The critical exponent for semilinear [formula omitted]-evolution equations with a strong non-effective damping

Abstract

In this paper, we find the critical exponent for the existence of global small data solutions to: utt+(−Δ)σu+(−Δ)θ2ut=f(u,ut),t≥0,x∈Rn,(u,ut)(0,x)=(0,u1(x)),in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1<α<α̃. Assuming initial data to be small in L1 or in some other Lp space, p∈(1,2), in addition to the energy space, the critical exponent only depends on the ratio n/(σp). We also prove the global existence of small data solutions in high space dimension for α>ᾱ, but we leave open to determine if a counterpart nonexistence result for α<ᾱ holds or not.