The critical exponent for semilinear σ-evolution equations with a strong non-effective damping. (February 2022) - D'Abbicco, M.; Ebert, M.R.
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.