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Time estimates for the Cauchy problem for a third-order hyperbolic equation. (English) Zbl 1022.35025

Summary: A classical solution is considered for the Cauchy problem: \((u_{tt}-\Delta u)_t+u_{tt}-\alpha\Delta u=f(x,t)\), \(x\in\mathbb{R}^3\), \(t>0\); \(u(x,0)=f_0(x)\), \(u_t(x, 0)=f_1(x)\), and \(u_{tt}(x)=f_2(x)\), \(x\in \mathbb{R}^3\), where \(\alpha = \text{const}\), \(0<\alpha <1\). The above equation governs the propagation of time-dependent acoustic waves in a relaxing medium. A classical solution of this problem is obtained in the form of convolutions of the right-hand side and the initial data with the fundamental solution of the equation. Sharp time estimates are deduced for the solution in question which show polynomial growth for small times and exponential decay for large time when \(f=0\). They also show the time evolution of the solution when \( f\neq 0\).

MSC:

35L30 Initial value problems for higher-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
76Q05 Hydro- and aero-acoustics
35C15 Integral representations of solutions to PDEs
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