Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

The zeros of the characteristic polynomial of many important equations in mathematical physics (e.g. the wave equation, the Schrödinger equation) are situated on the imaginary axis. This causes a very slow decay in the time variable of the solution driven by initial conditions of such equations. In this article we show that by displacing (by feedback) the zeros to the left of the imaginary axis so that they approach this axis asymptotically, one can change drastically the above situation. Indeed, one can achieve apolynomial decay of arbitrary degree in the time variableof the absolute value of the solution,uniformly in the space variable, provided the initial conditions are smooth enough. For such equations “smoothness in space implies decay in time.” The relation between smoothness and decay is established in a quantitative way. The systems under investigation are linear undamped partial differential equations with constant coefficients, in multidimensional space. We provide also natural conditions for the exponential decay of the absolute value of the solution.