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

Abstract

Abstract. In this note we study the global existence of small data solutions to the Cauchy problem for the semi-linear wave equation with a not effective scale-invariant damping term, namelywhere p > 1, n ≥ 2. We prove blow-up in finite time in the subcritical range p ∈ (1, p 2 (n)] and an existence result for p > p 2 (n), n = 2, 3. In this way we find the critical exponent for small data solutions to this problem. All these considerations lead to the conjecture p 2 (n) = p 0 (n + 2) for n ≥ 2, where p 0 (n) is the Strauss exponent for the classical wave equation.

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

“…Remark In the paper the number was clarified as follows: (1), (2) (3) if …”

“…If the coefficient μ 1 of the damping term is small, that is, smaller than for large dimensions, it is expected that the critical exponent has somehow a relation to the Strauss exponent. Indeed, it is proved in that the critical exponent for is a shift of Strauss exponent to .…”

The threshold of effective damping for semilinear wave equations

“…In order to prove Theorem 6, we follow the approach in [4,11]. For the sake of brevity, we only sketch the main ideas, highlighting the differences due to the presence of the propagation speed λ(t).…”

“…In order to manage the last two terms we use a Gagliardo-Nirenberg type inequality (see Lemma 2.3 in [11] and Lemma 9 in [4]) and we get…”

“…The same exponent was first obtained in [20] by using a modified potential well technique. It has been recently proved [4] that the exponent 1 + 2/n remains critical if we consider the wave equation with a time-dependent effective damping b(t)u t satisfying suitable assumptions. We say that the damping term is effective for the wave equation if the linear estimates have the same decay rate of the corresponding heat equation b(t)u t − △u = 0 (see [26,28,29,30]).…”