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This work deals with the blow-up of solutions for a new class of quasilinear wave equation with variable exponent nonlinearities. To clarify more, we prove in the presence of dispersion term − Δ u t t a finite-time blow-up result for the solutions with negative initial energy and also for certain solutions with positive energy. Our results are extension of the recent work (Appl Anal. 2017; 96(9): 1509-1515).

In this article, we study the unconstrained minimization problem (P) min ff (x) : x 2 R n g : where f : R n ! R is a continously di¤erentiable function. We introduce a new algorithm which accelerates the convergence of the steepest descent method. We study the global convergence of the new algorithm, named the Wolfe epsilon steepest descent algorithm, by using Wolfe inexact line search ([35],[36]). In [16], [33], Benzine, Djeghaba and Rahali studied the same problem by using exact and Armijo inexact line serachs. Numerical tests show that Wolfe Epsilon steepest descent Algorithm accelerates the convergence of the gradient method and is more successful than Armijo Epsilon Steepest descent, Exact Epsilon steepest descent algorithm and Steepest descent algorithm.

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