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 term 21+t∂tv and a cubic convolution (|x|−γ⁎v2)v with γ∈(−12,3) in three spatial dimension for initial data (v(x,0),∂tv(x,0))∈C2(R3)×C1(R3) with a compact support, where v=v(x,t) is an unknown function to the problem on R3×[0,T). Here T denotes the maximal existence time of v.The first aim of the present paper is to prove unique global (in time) existence and asymptotic behavior of the solutions of associated integral equations to the Cauchy problem in the supercritical case γ∈(0,3), and show a lower estimate of the lifespan in the critical or subcritical case γ∈(−12,0]. The essential part for their proofs is to derive a weaker estimate under a weaker condition (Proposition 3.1 and Theorem 3.1) than the case without damping (see [22]) and to recover the weakness by the effect of the dissipative term.The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case γ∈(−12,0). The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in [4] to obtain an upper estimate of the lifespan.Our two results determine that a critical exponent γc which divides global existence and blow-up for small solutions is 0, namely γc=0. As the result, we can see that the critical exponent shifts from 2 to 0 due to the effect of the scale invariant damping term.