The following Cauchy problem for parabolic equations is considered in the half-space D ¯ = ℝ N × 0 ∞ $$ overline{D}={mathrm{mathbb{R}}}^Nimes left[0,infty ight) $$ , N ≥ 3: L 1 u ≡ Lu + c x t u − u t = 0 , x t ∈ D , u x 0 = u 0 x , x ∈ ℝ N . $$ {L}_1uequiv Lu+cleft(x,tight)u-{u}_t=0,kern0.5em left(x,tight)in D,kern0.5em uleft(x,0ight)={u}_0(x),kern0.5em xin {mathrm{mathbb{R}}}^N. $$ It is proved that for any bounded and continuous in ℝ_( N )initial function u ~(0)( x ) , the solution of the above Cauchy problem stabilizes to zero uniformly with respect to x from any compact set K in ℝ_( N )either exponentially or as a power (depending on the estimate for the coefficient c ( x, t ) of the equation).
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机译：在半空间D≥N×0∞$$ overline {D} = { mathrm { mathbb {R}}} ^ N times中，考虑了抛物线方程的下列柯西问题 left [0， infty right）$$，N≥3：L 1 u≡Lu + cxtu − ut = 0，xt∈D，ux 0 = u 0 x，x∈N。 $$ {L} _1u equiv Lu + c left（x，t right）u- {u} _t = 0， kern0.5em left（x，t right）在D中， kern0.5em u left（x，0 right）= {u} _0（x）， kern0.5em x in { mathrm { mathbb {R}} } ^ N。 $$证明了对于ℝ_（N）初始函数u〜（0）（x）中的任何有界和连续性，上述柯西问题的解相对于x从ℝ_（（ N）呈指数形式或作为幂（取决于方程的系数c（x，t）的估计值）。
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