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We consider, in this paper, the following nonlinear equation with variable exponents: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. KEYWORDS blowup, nonlinear damping, Sobolev spaces with variable exponentsA > 0, 0 < < 1. The term Δ r(·) u = div(|∇u| r(·)−2 ∇u) is called r(·)− Laplacian.

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The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. Mathematics Subject Classification