Time estimates for the Cauchy problem for a third-order hyperbolic equation (Q1811886)

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 \( f\neq 0\).