Integral representation formulae for the solution of a wave equation with timedependent damping and mass in the scaleinvariant case
Abstract
This paper is devoted to derive integral representation formulae for the solution of an inhomogeneous linear wave equation with timedependent damping and mass terms, that are scaleinvariant with respect to the socalled 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.02408
 Bibcode:
 2019arXiv190502408P
 Keywords:

 Mathematics  Analysis of PDEs