Critical exponent for the semilinear wave equation with time-dependent damping

The threshold of effective damping for semilinear wave equations

“…In this special case, a global existence result has been obtained in [16,19]. On the other hand, if b(t) is a sufficiently smooth function satisfying lim sup t→∞ tb(t) < 1 then the dissipation is non effective [27].…”

“…By assuming compactly supported data, Y. Wakasugi recently extended the result in [16] to prove that if f (u) = |u| p with p > 1 + 2/n then there exists…”

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].…”

The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation

“…This effect is a consequence of the diffusion phenomenon: the asymptotic profile as t → ∞ of the solution to the damped wave is described by the solution to a heat equation. The situation remains the same if the damping term b(t)u t is added to the equation in (7), for a quite large class of coefficients b(t) (see [10,15,36,51]). However, an interesting transition model has been found and studied in [14,16]: if b(t) = 2/(1 + t), the critical exponent is given by 1 + 2/n if n = 1, 2 and by p 0 (n + 2) if n ≥ 3, is odd (here p 0 is as in (8)).…”