Integral representation formulae for the solution of a wave equation with time‐dependent damping and mass in the scale‐invariant case

Abstract

This paper is devoted to deriving integral representation formulae for the solution of an inhomogeneous linear wave equation with time‐dependent damping and mass terms that are scale invariant with respect to the so‐called hyperbolic scaling. Yagdjian's integral transform approach is employed for this purpose. The main step in our argument consists in determining the kernel functions for the different integral terms, which are related to the source term and to initial data. We will start with the one‐dimensional case (in space). We point out that we may not apply in a straightforward way Duhamel's principle to deal with the source term since the coefficients of lower order terms make our model not invariant by time translation. On the contrary, we shall begin with the representation formula for the inhomogeneous equation with vanishing data by using a revised Duhamel's principle. Then, we will derive the representation of the solution in the homogeneous case with nontrivial data. After deriving the formula in the one‐dimensional case, the classical approach by spherical means is used in order to deal with the odd‐dimensional case. Finally, using the method of descent, the representation formula in the even‐dimensional case is proved.

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

“…Let us illustrate our strategy in the proof of Theorems 2.1 and 2.2: our approach in the proof of the blow-up results is based on the work [64] for the classical wave equation with nonlinearity of derivative type; therefore, as main tool we need to employ an integral representation formula for the linear and onedimensional problem associated to (1), which generalize d'Alembert's formula in the case of the free wave equation. This formula has been proved really recently in [38]. Applying such formula, we end up with a nonlinear ordinary integral inequality (OII) for the single equation (1) and a system of OIIs for the weakly coupled system (5), respectively.…”

“…= 2 −((4+ µ 2 2 )p+1) CK p (pq − 1). Repeating the same argument as in Subsection 4.2 (application of the logarithmic function and iterative use of the resulting inequality), we find that log D j ≥ (pq) j log( Dε), (38) for j ≥ j 34), ( 37) and ( 38) we get…”

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

“…Integral representation formula. In the series of papers [22,23,27,28,24,25,13,26], several integral representation formulae for solutions to Cauchy problems associated with linear hyperbolic equations with variable coefficients have been derived and applied both to study the necessity and the sufficiency part concerning the problem of the global (in time) existence of solutions. The general scheme to determine an integral representation in the above cited literature is the following: the desired formula is obtained by considering the composition of two operators.…”

“…We point out that, even though in this section we will focus on the case n = 1, analogously to what is done in [22,23,27,28,13] it is possible to extend the integral representation even to the higher dimensional case by using the spherical means and the method of descent.…”

“…Proof. We are going to prove the representation formula in (2.3) by means of a suitable change of variables that transforms (2.1) in a linear wave equation with scale-invariant damping and mass terms and allows us to employ a result from [13].…”