Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Abstract

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied. Keywords Blow-up, Glassey exponent, Nonlinearity of derivative type, Time-dependent and scale-invariant lower order terms, Integral representation formula, Upper bound estimates of the lifespan AMS Classification (2010) Primary: 35B44, 35L71; Secondary: 35B33, 35C15

A global existence result for a semilinear scale‐invariant wave equation in even dimension

“…Similarly, combining Theorem with the blow‐up result stated in Theorem 2.6 from Nunes do Nascimento et al and the global existence results from Palmieri, we have that p crit ( n , μ ) is critical exponent for the model assuming for n = 1 and for n ≥ 3 in the radial symmetric case with μ ≤ M ( n ). The two‐dimensional case is still open, even though from the necessity part, we expect to be critical.…”

“…Let us define the quantity δ ≐( μ − 1) 2 − 4 ν 2 , which describes the interplay between the damping term and the mass term . For further considerations on how the quantity δ describes the interplay between the damping term and the mass term one can see …”

“…Supposing the validity of in Nunes do Nascimento has been proved a blow‐up result for provided that data are nonnegative and compactly supported (see also Theorem 2.6 in Nunes do Nascimento). Very recently, in Palmieri, the exponent p crit ( n , μ ) is shown to be critical in the odd dimensional case n ≥ 1. In particular, following the approach of Kubo and Kubo and Kubota, in the odd dimensional case n ≥ 3 the radial symmetric case is considered, but an upper bound for μ has to be required.…”

“…Since in even dimension Huygens' principle is no longer valid, it is clear that something has to be modified with respect to the approach in Palmieri, in order to study the even dimensional case. Purpose of this work is to study the sufficiency part for under the assumption when n ≥ 4 is even.…”

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

“…On the other hand it is possible (see ) to prove a global (in time) existence result for in Sobolev spaces with exponential weights for such that and Then, using Theorem , we can enlarge the range of admissible exponents p for which we have global existence results in the case if we assume stronger conditions on δ.…”