Jin Takahashi

Jin Takahashi
Tokyo Institute of Technology | TITech

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27
Publications
632
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102
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Introduction
Skills and Expertise

Publications

Publications (27)
Preprint
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem...
Preprint
We study the solvability of the initial value problem for the semilinear heat equation $u_t-\Delta u=u^p$ in a Riemannian manifold $M$ with a nonnegative Radon measure $\mu$ on $M$ as initial data. We give sharp conditions on the local-in-time solvability of the problem for complete and connected $M$ with positive injectivity radius and bounded sec...
Preprint
We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range $p>p_S$...
Article
Full-text available
We consider the semilinear heat equation ∂tu-Δu=uN/(N-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t u -\Delta u=u^{N/(N-2)}$$\end{document} in Ω\documen...
Preprint
The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, these solutions either solve the original equation in the distributional sense, or they are not locally integra...
Preprint
Full-text available
We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.
Article
We consider the Cauchy problem for the Hardy parabolic equation ∂tu−Δu=|x|−γup with initial data u0 singular at some point z. Our main results show that, if z≠0, then the optimal strength of the admissible singularity of u0 at z for the solvability of the equation is the same as that of the Fujita equation ∂tu−Δu=up. Moreover, if z=0, then the opti...
Article
Full-text available
Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexist...
Preprint
Full-text available
We construct solutions of the fast diffusion equation, which exist for all $t\in\mathbb{R}$ and are singular on the set $\Gamma(t):= \{ \xi(s) ; -\infty <s \leq ct \}$, $c>0$, where $\xi\in C^3(\mathbb{R};\mathbb{R}^n)$, $n\geq 2$. We also give a precise description of the behavior of the solutions near $\Gamma(t)$.
Preprint
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-\Delta u=|x|^{-\gamma}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t...
Article
We consider nonnegative solutions of a semilinear heat equation ut−Δu=up in RN (N≥3) with p=N/(N−2) and a nonnegative initial data u0∈LN/(N−2)(RN) which has a singularity at ξ0∈RN. We prove that there exists u0 such that, for any ξ∈Cα([0,∞);RN) with α∈(1/2,1] and ξ(0)=ξ0, the problem admits a nonnegative solution uξ∈C([0,Tξ];LN/(N−2)(RN)) for some...
Preprint
Full-text available
Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutio...
Article
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
Article
We construct solutions with prescribed moving singularities for equations of porous medium type in two space dimensions. This complements a previous study of the problem where only dimensions higher than two were considered.
Preprint
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
Article
In this paper we consider the fast diffusion equation ∂tu=Δ(um) (x∈Ω, t>0) with a nonlinear boundary condition ∂νum=up (x∈∂Ω, t>0), where 0<m<1, p>0, Ω⊂RN is a smooth domain and N≥1. We prove that p0=(m+1)/2 is the critical global existence exponent for the cases Ω=RN∖B1‾ (N≥2) and Ω=B1:={x∈RN:|x|<1} (N≥1).
Article
We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions.
Article
Full-text available
This paper is concerned with the existence of singular solutions of a superlinear parabolic equation. It is shown that under some growth conditions on the nonlinearity, there exists a solution whose singularity forms a one or higher-dimensional time-dependent set. Such solutions are constructed by modifying singular solutions of the linear heat equ...
Article
We consider the heat equation with a superlinear absorption term ∂tu−Δu=−up in Rⁿ and study the existence of nonnegative solutions with an m-dimensional time-dependent singular set, where n−m≥3. We prove that if 1<p<(n−m)/(n−m−2), then there are two types of singular solutions. Moreover, we show the uniqueness of the solutions and specify the exact...
Preprint
Full-text available
We consider the heat equation with a superlinear absorption term $\partial_{t} u-\Delta u= -u^{p}$ in $\mathbb{R}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set, where $n-m\geq 3$. First, we prove that if $p\geq (n-m)/(n-m-2)$, then there is no singular solution. We next prov...
Article
We study the existence of solutions of the semilinear parabolic equation ∂tu−Δu=up+Λ in D′(RN×I). Here N≥2, p>1, (N−2)p<N and I⊂R is an open interval. It is assumed that Λ∈D′(RN×I) is supported on {(ξ(t),t)∈RN+1;t∈I} for some ξ∈C1/2(I‾;RN) and is given by Λ(φ)=∫Iφ(ξ(t),t)dμ(t), where μ is a Radon measure on I satisfying μ((a,b))≤ϕ(b−a) (a,b∈I) for...
Chapter
We study a sharp condition for the solvability of the Cauchy problem \(u_t-\varDelta u=u^p\), \(u(\cdot ,0)=\mu \), where \(N\ge 1\), \(p\ge (N+2)/N\) and \(\mu \) is a Radon measure on \(\mathbf {R}^N\). Our results show that the problem does not admit any local nonnegative solutions for some \(\mu \) satisfying \(\mu (\{y\in \mathbf {R}^N; |x-y|<...
Article
Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is ∂tu-δu=up, x∈RN(set minus)(ξ(t)), t∈I. Here N≥2, p>1, I⊂R is an open interval and ξ∈Cα(I;RN) with α>1/2. For this equation it is shown that every nonnegative solution u satisfies ∂tu-δu=up+Λ in D'(RN×I) for some measure Λ whose support is contained in...
Article
This paper concerns solutions with time-dependent singularities for a semilinear parabolic equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on time. It is shown that if the power of the nonlinearity is in some range, then any singularity...
Article
Let N >= 2, T is an element of(0, infinity] and xi is an element of C(0, T;R-N). Under some regularity condition for xi it is known that the heat equation u(t) - Delta u = 0, x is an element of R-N\{xi(t)}, t is an element of (0, T) has a solution behaving like the fundamental solution of the Laplace equation as x -> xi(t) for any fixed t. In this...
Article
We consider solutions of the linear heat equation with time-dependent singularities. It is shown that if a singularity is weaker than the order of the fundamental solution of the Laplace equation, then it is removable. We also consider the removability of higher dimensional singular sets. An example of a non-removable singularity is given, which im...

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