arXiv subjects

Yohei Fujishima

Publications and source records attributed to Yohei Fujishima.

13 recordsLinked to original sources

arXiv2025

Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms

This paper investigates the multiplicity of singular solutions for the nonlinear elliptic equation $-\Delta u =f(u)$ near the origin. Applying the classification of nonlinear functions and the transformation, which were developed by the authors, we generalize the multiplicity results known for the concrete model nonlinearity $f(u)=u^p$ with $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. Our result applies to various nonlinearities, such as $f(s)=s^p+s^r$ with $0<r<p$, $f(s)=s^p(\log s)^r$ with $r\in \mathbb{R}$, $f(s)=s^p\exp((\log s)^r)$ with $0<r<1$ and $f(s)=s^p+s^r(\log s)^{\beta}$ with $0<r<p$ and $\beta \in \mathbb{R}$, for $\frac{N}{N-2}<p<\frac{N+2}{N-2}$.

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arXiv2025

Non-uniqueness of mild solutions for 2d-heat equations with singular initial data

In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in $\mathbb R^2$ which are also distributional solutions on the whole disc. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: ${u_s}(t,x) \equiv U(x)$ is a stationary solution, and there exists also a solution ${u_r}(t,x)$ departing from $U$ which is bounded for $t > 0$. While such non-uniqueness results have been known in higher dimensions by Ni--Sacks [33], Terraneo [40] and Galaktionov--Vazquez [16], only two very specific results have recently been obtained in two dimensions by Ioku--Ruf--Terraneo [22] and Ibrahim--Kikuchi--Nakanishi--Wei [21].

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arXiv2025

Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid Growth

We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical $L^q$-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into $L^{\infty}$. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness.

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arXiv2024

Uniform boundedness and blow-up rate of solutions in non-scale-invariant superlinear heat equations

For superlinear heat equations with the Dirichlet boundary condition, the $L^\infty$ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than $9$, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than $2$ and smaller than $10$, our results are applicable for nonlinear terms growing extremely faster than the exponential function.

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arXiv2024

Existence of solutions for a semilinear parabolic system with singular initial data

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0 1$, and $(\mu,\nu)$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.

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arXiv2024

Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions

By introducing a new classification of the growth rate of exponential functions, singular solutions for semilinear elliptic equations in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity which admits an explicit singular solution. Then, using a transformation as in [8], one obtains an approximate singular solution, and then one concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way. As a special case, our result contains a pioneering contribution by Ibrahim--Kikuchi--Nakanishi--Wei [15] for the Moser--Trudinger type nonlinearity.

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arXiv2023

Local solvability and dilation-critical singularities of supercritical fractional heat equations

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

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arXiv2022

Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0, &x\in \mathbb{R}^N, \end{cases} \end{equation} where $N\ge 1$, $u_0$ is a nonnegative initial function and $f\in C^1([0,\infty)) \cap C^2((0,\infty))$ denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem~\eqref{eq:P}. The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions $W$ of \begin{equation} \notag \Delta W + \frac{y}{2}\cdot \nabla W + f(W)F(W) + f(W) + \frac{|\nabla W|^2}{f(W)F(W)} \Bigl[ q - f'(W)F(W) \Bigr] = 0, \quad y \in \mathbb{R}^N, \end{equation} where $F(s):=\displaystyle\int_s^{\infty}\dfrac{1}{f(\eta)}d\eta$ and $q\ge 1$.

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arXiv2022

Solvability of Superlinear Fractional Parabolic Equations

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.

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arXiv2021

Initial traces and solvability of Cauchy problem to a semilinear parabolic system

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1Δu+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2Δv+u^q\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & \quad\mbox{in}\quad{\bf R}^N, \end{cases} \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0 1$ and $(μ,ν)$ is a pair of Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study qualitative properties of the initial trace of the solution $(u,v)$ and obtain necessary conditions on the initial data $(μ,ν)$ for the existence of solutions to problem (P).

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arXiv2020

Optimal singularities of initial functions for solvability of a semilinear parabolic system

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0 1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.

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arXiv2017

Critical exponent for the global existence of solutions to a nonlinear degenerate/singular parabolic equation

We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.

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arXiv2016

Existence and nonexistence of solutions for the heat equation with a superlinear source term

Classification theory on the existence and non-existence of local in time solutions for initial value problems of nonlinear heat equations are investigated. Without assuming a concrete growth rate on a nonlinear term, we reveal the threshold integrability of initial data which classify existence and nonexistence of solutions via a quasi-scaling and its invariant integral. Typical nonlinear terms, for instance polynomial type, exponential type and its sum, product and composition, can be treated as applications.

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