)+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u $$ in a smoothly bounded domain $Omegasubseteq R^{n}, ngeq1$, where $r(.), m(.)$ and $p(.)$ are continuous and measurable functions. Under suitable conditions on variable exponents and initial data, the blow-up of solutions is proved with negative initial energy as well as positive.