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