21,473 research outputs found
On a non-homogeneous and non-linear heat equation
We consider the Cauchy-problem for a parabolic equation of the following
type:
\begin{equation*}
\frac{\partial u}{\partial t}= \Delta u+ f(u,|x|),
\end{equation*} where is supercritical. We supply this equation
by the initial condition , and we allow to be either
bounded or unbounded in the origin but smaller than stationary singular
solutions. We discuss local existence and long time behaviour for the solutions
for a wide class of non-homogeneous non-linearities . We show
that in the supercritical case, Ground States with slow decay lie on the
threshold between blowing up initial data and the basin of attraction of the
null solution. Our results extend previous ones allowing Matukuma-type
potential and more generic dependence on .
Then, we further explore such a threshold in the subcritical case too. We
find two families of initial data and which are
respectively above and below the threshold, and have arbitrarily small distance
in norm, whose existence is new even for . Quite
surprisingly both and have fast decay (i.e. ), while the expected critical asymptotic behavior is slow decay
(i.e. ).Comment: 2 figure
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