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

Document Type : Research Paper


Department of Mathematics, Jahrom University, Jahrom, Iran


In this paper, we consider the nonlinear $r(x)-$Laplacian Lam'{e} equation
u_{tt}-\Delta_{e}u-div\big(|\nabla u|^{r(x)-2}\nabla u\big)+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u
in a smoothly bounded domain $\Omega\subseteq R^{n},\ n\geq1$, 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.