arXiv subjects

Yūki Naito

Publications and source records attributed to Yūki Naito.

3 recordsLinked to original sources

arXiv2026

Existence of infinitely many singular self-similar solutions for the semilinear heat equation

We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation $\partial_t u = Δu + u^p$ for $x \in \R^N$, $t > 0$, where $N \geq 3$ and $N/(N-2) \leq p \leq (N+2)/(N-2)$. We also investigate the asymptotic behavior and the bound of the profiles of the solutions as $|x| \to 0$ and $|x| \to \infty$, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with $p = N/(N-2)$ by using singular self-similar solutions.

math.AP↗

arXiv2026

Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions

We study the simplest parabolic-elliptic model of chemotaxis in ${\bf R^N}$, and show the optimal criteria for the existence of infinite time blow-up of solutions with the initial data below the Chandrasekhar singular solution when $N \geq 10$. In particular, we exhibit the differences of the criteria between the cases $N \geq 11$ and $N = 10$. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution and the asymptotic expansions of solutions to the Liouville equation.

math.AP↗

arXiv2026

Classification of the structures of stable radial solutions for semilinear elliptic equations in $\bf R^N$

We study the stability of radial solutions of the semilinear elliptic equation $Δu +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with respect to the stability of radial solutions, and establish criteria for the existence and nonexistence of stable radial solutions in terms of the limits of $f'(u)F(u)$ as $u \to 0$ or $\infty$, where $F(u) = \int^{\infty}_u 1/f(t)dt$. Furthermore, we show the relation between the existence of singular stable solutions and the solution structure.

math.AP↗