arXiv subjects

Yasuhito Miyamoto

Publications and source records attributed to Yasuhito Miyamoto.

At least 19 recordsLinked to original sources

arXiv2026

Monotonicity of the bifurcation curve for supercritical elliptic problems in the borderline dimension $N=10$

We study the global structure of bifurcation diagrams for semilinear elliptic Dirichlet problems with supercritical nonlinearities in the unit ball. In particular, we focus on the borderline dimension $N = 10$, where the qualitative behavior of the bifurcation diagram is not determined solely by the growth rate of the nonlinearity. We show that the bifurcation curve is monotone, yielding uniqueness of classical solutions, for a class of nonlinearities including $f(u) = \exp((u+1)^p)$ with $p > 1$ and iterated exponential functions. Our approach is based on the construction of suitable singular subsolutions that satisfy a Hardy-type stability condition, avoiding the need for explicit representations of singular solutions. As a consequence, we show that, in dimension $N = 10$, these nonlinearities exhibit the same qualitative bifurcation diagram as the classical Gel'fand problem. We also characterize the monotonicity of the bifurcation curve in terms of the existence of global-in-time unbounded solutions to the associated parabolic problem.

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arXiv2026

Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}(\mathbb{R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^γ_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le γ<N(p_f-1)/2$, where $p_f:=\lim_{u\to\infty}uf'(u)/f(u)$ is a growth rate of $f$. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of $u(t)$ is based on the monotonicity argument. Transformations of forward self-similar solutions for $f(u)=u^p$ and $e^u$ play a crucial role.

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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.

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arXiv2025

The Gierer-Meinhardt system in the entire space with non-local proliferation rates

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: $$ \begin{cases} \displaystyle -Δu+λu=\frac{J*u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 1,\\[0.1in] \displaystyle -Δv+μv=\frac{J*u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N.\\[0.1in] \end{cases} $$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modelling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J*u^p, J*u^m$ where the $*$-symbol denotes the convolution operation in $\mathbb{R}^N$ with a kernel $J\in C^1(\mathbb{R}^N\setminus\{0\})$. In the system, we assume that $0<ρ\in C^{0, γ}(\mathbb{R}^N)$ with $γ\in (0,1)$, while the parameters satisfy $λ, μ, q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel $J$, we establish the existence and non-existence of classical positive solutions in the function space $C^{2, δ}_{loc}(\mathbb{R}^N).$ These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.

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arXiv2025

Threshold property of a singular stationary solution for semilinear heat equations with exponential growth

Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ such that $u^*(x)\to\infty$ as $|x|\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ and $u_0\not\equiv u^*$, while the problem has no nonnegative local-in-time solutions $u$ such that $u\ge u^*$ if $u_0\ge u^*$ and $u_0\not\equiv u^*$.

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arXiv2025

Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $Δu+K(|x|)u^p+μf(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $μ>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^α$, $α>-2$, near $r=0$, $K(r)\sim r^β$, $β>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $μ$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(α)$ and Joseph-Lundgren exponent $p_{JL}(α)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(α) -2$.

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arXiv2025

Exact solutions describing very slow layer oscillations in a shadow reaction-diffusion system

We show in a rigorous way that a stable internal single-layer stationary solution is destabilized by the Hopf bifurcation as the time constant exceeds a certain critical value. Moreover, the exact critical value and the exact period of oscillatory solutions can be obtained. The exact period indicates that the oscillation is very slow, i.e., the period is of order $O(e^{C/\varepsilon})$. We also rigorously prove that Hopf bifurcations from multi-layer stationary solutions occur. In this case anti-phase horizontal oscillations of layers are shown by formal calculations. Numerical experiments show that the exact period agrees with the numerical period of a nearly periodic solution near the Hopf bifurcation point. Anti-phase (out of phase) horizontal oscillations of layers are numerically observed.

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arXiv2021

Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_α\ast\big((I_β\ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $α, β\in (0,N)$, $p, q>0$ and $I_α$, $I_β$ denote the Riesz potentials of order $α$ and $β$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $α$, $β$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

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arXiv2021

Polyharmonic inequalities with nonlocal terms

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm Δ^m u \geq \big(Ψ(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $Δ^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $Ψ>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

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arXiv2021

Thresholds on growth of nonlinearities and singularity of initial functions for semilinear heat equations

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equation \[ \begin{cases} \partial_tu=Δu+f(u) & \textrm{in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N \end{cases} \] under mild assumptions on $f$. A relationship between a growth of $f$ and an integrability of $u_0$ is studied in detail. Our existence theorem gives a sharp integrability condition on $u_0$ in a critical and subcritical cases, and it can be applied to a regularly or rapidly varying function $f$. In a doubly critical case existence and nonexistence of a nonnegative solution can be determined by special treatment. When $f(u)=u^{1+2/N}[\log(u+e)]^β$, a complete classification of existence and nonexistence of a nonnegative solution is obtained. We also show that the same characterization as in Laister et. al. [11] is still valid in the closure of the space of bounded uniformly continuous functions in the space $L^r_{\rm ul}(\mathbb{R}^N)$. Main technical tools are a monotone iterative method, $L^p$-$L^q$ estimates, Jensen's inequality and differential inequalities.

