Vol 66, No 1 (2020): Partial Differential 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 ⊂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:

1. bounded in R^N,
2. with power growth rate at the infinity in R^N,
3. with exponential order at the infinity.

Using examples, we show that sufficient conditions are sharp and, moreover, do not allow the uniform in 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 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 R^N \ 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 R^N \ Q is a cone for t =0.