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DCDS

Let $u$ be a global in time solution of
the Cauchy-Dirichlet problem for a semilinear heat equation,
$$
\left\{
\begin{array}{ll}
\partial_t u=\Delta u+u^p,\quad &
x\in\Omega,\,\, t>0,\\
u=0,\quad & x\in\partial\Omega,\,\,t>0,\\
u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega,
\end{array}
\right.
$$
where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$,
$\Omega$ is a smooth domain in ${\bf R}^N$,
and $\phi\in L^\infty(\Omega)$.
In this paper we give a sufficient condition for the solution $u$
to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$,
and give a classification of the large time behavior of the solution $u$.

PROC

In this paper, we discuss the asymptotic
behavior of some solutions for nonlinear parabolic equation in
${\Bbb R}^N$ involving critical Sobolev exponent.
For the subcritical problem (with bounded domain), it is
well-known that the solution which intersects the "stable set"
must be a global one. But for the critical problem, it is not
known whether the same conclusion holds or not.
In this paper, we shall show that, in the critical case, the same
conclusion actually holds true. The proof requires the
concentration compactness type argument.

DCDS

The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights.
Time local and global solutions are shown to exist in the energy class.
The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.

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