LARGE TIME ASYMPTOTICS FOR A CLASS OF WAVE-EQUATIONS OF HIGHER-ORDER WITH A VARIABLE-COEFFICIENT AND TIME-INDEPENDENT INCITATION

We consider the equation (- 1)(m) del(m)(p del(m)u) + partial derivative(t)(2)u = f in R(n)x [0 x infinity) for arbitrary positive integers m and n and under the assumptions p - 1, f is an element of C-0(infinity) (R(n)) and p > 0. Under the additional assumption that the differential operator( - 1)(m) del(m)(p del(m)u) has no eigenvalues we derive an asymptotic expansion for u(x, t) as t --> infinity including all terms up to order o(1). In particular, we show that for 2m greater than or equal to n terms of the orders t(alpha), log t, (log t)(2) and t(beta)-log t as t --> infinity may occur.