Blow-up Result for a Semilinear Wave Equation with a Nonlinear Memory Term

Abstract

In this note, we study the blow-up dynamic of a semilinear Cauchy problem for the wave equation with a nonlinear memory term. More precisely, we consider as memory term the Riemann-Liouville fractional integral of order 1 − γ of the p power of the solution, where γ ∈ (0, 1). We prove two blow-up results by using an iteration argument. In the subcritical case we show the blow-up in finite time of the space average of a local in time solution, under certain integral sign assumptions for the initial data. In the result for the limit case, we refine this approach by considering a weighted average of a local solution instead and applying the so-called slicing method.