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DCDS

For any nonnegative Radon measure $\mu$,
we prove the existence of solutions
for the Cauchy problem:

$ u_t =\Delta\phi(u)\qquad\text{in}\quad R^N\times(0,T);\qquad u(\cdot,0) =\mu(\cdot)\ge 0\quad \text{in}\quad R^N, $

where $\phi'(s)$ ~ $\log^m s$, $m<-1$, as $s\to\infty$. On the other hand, for the case $m\ge -1$, we give a sufficient condition for the solvability of the Cauchy problem.

CPAA

We consider the Cauchy problem of the semilinear heat equation,

$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$

$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$

where $N \geq 1$, $f \in C^1([0,\infty))$, and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$. We study the asymptotic behavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$, by using the relative entropy methods.

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 $u$.

DCDS

We establish the local existence and the uniqueness of solutions
of the heat equation with a nonlinear boundary condition
for the initial data in uniformly local $L^r$ spaces.
Furthermore, we study the sharp lower estimates of the blow-up time
of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$
and the lower blow-up estimates of the solutions.

DCDS-S

We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity
and the notion of parabolic quasi-cocavity introduced in [18]. Then we investigate the $\alpha$-parabolic quasi-concavity of solutions to parabolic problems with vanishing initial datum. The results here obtained are generalizations of some of the results of [18].

DCDS-S

We consider the Cauchy problem of the two dimensional heat equation with a radially symmetric,
negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$,
for some $\kappa > 2$. We study the rate and the direction for hot spots to tend to the spatial infinity.
Furthermore we give a sufficient condition for hot spots to consist of only one point
for any sufficiently large $t>0$.

CPAA

We study the large time behavior of positive solutions for
the Laplace equation on the half-space with a nonlinear dynamical boundary condition.
We show the convergence to the Poisson kernel in a suitable sense provided initial data
are sufficiently small.

DCDS-S

We study
the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel
\begin{eqnarray*}
& & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\
& & \qquad\quad
+\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds,
\end{eqnarray*}
where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$.
The arguments of this paper are applicable to
the Cauchy problem for various nonlinear parabolic equations
such as fractional semilinear parabolic equations, higher order semilinear parabolic equations
and viscous Hamilton-Jacobi equations.

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