In a cylindric domain D = (0,∞) ×Ω, where Ω is contained in R_(n+1) is an unbounded domain, the first mixed problem for a high-order parabolic equation ut+(-1)~kD_x~k(a(x,y)D_x~ku)+Σ_(i=1)~mΣ_(|α|=|β|=i)(-1)~iD_y~α(bα β(x,y)D_y~βu)=0 1 ≤m, k,l,m ∈ N, is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of the new geometrical characteristics of the domain, the upper estimate of L_2-norm ||u(t)|| of the solution to the problem is established. In particular, in domains {(x,y) ∈ R_(n+1) | x > 0, |y_1| < x~a}, 0 < a < q/l, under the assumption that the upper and lower symbols of the operator L are separated from zero, this estimate takes the form ‖u(t)‖≤M exp(-n2t~b)‖(&)‖,b=k-la/k-la+2lak. This estimate is determined by minor terms of the equation. The sharpness of the estimate for a wide class of unbounded domains is proved in the case k = l = m = 1.
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