Disappearing solutions for dissipative hyperbolic systems of constant multiplicity (Q1109229)

Let \[ (\partial_ t-\sum^{n}_{i=1}A_ j\partial_ j)u=0\quad on\quad (0,\infty)\times \Omega,\quad \Lambda (x)u=0\quad on\quad (0,\infty)\times \partial \Omega,\quad u(0,x)=f(x), \] where \(A_ j\), \(\Lambda\) (x) are \(r\times r\) matrices, \(\Lambda\) (X) is real analytic and \(f(x)\in L^ 2(\Omega;C^ r)\), and \(\Omega\) is an open domain in \(R^ n\) with a smooth boundary. The essential assumptions \(are:\) A\({}_ j\) are constant Hermitian matrices, the eigenvalues of \(A(\xi)=\sum A_ j\xi_ j\) have constant multiplicity for \(\xi \in R^ n.\) A solution of the above problem is called disappearing if \(f\neq 0\) and there exists \(T_ 0>0\) such that \(u(t,x)=0\) for \(t\leq T_ 0.\) The author investigates the conditions for the solution to be disappearing in connection with the boundary operator \(\Lambda\) (x) and f(x) throughout the paper in great detail.