The energy decay problem for hyperbolic equations of second order with dissipative terms
Let \[ L_ A[u]=\{D^ 2+(A(x)/t)D+B(x)-\sum^{n}_{i,j=1}\partial_ ia_{ij}(x)\partial_ j\}u \] and \((L_ A+f)[u]=L_ A[u]+f(u).\) A bounded or unbounded domain of \(R^ n\) is denoted by \(\Omega\), \(\partial \Omega\) is its boundary. The total energy of u is defined \[ E(t;A)=\int_{\Omega}\{| Du|^ 2+\sum a_{ij}(x)\partial_ iu\partial_ ju+B(x)u^ 2+2F(u)\}dx+\int_{\partial \Omega}\Gamma (u)dS, \] where \(\Gamma\) is a certain quadratic form derived from the boundary condition by integration by part. Here the author investigates the energy decay properties of initial- or boundary-value problems for hyperbolic partial differential equations of second order with dissipative term A(x)/t: \((L_ A+f)[u]=0.\) Then he obtains Theorem 1. Under the assumptions A(x) be non-negative with additional assumptions, for any \(t\geq T\) \(E(t;A)\leq C/t^{\mu}\) holds where \(\alpha =\inf A(x)\) and \(\mu =Min\{2,\alpha \}.\) Theorem 2. Under some suitable assumptions and inf B(x)\(>0\), then \(E(t;A)=C/t^{\alpha}\) for \(t\geq T\).
scientific article; zbMATH DE number 4071483