On the stabilization rate of solutions of the Cauchy problem for a~parabolic equation with lower-order terms

V. N. Denisov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, Tome 59 (2016), pp. 53-73

Résumé

For a parabolic equation in the half-space $\overline D=\mathbb R^N\times[0,\infty)$, $N\geqslant3$, we consider the Cauchy problem \begin{gather*} L_1u\equiv Lu+c(x,t)u-u_t=0,\quad (x,t)\in D,\\ u(x,0)=u_0(x),\quad x\in\mathbb R^N. \end{gather*} Depending on estimates on the coefficient $c(x,t),$ we establish power or exponential rate of stabilization of solutions of the Cauchy problem равномерно по $x$ на каждом компакте $K$ в $\mathbb R^N$ для произвольной ограниченной непрерывной в $\mathbb R^N$ начальной функции $u_0(x)$.