Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

Abstract

We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.

Strauss exponent for semilinear wave equations with scattering space dependent damping

“…, and θ = 2, we have p 1 = p 2 = 1, p 3 = 4/3, and thus recover the blow-up result in [4] for 1 < p < p S (5).…”

“…In [4], Georgiev, Kubo and Wakasa showed that the critical power for the radial solutions is the shifted Strauss exponent p = p S (3 + 2), for a special case in R 3 , with damping and potential coefficients satisfying the relation…”

Blow-up for Strauss type wave equation with damping and potential

“…We remark that as for regular damping and potential terms, it has been shown in [3] that p c (3, 2, 2) = p c (5, 0, 0) and the upper and lower bounds for T ε,2,2 (3, p) coincides with those of T ε,0,0 (5, p). Now, we are in a position to state our main result in this paper about the upper bound of lifespan of solutions to (1.1), under the following technical requirements.…”

“…The former one is easy to verify by a direct calculation, so we leave it to readers. As for the latter one, we use the idea comes from [3]. To begin with, we introduce a proposition.…”

“…To begin with, we introduce a proposition. Proposition 5.4 (Lemma 3.2 of [3]). Let C 1 , C 2 > 0, α, β ≥ 0, κ ≤ 1, ε ∈ (0, 1], and p > 1.…”