Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

Abstract

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping 2 1+t ∂ t v and a cubic convolution (|x|) is an unknown function to the problem on R 3 × [0, T ). Here T denotes a maximal existence time of v.The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case γ ∈ (0, 3), and show a lower estimate of the lifespan in the critical or subcritical case γ ∈ − 1 2 , 0 . The