On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Terms

Abstract

For the Cauchy problem

$$ {\displaystyle \begin{array}{c}{L}_1u\equiv Lu+\left(b,\nabla u\right)+ cu-{u}_t=0,\kern0.5em \left(x,t\right)\in D,\\ {}u\left(x,0\right)={u}_0(x),\kern1em x\in {\mathbb{R}}^N,\end{array}} $$

for a nondivergent parabolic equation with a growing lower-order term in the half-space $\overline{D}={\mathbb{R}}^N\times \left[0,\left.\infty \right)\right.$, N ≥ 3, we prove sufficient conditions guaranteeing the exponential rate of the (uniform with respect to x on each compact domain K of ℝ^N) stabilization of the solution as t → +∞ under the assumption that the initial function u0(x) is bounded and continuous in ℝ^N