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Jin Takahashi

Publications and source records attributed to Jin Takahashi.

14 recordsLinked to original sources
arXiv2026

Infinite time blow-up and slow decay for the six dimensional energy-critical heat equation with self-similarly decaying initial data

We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions that decay strictly more slowly than the self-similar rate. Moreover, the blow-up and decay rates are not uniquely determined by the decay rate of the initial data, but exhibit a certain flexibility depending on the construction. The proof is based on gluing suitably rescaled bubbles to forward self-similar solutions.

math.AP↗
arXiv2025

Blow-up rate for the subcritical semilinear heat equation in non-convex domains

We consider the semilinear heat equation $u_t=Δu+|u|^{p-1} u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range $(n-2)p<n+2$. This resolves a long-standing open question dating back to the 1980s and also deduces the blow-up of the scaling critical norm.

math.AP↗
arXiv2024

Critical norm blow-up rates for the energy supercritical nonlinear heat equation

We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded $L^{n(p-1)/2,\infty}(\mathbf{R}^n)$-norm up to the blow-up time. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative $\varepsilon$-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

math.AP↗
arXiv2024

Moving gradient singularity for the evolutionary $p$-Laplace equation

We consider the evolutionary $p$-Laplace equation in $\mathbb{R}^n$. For $p>n$, we construct a solution $u$ with a moving gradient singularity in the sense that $|\nabla u(x,t)|\to \infty$ for each $t$ as $x\toξ(t)$, where $ξ:[0,\infty)\to\mathbb{R}^n$ is a given curve.

math.AP↗
arXiv2023

Critical norm blow-up for the energy supercritical nonlinear heat equation

We address the critical norm blow-up problem for the nonlinear heat equation $u_t-Δu=|u|^{p-1}u$ in $\mathbf{R}^n\times(0,T)$. In the supercritical range $p>(n+2)/(n-2)$, we prove that if the maximal existence time $T$ is finite, then $\lim_{t\to T}\|u(\cdot,t)\|_{L^{n(p-1)/2}(\mathbf{R}^n)} =\infty$ without assuming extra conditions such as radial symmetry or the type of blow-up.

math.AP↗
arXiv2023

Blow-up of the critical norm for a supercritical semilinear heat equation

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ 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 $p>p_S$ is optimal in view of the existence of type II blow-up solutions with bounded critical norm for $p=p_S$.

math.AP↗
arXiv2022

Initial traces and solvability for a semilinear heat equation on a half space of ${\mathbb R}^N$

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. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.

math.AP↗
arXiv2022

Solvability of a semilinear heat equation on Riemannian manifolds

We study the solvability of the initial value problem for the semilinear heat equation $u_t-Δu=u^p$ in a Riemannian manifold $M$ with a nonnegative Radon measure $μ$ 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 sectional curvature.

math.AP↗
arXiv2022

Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

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, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.

math.AP↗
arXiv2021

Infinite-time incompleteness of noncompact Yamabe flow

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.

math.DG↗
arXiv2021

Optimal singularities of initial data for solvability of the Hardy parabolic equation

We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ 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 $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.

math.AP↗
arXiv2017

Solutions with time-dependent singular sets for the heat equation with absorption

We consider the heat equation with a superlinear absorption term $\partial_{t} u-Δu= -u^{p}$ 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 prove that, if $1<p<(n-m)/(n-m-2)$, then there are two types of singular solution. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.

math.AP↗
arXiv2013

Removability of time-dependent singularities in the heat equation

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 implies the optimality of the condition for removability.

math.AP↗