arXiv subjects
Publications and source records attributed to Kazuhiro Ishige.
In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.
This paper is concerned with the elliptic problem for a scalar field equation with a forcing term \begin{equation} \tag{P}-Δu+u=u^p+ κμ\quad \mbox{in} \quad{\bf R}^N, \quad u>0 \quad \mbox{in} \quad {\bf R}^N, \quad u(x)\to 0\quad \mbox{as} \quad |x| \to \infty, \end{equation} where $N\ge 2$, $p>1$, $κ>0$ and $μ$ is a Radon measure in ${\bf R}^N$ with a compact support. Under a suitable integrability condition on $μ$, we give a complete classification of the solvability of problem~(P) with $1<p<p_{JL}$. Here $p_{JL}$ is the Joseph-Lundgren exponent defined by $$ p_{JL} :=\infty\quad\mbox{if}\quad N\le 10, \qquad p_{JL}:=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\quad \text{if} \quad N\ge 11. $$
We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
Consider the Cauchy problem for a nonlinear diffusion equation \begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=Δu^m+u^α& \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0)=λ+φ(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} where $m>0$, $α\in(-\infty,1)$, $λ>0$ and $φ\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N)$ with $1\le r<\infty$ and $\inf_{x\in{\bf R}^N}φ(x)>-λ$. Then the positive solution to problem (P) behaves like a positive solution to ODE $ζ'=ζ^α$ in $(0,\infty)$ and it tends to $+\infty$ as $t\to\infty$. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.
We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.
We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-Δu+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of $u$ and reveal the relationship between the behavior of the hot spots and the harmonic functions for $-Δ+V$.
Let $H$ be a norm of ${\bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Δ_H$ the Finsler-Laplace operator defined by $Δ_Hu:=\mbox{div}\,(H(\nabla u)\nabla_ξH(\nabla u))$. In this paper we prove that the Finsler-Laplace operator $Δ_H$ acts as a linear operator to $H_0$-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation $$ \partial_t u=Δ_H u,\qquad x\in{\bf R}^N,\quad t>0, $$ where $N\ge 1$ and $\partial_t:=\partial/\partial t$.
Let $L:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ 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 $e^{-tL}φ$, where $φ\in L^2({\bf R}^N,e^{|x|^2/4}\,dx)$.
We obtain necessary conditions and sufficient conditions for the solvability of the heat equation in a half-space of ${\bf R}^N$ with a nonlinear boundary condition. Furthermore, we study the relationship between the life span of the solution and the behavior of the initial function.
In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-Δ)^{\fracθ{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=φ(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<θ<2$ and $φ\in L_K:=L^1({\bf R}^N,\,(1+|x|)^K\,dx)$ with $K\ge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ \partial_t u+(-Δ)^{\fracθ{2}}u=|u|^{p-1}u\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), $$ where $0<θ<2$ and $p>1+θ/N$.
In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-Δ)^{\fracθ{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=μ\ge 0\quad\mbox{in}\quad{\bf R}^N, $$ where $N\ge 1$, $0<θ\le 2$, $p>1$ and $μ$ is a Radon measure or a measurable function in ${\bf R}^N$. Our conditions lead optimal estimates of the life span of the solution with $μ$ behaving like $λ|x|^{-A}$ ($A>0$) at the space infinity, as $λ\to +0$.
We consider the Schr{ö}dinger operator H = --$Δ$ + V (|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x, y, t) of the type 0 \textless{} p(x, y, t) $\le$ C t -- N 2 U (min{|x|, $\sqrt$ t})U (min{|y|, $\sqrt$ t}) U ($\sqrt$ t) 2 exp -- |x -- y| 2 Ct for all x, y $\in$ R N and t \textgreater{} 0, where U is a positive harmonic function of H. Third, if U 2 is an A 2 weight on R N , then we prove a lower bound of a similar type.
In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=Δu+v^p,\qquad & x\inΩ,\,\,\,t>0,\\ \partial_t v=Δv+u^q, & x\inΩ,\,\,\,t>0,\\ u=v=0, & x\in\partialΩ,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\inΩ, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $Ω$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $Ω$. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions.
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 $λψ$ as $λ\to 0$ or $λ\to\infty$ and the lower blow-up estimates of the solutions.
Let $H:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric function decaying quadratically at the space infinity. In this paper we consider the Schrödinger heat semigroup $e^{-tH}$, and make a complete table of the decay rates of the operator norms of $e^{-tH}$ in the Lorentz spaces as $t\to\infty$.
This paper is concerned with a nonlinear integral equation $$ (P)\qquad u(x,t)=\int_{{\bf R}^N}G(x-y,t)φ(y)dy+\int_0^t\int_{{\bf R}^N}G(x-y,t-s)f(y,s:u)dyds, \quad $$ where $N\ge 1$, $φ\in L^\infty({\bf R}^N)\cap L^1({\bf R}^N,(1+|x|^K)dx)$ for some $K\ge 0$. Here $G=G(x,t)$ is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of $(P)$ behaving like a multiple of the integral kernel $G$ as $t\to\infty$.
This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=Δu +f(x,t,u,\nabla u) & \mbox{in}\quadΩ\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial Ω\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quadΩ, \end{array} \right. \end{equation} where $Ω$ is a bounded convex domain in ${\bf R}^n$ and $f$ is a nonnegative continuous function in $Ω\times(0,\infty)\times{\bf R}\times{\bf R}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\barΩ\times[0,\infty)$.
Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \partial_t u=Δu+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=φ(x)\quad in \quad{\bf R}^N, $$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $φ$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations.