On the Stabilization Rate of Solutions of the Cauchy Problem for a Parabolic Equation with Lower-Order Terms

Abstract

The following Cauchy problem for parabolic equations is considered in the half-space $\overline{D}={\mathrm{\mathbb{R}}}^N\times \left[0,\infty \right)$, N ≥ 3:

$$ {L}_1u\equiv Lu+c\left(x,t\right)u-{u}_t=0,\kern0.5em \left(x,t\right)\in D,\kern0.5em u\left(x,0\right)={u}_0(x),\kern0.5em x\in {\mathrm{\mathbb{R}}}^N. $$

It is proved that for any bounded and continuous in ℝ^N initial function u₀(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).