We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifurcation theory, which we outline in a separate section for the readers' convenience.
We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of $\mathbb{R}^n$.
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+…
We consider the Neumann problem for the equation $u_{xx}+λf(u)=0$ in the punctured interval $(-1,1) \setminus \{0\}$, where $λ>0$ is a bifurcation parameter and $f(u)=u-u^3$. At $x=0$, we impose the conditions $u(-0)+au_x(-0)=u(+0)-au_x(+0)$ and $u_x(-0)=u_x(+0)$ for a constant $a>0$ (the symbols $+0$ and $-0$ stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval $(-1,1)$. First we prove that odd solutions and even solutions form families of branches $\{ \mathcal{C}^o_k\}_{k \in \mathb…
Large-time behavior and grow-up rates of inhomogeneous semilinear heat equations
Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）；Mathematical Institute and Graduate School of Science, Tohoku University（东北大学数学研究所及理学研究科）
We study solutions of $Δu - F'(u)=0$, where the potential $F$ can have an arbitrary number of wells at arbitrary heights, including bottomless wells with subcritical decay. In our setting, ground state solutions correspond to unstable solutions of least energy. We show that in convex domains of $\mathbb{R}^N$ and manifolds with $\operatorname{Ric}\geq 0$, ground states are always of mountain-pass type and have Morse index 1. In addition, we prove symmetry of the ground states if the domain is either an Euclidean ball or the entire sphere $S^{N}$. For the Allen-Cahn equation $\varepsilon^2Δu - …
We construct positive singular solutions for the problem $-Δu=λ\exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Differential Equations \textbf{264} (2018), 2684--2707] for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which…
We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $λ> 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_λ$ and determine the monotonicity and co…
Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence
在严格的第二类星形域上，我们找到了自然的充分条件，这些条件允许解决与下面的平均场方程$\prl$密切相关的两个长期存在的公开问题。一方面我们得到了二维湍流的Onsager描述中平均场微正则变分原理（简称MVP）熵的全局行为。这是Caglioti等人在Comm.Math.Phys.（1995）中首次建立的著名结果的完善。除其他外，我们发现一个完整的无界区间严格凸的熵。另一方面，为了实现这一目标，我们必须提供一个详细的定性描述的全球分支的解决方案$\prl$源自$\lm=0$和交叉$\lm=8π$。这是Suzuki A.I.H.P.（1992）和Chang et al. New Stud. Adv. Math.（2003）中首次建立的$\lm\leq 8π$的著名结果的完善，并且它具有独立的数学意义，因为半线性椭圆型方程的全局分支，除了很少的著名例外，知之甚少。（MVP）建议使用正确的变量（即能量）来获得$\prl$解的全局参数化。一个关键的频谱简化得到的事实，即通过定义，解决方案（MVP）最大化的熵在固定的能量和总涡度。
