INTRODUCTION

In this chapter, we study the wave equation in higher (space) dimensions andanalyze different problems associated with it: the Cauchy problem (initialvalue problem [IVP]), initial-boundary value problem in half-space, and soon. As noted in the introductory chapter, the wave equation arises in manyphysical contexts and it is a fundamental equation that has influenced theanalysis of solutions of general hyperbolic equations and systems. Unlikethe heat equation, the nature of solution of the wave equation depends onbeing the (space) dimension odd or even, except for one-dimensional case.This is also the reason to take up separately the study of the wave equationin higher dimensions. We also learn that the solution of the wave equationin even dimensions may be obtained from the solution in odd dimensions, bythe method of descent. We begin with the following Cauchyproblem for the homogeneous wave equation in the free spaceℝn:

Here n ≥ 2 is an integer, the (spatial)dimension, c > 0 is a constant, thespeed of propagation and u0;u1 are given smooth functions, the initial values. We describetwo methods to find a formula for the solution of (10.1) and (10.2).

The general references for this chapter are Ladyzhenskaya (1985), Rauch(1992), Mitrea (2013), Pinchover and Rubinstein (2005), McOwen (2005),Trèves (2006), Courant and Hilbert (1989), John (1971, 1975, 1978),DiBenedetto (2010), Renardy and Rogers (2004), Prasad and Ravindran (1996),Salsa (2008), Mikhailov (1978), Benzoni-Gavage and Serre (2007), Evans(1998), Kreiss and Lorenz (2004), and Vladimirov (1979, 1984).

To get an idea how this method works, we first consider a special case.Suppose the functions u0 andu1 are radial functions,that is,

THREE-DIMENSIONAL WAVE EQUATION: METHOD OF SPHERICAL MEANS

To get an idea how this method works, we first consider a special case.Suppose the functions u0 and u1 are radial functions,that is,

where. Also, extend u0 andu1 for by definingu0(−r =u0(r) andu1(−r) =u1(r). In this situation,we can expect a solution u of (10.1) also to be radial inx, that is u(x, t) =u(r, t). For such a functionu, we have

Therefore, (10.1) reduces to

Consider the case n = 3. Then, the functionv = ru satisfies the one-dimensionalequation

with initial conditions and, as follows from (10.3).