We study the stabilization of solutions of the Cauchy problem for
second-order parabolic equations depending on the behavior of the
lower-order coefficients of equations at the infinity and on the growth
rate of initial functions. We also consider the stabilization of
solution of the first boundary-value problem for a parabolic
equation without lower-order coefficients depending on the
domain $Q$ where the initial function is defined for $t=0.$ In the first chapter, we study sufficient conditions for uniform
in $x$ on a compact $K\subset\mathbb{R}^N$ stabilization to zero of the
solution of the Cauchy problem with divergent elliptic operator
and coefficients independent of $t$ and depending only on $x.$ We
consider classes of initial functions:
bounded in $\mathbb{R}^N$,
with power growth rate at the infinity in $\mathbb{R}^N$,
with exponential order at the infinity.
\noindent Using examples, we show that sufficient conditions are
sharp and, moreover, do not allow the uniform in $\mathbb{R}^N$
stabilization to zero of the solution of the Cauchy problem.
In the second chapter, we study the Cauchy problem with elliptic
nondivergent operator and coefficients depending on $x$ and $t.$
In different classes of growing initial functions we obtain exact
sufficient conditions for stabilization of solutions of the
corresponding Cauchy problem uniformly in $x$ on any compact $K$
in $\mathbb{R}^N$. We consider examples proving the sharpness of these
conditions.
In the third chapter, for the solution of the first boundary-value
problem without lower-order terms, we obtain necessary and
sufficient conditions of uniform in $x$ on any compact in $Q$
stabilization to zero in terms of the domain $\mathbb{R}^N \setminus Q$
where $Q$ is the definitional domain of the initial function for
$t=0.$ We establish the power estimate for the rate of
stabilization of the solution of the boundary-value problem with
bounded initial function in the case where $\mathbb{R}^N \setminus Q$ is a
cone for $t=0$.
