arXiv subjects

Kenta Kumagai

Publications and source records attributed to Kenta Kumagai.

9 recordsLinked to original sources

arXiv2026

Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension

This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the large-time behavior of a solution changes at the critical dimension $N=10$, and the grow-up phenomenon occurs for the case $N\ge 10$. For the inhomogeneous case with $N\ge 10$, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once $f$ exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case $N\ge 11$ studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case $f=0$, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].

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arXiv2026

Large-time behavior and grow-up rates of inhomogeneous semilinear heat equations

We consider the semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the bifurcation structure of the stationary problem undergoes a qualitative change at the critical dimension $N=10$. This change affects the large-time behavior of solutions to the heat equation, and in particular, the grow-up phenomenon occurs for $N\ge 10$. In this paper, we show that once $f$ exceeds a threshold, the bifurcation structure changes to a type that does not appear in the case $f=0$. The change in the bifurcation structure leads to the disappearance of the grow-up phenomenon beyond the threshold. Moreover, we provide a quantitative characterization of this transition by determining the sharp grow-up rates for $N\ge 11$. In particular, we identify a new dimension-specific phenomenon in the threshold case: a log-log type correction term emerges in the grow-up rate only for $N=11$.

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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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arXiv2025

Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions

In this paper, we investigate semilinear elliptic equations with general exponential-type nonlinearities in two dimensions. For such nonlinearities, we establish two main results. The first is the construction of a singular solution. Recently, Fujishima, Ioku, Ruf, and Terraneo [10] proved the existence of singular solutions under certain assumptions for nonlinearities. We succeed in relaxing these conditions by providing the precise asymptotic form of a singular solution. Our second result concerns the bifurcation diagram of regular solutions. While the bifurcation structure has been extensively studied in three or higher dimensions, comparatively little was known in two dimensions until recently. In [18], the second author proved that the bifurcation curve possesses infinitely many turning points for supercritical analytic nonlinearities. In the present work, we refine this analysis by showing the bifurcation curve oscillates infinitely many times around some point, without assuming analyticity of the nonlinearities. The novelty of our approach lies in the introduction of a generalized Emden-type transformation.

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arXiv2025

Classification of bifurcation structure for semilinear elliptic equations in a ball

We consider the Gelfand problem with Sobolev supercritical nonlinearities $f$ in the unit ball. In the case where $f$ is a power type nonlinearity or the exponential nonlinearity, it is well-known that the bifurcation curve has infinitely many turning points when the growth rate of $f$ is smaller than that of the specific nonlinearity (called the Joseph-Lundgren critical nonlinearity), while the bifurcation curve has no turning point when the growth rate of $f$ is greater than or equal to that of the Joseph-Lundgren critical nonlinearity. In this paper, we give a new type of nonlinearity $f$ such that the growth rate is greater than or equal to that of the Joseph-Lundgren critical nonlinearity, while the bifurcation curve has infinitely many turning points. This result shows that the bifurcation structure is not determined solely by the comparison between the growth rate of $f$ and that of the Joseph-Lundgren critical nonlinearity. In fact, we find a general criterion which determines the bifurcation structure; and give a classification of the bifurcation structure.

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arXiv2025

Uniformly boundedness of finite Morse index solutions to semilinear elliptic equations with rapidly growing nonlinearities in two dimensions

We consider the Gelfand problem with rapidly growing nonlinearities in the two-dimensional bounded strictly convex domains. In this paper, we prove the uniformly boundedness of finite Morse index solutions. As a result, we show that there exists a solution curve having infinitely many bifurcation/turning points. These results are recently proved by the present author for supercritical nonlinearities when the domain is the unit ball via an ODE argument. Instead of the ODE argument, we apply a new method focusing on the interaction between the growth condition of the nonlinearities and the shape of the fundamental solution. As a result, we clarify the bifurcation structure for general convex domains.

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arXiv2024

Bifurcation diagrams for semilinear elliptic equations with singular weights in two dimensions

We consider the bifurcation diagram of radial solutions for the Gelfand problem with a positive radially symmetric weight in the unit ball. We deal with the exponential nonlinearity and a power-type nonlinearity. When the weight is constant, it is well-known that the bifurcation curve exhibits three different types depending on the dimension and the exponent of power for higher dimensions, while the curve exhibits only one type in two dimensions. In this paper, we succeed in realizing in two dimensions a phenomenon such that the bifurcation curve exhibits all of the three types, by choosing the weight appropriately. In particular, to the best of the author's knowledge, it is the first result to establish in two dimensions the bifurcation curve having no turning points.

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arXiv2024

Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions

We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $λ$. The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $λ=0$. As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in $N$ dimensions with $3\le N \le 9$ and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $2\le N\le 9$.

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arXiv2023

Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension

We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a radially symmetric weight $0<a(|x|)\in C^2(\overline{B_1})$ in the unit ball. When the weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension $N\le 9$, and it has no turning points if $N\ge 10$. In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when $N\le 9$. Moreover, we find specific radial singular solutions with specific weights and study the Morse index of the solutions. As a consequence, we prove that the perturbation affects the bifurcation structure in the critical dimension $N=10$. Moreover, we give an optimal classification of the bifurcation diagrams in the critical dimension.

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