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arXiv2020

The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

In this paper, we consider the following minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = α_1, \ \| u_2 \|_{L^2}^2 = α_2 \right\}, \] where $α_1,α_2 > 0$ and $E(u)$ is defined by \[ E(u) := \int_{\mathbf{R}^N} \left\{\frac{1}{2} \sum_{i=1}^2 \left( |\nabla u_1|^2 + V_i (x) |u_i|^2 \right) - \sum_{i=1}^2 \frac{μ_i}{2p_i+2} |u_i|^{2p_i+2} - \fracβ{p_3+1} |u_1|^{p_3+1} |u_2|^{p_3+1} \right\} \mathrm{d} x. \] Here $N \geq 1$, $ μ_1,μ_2,β> 0$ and $V_i(x)$ $(i=1,2)$ are given functions. For $V_i(x)$, we consider two cases: (i) both of $V_1$ and $V_2$ are bounded, (ii) one of $V_1$ and $V_2$ is bounded. Under some assumptions on $V_i$ and $p_j$, we discuss the compactness of any minimizing sequence.

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arXiv2020

Radial regular and rupture solutions for a MEMS model with fringing field

We investigate radial solutions for the problem \[ \begin{cases} \displaystyle -ΔU=\frac{λ+δ|\nabla U|^2}{1-U},\; U>0 & \textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset \mathbb{R}^N$ $(N\geq 2)$ denotes the open unit ball and $λ, δ>0$ are real numbers. Two classes of solutions are considered in this work: (i) {\it regular solutions}, which satisfy $0 0$ is also discussed.

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arXiv2020

Radial single point rupture solutions for a general MEMS model

We study the initial value problem $$ \begin{cases} r^{-(γ-1)}\left(r^α|u'|^{β-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0 0 & \textrm{for}\ 0 α>β\geq 1$ and $f\in C[0,\bar u)\cap C^2(0,\bar u)$, $f(0)=0$, $f(u)>0$ on $(0, \bar u)$ and $f$ satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution $u^*$ to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions $u(\,\cdot\,,a)$ (with $u(0,a)=a$) as $a\to 0$. In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.

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arXiv2020

Fractional semilinear heat equations with singular and nondecaying initial data

We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a new supersolution which plays a crucial role. Our method does not rely on a change of variables, and hence it can be applied to a wide class of nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$ or $e^u$, a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen's inequality and $L^p$-$L^q$ type estimates.

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arXiv2020

Structure of the positive radial solutions for the supercritical Neumann problem $\varepsilon^2Δu-u+u^p=0$ in a ball

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2Δu-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge 3$. We show that there exists a sequence $\{\varepsilon_n^*\}_{n=1}^{\infty}$ ($\varepsilon_1^*>\varepsilon_2^*>\cdots\rightarrow 0$) such that this problem has infinitely many singular solutions $\{(\varepsilon_n^*,U_n^*)\}_{n=1}^{\infty}\subset\mathbb{R}\times (C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist of infinitely many smooth curves in the $(\varepsilon,U(0))$-plane. It is shown that each curve blows up at $\varepsilon_n^*$ and if $p_{\rm{S}} 0$ such that the problem has no nonconstant regular solution if $\varepsilon>\bar{\varepsilon}$. The main technical tool is the intersection number between the regular and singular solutions.

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arXiv2020

Exact eigenvalues and eigenfunctinos for a one-dimensional Gel'fand problem

It is known that every positive solution of a one-dimensional Gel'fand problem can be written explicitly. In this paper we obtain exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We also study asymptotic behaviors of eigenvalues and eigenfunctions as the $L^{\infty}$-norm of the solution goes to the infinity.

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arXiv2019

A doubly critical semilinear heat equation in the $L^1$ space

We study the existence and nonexistence of a Cauchy problem of the semilinear heat equation $\partial_tu=Δu+|u|^{p-1}u$ in $\mathbb{R}^N\times(0,T)$, $u(x,0)=ϕ(x)$ in $\mathbb{R}^N$, in $L^1(\mathbb{R}^N)$. Here, $N \ge 1$, $p=1+2/N$ and $ϕ\in L^1( \mathbb{R}^N)$ is a possibly sign-changing initial function. Since $N(p-1)/2=1$, the $L^1$ space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every $ϕ\in L^1(\mathbb{R}^N)$. Let $X_q:=\{ ϕ\in L^1_{\rm{loc}}(\mathbb{R}^N)\ |\ \int_{\mathbb{R}^N}|ϕ| \left[\log (e+|ϕ|)\right]^qdx<\infty \} (\subset L^1(\mathbb{R}^N))$. In this paper we construct a local-in-time mild solution in $L^1(\mathbb{R}^N)$ for $ϕ\in X_q$ if $q\ge N/2$. We show that, for each $0\le q<N/2$, there is a nonnegative initial function $ϕ_0\in X_q$ such that the problem has no nonnegative solution, using a necessary condition given by Baras-Pierre [Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 185--212]. Since $X_q\subset X_{N/2}$ ($q\ge N/2$), $X_{N/2}$ becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.

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arXiv2019

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equation $Δu+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $Δu+u^p=0$ and $Δu+e^u=0$.

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