Semi-linear wave equations with effective damping

Abstract

The authors study the Cauchy problem for the semi-linear damped wave equation\ud in any space dimension. It is assumed that the time-dependent damping term is effective. The global existence of small energy\ud data solutions for polynomial nonlinear term in the supercritical case is proved.The authors study the Cauchy problem for the semi-linear damped wave equation utt - Δu + b(t)ut = f(u), u(0,x) =u0(x), ut(0,x) = u1(x) in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for pipef(u)pipe ≈ pipeupipep in the supercritical case of p > 1 + 2/n and p ≤ n/n-2 for n ≥ 3 is proved. © 2013 Fudan University and Springer-Verlag Berlin Heidelberg

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

“…To complete our overview, we mention that the critical exponent for the wave equation with timedependent damping µ t κ u t is 1 + 2/n if κ ∈ (−1, 1) (see [6,18,19]), whereas global existence of small data solutions for p > 1 + 2/(n − α) for the wave equation with damping µ x −α t −β , if α, β > 0 and α + β < 1 has been derived in [25].…”

$$L^1$$ L 1 – $$L^p$$ L p estimates for radial solutions of the wave equation and application

“…for any u, v ∈ X (see, for instance, [3]). Now, we recall that in Theorem 2 we already proved that u (0) (t, x) = P(u (−1) ) = K 1 (t, x) * (x) g(x) ∈ X .…”

“…Moreover, the asymptotic behavior on t of S t follows by homogeneity, namely S t = t 1−n/ p+n/q S 1 , i.e., there exists a positive constant C such that u(t, ·) L q ≤ C t 1−n/ p+n/q g L p (3) hold uniformly for any t > 0. The Strichartz's line condition [21] and duality arguments are the reasons why (3) is not obtained outside the triangle P 1 P 2 P 3 .…”

Global existence for semi-linear structurally damped σ-evolution models

“…Thus, it was reasonable to generalize the results from [16] to the family of effective damping terms b(t)u t . This is done in [6]. The next step was to use other damping mechanisms, for example, structural damping terms of the form (−Δ) δ u t .…”