We prove that the solutions to the initial-value problem for the 2-dimensional Schrödinger maps are unique in $$ C_tL^\infty_x \cap L^\infty_t (\dot{H}^1_x\cap\dot{H}^2_x) . $$ For the proof, we follow McGahagan's argument with improving its technical part, combining Yudovich's argument.

In this paper, we study a Kirchhoff type elliptic problem, \[ \begin{cases} \displaystyle - \Big ( 1+\alpha \int_{\Omega}|\nabla u|^2dx \Big ) \Delta u =\lambda u^q+u^3,\ u > 0 \text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \] where $\Omega\subset \mathbb{R}^4$ is a bounded domain with smooth boundary $\partial \Omega$ and we assume $\alpha,\lambda > 0$ and $1\le q < 3$. For $q=1$, we prove the existence of possibly multiple solutions for $\alpha > 0$ and $\lambda\ge \lambda_1$ in suitable intervals, where $\lambda_1 > 0$ is the first eigenvalue of $-\Delta$ on $\Omega$. On the other hand for $q\in (1,3)$, we show the existence of a solution for all small $\alpha > 0$ and all $\lambda > 0$. We establish the results by the method of the Nehari manifold and the concentration compactness analysis for Palais-Smale sequences.

We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearities. It will turn out that the decay rate which is previously found in some models is optimal.

We consider the nonlinear eigenvalue problem $$ [D(u)u']' + \lambda f(u) = 0, \ \ u(t) > 0, \ \ t \in I := (0,1), \ \ u(0) = u(1) = 0, $$ where $D(u) = u^p$, $f(u) = u^{q} + \sin (u^n)$ and $\lambda > 0$ is a bifurcation parameter. Here, $p \ge 0$, $n > 0$ and $q > 0$ are given constants and $k:=(p + q + 1)/2 \in \mathbb{N}$. This equation is motivated by the mathematical model of animal dispersal and invasion and $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(\alpha)$. We establish new precise asymptotic expansion formulas for $\lambda(\alpha)$ as $\alpha \to \infty$. In particular, we obtain the precise asymptotic behavior of $\lambda(\alpha)$ which tends to $0$ with oscillation as $\alpha \to \infty$.

The present paper is devoted to the investigation of noisy impacts on the dynamics of periodic ordinary differential equations (ODEs). To do so, we consider a family of stochastic differential equations resulting from a periodic ODE perturbed by small white noises, and study noise-vanishing behaviors of their “steady states” that are characterized by periodic probability solutions of the associated Fokker-Plank equations. By establishing noise-vanishing concentration estimates of periodic probability solutions, we prove that any limit measure of periodic probability solutions must be a periodically invariant measure of the ODE and that the global periodic attractor of a dissipative ODE is stable under general small noise perturbations. For local periodic attractors (resp. local periodic repellers), small noises are constructed to stabilize (resp. de-stabilize) them. Our study provides an elementary step towards the understanding of stochastic stability of periodic ODEs.