Absence of global solutions to a class of nonlinear parabolic inequalities
Volume 94 / 2002
Abstract
We study the absence of nonnegative global solutions to parabolic
inequalities of the type $ u_t \geq -(-{\mit \Delta })^{{\beta /2}}
u - V(x)u + h(x,t) u^{p}$, where $ (-{\mit \Delta })^{{\beta /2}}$,
$0 < \beta \leq 2 $, is the $ \beta /2 $ fractional power of the
Laplacian. We give a sufficient condition which implies that the
only global solution is trivial if $ p > 1 $ is small. Among other
properties, we derive a necessary condition for the existence of
local and global nonnegative solutions to the above problem for the
function $ V $ satisfying $ V_+(x)\sim a |
x| ^{-b}$, where $a \geq 0$, $b > 0$, $p > 1 $ and $ V_+(x):=
\max\{V(x),0\}$. We show that the existence of solutions depends on the
behavior at infinity of both initial data and $ h$.
In addition to our main results, we also discuss the nonexistence
of solutions for some degenerate parabolic inequalities like $ u_t \geq
{\mit \Delta } u^m + u^p $ and $ u_t \geq {\mit \Delta }_p u + h(x,t)u^p$.
The approach is based upon a duality argument combined with an appropriate
choice of a test function. First we obtain an a priori estimate and then we
use a scaling argument to prove our nonexistence results.
