Time decay rates for undamped constant coefficients linear partial differential equations (Q1910868)

The Cauchy problem for linear partial differential operators of order \(m\) of the type \[ T(\tau, D)= \sum^p_{j= 0} \tau^{p- j} R_j(D), \] \(\tau= \partial/\partial t\), \(D= (\partial/\partial x_1,\dots, \partial/\partial x_n)\), \(R_j\) being polynomials of degree \(d\leq m- p+ j\), is analyzed. The zeros of the characteristic polynomial \(T(s, i\xi)\) are assumed to be in the complex open left half plane, but not necessarily bounded away from the imaginary axis. It is shown that the solution \(y\) tends to zero uniformly in space, as \(t\to \infty\), and decay rates are described. Smoothness of the initial data \(\tau^j y(0, x)\), \(j= 0,\dots, p- 1\), imply fast decay, in general polynomial. For the proof the Fourier transform and explicit representation formulae for the solution are exploited. An example is presented which also illustrates the relation to feedback stabilization.