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On stabilization of solutions of the Cauchy problem for a parabolic equation with lower-order coefficients
V. N. Denisov
Abstract:
In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation
$$
\Delta u+c(x,t)u-u_t=0\ \ \text{for}\ \ x\in\mathbb R^N,\ \ t>0,
$$
under which its solution satisfying the initial condition
$$
u|_{t=0}=u_0(x)\ \ \text{for}\ \ x\in \mathbb R^N,
$$
stabilizes to zero, i.e., there exists the limit
$$
\lim_{t\to\infty}{u(x,t)}=0,
$$
uniform in
$x$ from every compact set
$K$ in
$\mathbb R^N$ for any function
$u_0(x)$ belonging to a certain uniqueness class of the problem considered and growing not rapidly than
$e^{a|x|^b}$ with
$a>0$ and
$b>0$ at infinity.
UDC:
517.955
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