Solution representations for a wave equation with weak dissipation

摘要

We consider the Cauchy problem for the weakly dissipative wave equationsquareupsilon + mu/1+t upsilont = 0, x is an element of R-n, t greater than or equal to 0 I + tparameterized by mu > 0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain L-p-L-q, estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L-2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of mu and that mu = 2 is critical. Copyright (C) 2004 John Wiley Sons, Ltd.