Energy decay for a degenerate hyperbolic equation with a dissipative term
The goal of the paper is to study decay properties of the total energy of solutions to hyperbolic equations. In opposite to other papers which prefer to study wave equations with mass and dissipation, the author is interested in the Cauchy problem \[ u_{tt}- a(t)\Delta u+ 2b(t) u_t+ m^2u= 0\quad\text{in }\mathbb{R}^{n+1}_+, \] \[ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x)\quad\text{on }\mathbb{R}^n, \] in two situations: 1) the equation is strictly hyperbolic, \(a(t)\geq c>0\), it is enough to study the behaviour of the Sobolev-type energy; 2) the equation is weakly hyperbolic, \(a(t)\geq 0\), one cannot expect the existence of Sobolev solutions, the regularity \(a(t){\i}C^1[0,\infty)\) guarantees the existence of Gevrey solutions in Gevrey spaces \(s< 3/2\), therefore it is necessary to study the infinite-order energy. The main results of the paper yield an exponential decay of both energies. The method is based on an approach, which was introduced by Zuazua. The main problem is to estimate the partial energies and to understand the influence of oscillations of the leading coefficient \(a= a(t)\). It would be interesting to compare the results with results, which are obtained by an application of the method of representation of the solution by Fourier multipliers.
scientific article; zbMATH DE number 1439855