DCDS
On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data
Kazuhiro Ishige
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.

keywords: Cauchy problem porous medium equation. Radon measure
CPAA
Asymptotic behavior of solutions for some semilinear heat equations in $R^N$
Kazuhiro Ishige Tatsuki Kawakami
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.

keywords: Cauchy problem semilinear heat equation Asymptotic behavior relative entropy methods.
DCDS
Global solutions for a semilinear heat equation in the exterior domain of a compact set
Kazuhiro Ishige Michinori Ishiwata
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 $u$.
keywords: global solutions. exterior domain Semilinear heat equation
DCDS
Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces
Kazuhiro Ishige Ryuichi Sato
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.
keywords: heat equation blow-up time blow-up rate. Nonlinear boundary condition uniformly local $L^{r}$ spaces
DCDS-S
On a new kind of convexity for solutions of parabolic problems
Kazuhiro Ishige Paolo Salani
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].
keywords: Parabolic equations convexity.
DCDS-S
Hot spots for the two dimensional heat equation with a rapidly decaying negative potential
Kazuhiro Ishige Y. Kabeya
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 $t>0$.
keywords: heat equation large time behavior. Hot spots
DCDS
Large time behavior of solutions of the heat equation with inverse square potential
Kazuhiro Ishige Asato Mukai

Let $L: = -Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N≥ 2$ and $V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}\varphi$, where $\varphi∈ L^2({\bf R}^N, e^{|x|^2/4}\, dx)$.

keywords: Schrödinger operator inverse square potential large time behavior
CPAA
Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition
Marek Fila Kazuhiro Ishige Tatsuki Kawakami
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.
keywords: dynamical boundary conditions Poisson kernel. Laplace equation
DCDS-S
Global solutions for a nonlinear integral equation with a generalized heat kernel
Kazuhiro Ishige Tatsuki Kawakami Kanako Kobayashi
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.
keywords: generalized heat kernel Global solutions nonlinear integral equation. weak $L^r$ space semilinear parabolic equations

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