Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation

In a cylindrical domain D = (0, infinity) x Omega, where Omega is an unbounded subdomain of Rn+1, one considers the first mixed problem for a higher order equation [GRAPHIC] with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an tipper estimate of the L-2-norm vertical bar vertical bar u(t)vertical bar vertical bar of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains {(x, y) epsilon Rn+1 : vertical bar y(1)vertical bar < x(a)}, 0 < a < q/l, for the operator L with symbol satisfying a certain condition this estimate takes the following form: vertical bar vertical bar u(l)vertical bar vertical bar <= M exp(-kappa 2t(b))vertical bar vertical bar phi, b = q - la/ q - la + 2laq The estimate is shown to be sharp in a broad class of unbounded domains for q = k = l = m = 1, that is, for second-order parabolic equations.