ON STABILIZATION OF SOLUTIONS OF THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION WITH LOWER-ORDER COEFFICIENTS

摘要

In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation △u + c(x,t)u-u_t = 0 for x ∈ R~N, t > 0, under which its solution satisfying the initial condition u|_(t=0) = u_0(x) for x ∈ R~N stabilizes to zero, i.e., there exists the limit _∞~lim u(x, t) = 0,t→ ∞ uniform in x from every compact set K in R~N for any function u_0(x) belonging to a certain uniqueness class of the problem considered and growing not rapidly than e~(a|x|~6) with a > 0 and b > 0 at infinity.