On large-time behavior of solutions of parabolic equations

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 $t=0$.