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On the Cauchy problem for an equation describing time-dependent waves in a medium with relaxation. (English. Russian original) Zbl 0685.35061
Sov. Math., Dokl. 39, No. 1, 145-148 (1989); translation from Dokl. Akad. Nauk SSSR 304, No. 4, 793-795 (1989).
In dimensionless variables the propagation of linear acoustic waves in a homogeneous medium with relaxation is described by \[ (u_{tt}-\Delta u)_ t+u_{tt}-\alpha \Delta u=0\quad (x\in {\mathbb{R}}^ 3,t>0),\quad u(x,0)=u_ 0(x),\quad u_ t(x,0)=u(x),u_{tt}(x,0)=u_ 2(x). \] In this note the existence and uniqueness of a classical solution are proved, and an integral representation and approximation for it are established.
Reviewer: N.Kazarinoff
MSC:
35L30 Initial value problems for higher-order hyperbolic equations
76Q05 Hydro- and aero-acoustics
35C15 Integral representations of solutions to PDEs
35A35 Theoretical approximation in context of PDEs
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