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 Schrodinger equa tion) are situated on the imaginary axis. This causes a very slow deca y in the time variable of the solution driven by initial conditions of such equations. In this article we show that by displacing (by feedba ck) the zeros to the left of the imaginary axis so that they approach this axis asymptotically, one can change drastically the above situati on. Indeed, one can achieve a polynomial decay of arbitrary degree in the time variable of 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 quantitativ e way. The systems under investigation are linear undamped partial dif ferential equations with constant coefficients, in multidimensional sp ace. We provide also natural conditions for the exponential decay of t he absolute value of the solution. (C) 1996 Academic Press, Inc.