The aim of this paper is to establish a comparison theorem for the Bellman equation of an ergodic control in $ \mathbb R ^d$. This comparison theorem gives a new characterization of the optimal value by sub/super--solutions of the Bellman equation. This result is applicable to compute both of lower and upper bound of the optimal value. Furthermore, from this comparison theorem, we also give a simple proof of the uniqueness result for the Bellman equation.

A uniqueness theorem for positive solutions is proved for a two point boundary value problem governed by a class of nonlinear second order differential equations with quadratic growth in the derivative. An application to the Dirichlet boundary value problem for a nonlinear elliptic equation on a two dimensional annulus is presented.

We study the blow-up behavior of solutions of the semilinear heat equations under the Dirichlet boundary condition or the Neumann boundary condition. We prove the nondegeneracy of blow-up of the solutions. Furthermore, we give a result on the blow-up rate of the solutions under the Neumann boundary condition.

We study the instability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with an attractive power nonlinearity $|u|^{p-1}u$ and a potential $V(x)$ in ${\mathbb{R}}^n$. Here, $\omega>0$ and $\phi_{\omega}(x)$ is a minimal action solution of the stationary problem. Under suitable assumptions on $V(x)$, we show that if $p>1+4/n$, $e^{i \omega t}\phi_{\omega}(x)$ is unstable for sufficiently large $\omega$. For example, our theorem covers a harmonic potential $V(x)=|x|^2$, to which the arguments in the previous papers [2], [14], and [19] are not directly applicable. As another application, we also prove a similar result for a nonlinear Schrödinger equation with a constant magnetic field.

A linear stability criterion for stationary curves is presented for the curvature-driven motion with a triple junction. First we perform formal asymptotic expansions and derive a linearized system. Second we investigate eigenvalues of the infinitesimal generator of the system. Finally, through numerical experiments, we demonstrate that the linear criterion determines the stability in the sense of Lyapunov.

We consider the small data global existence of solutions to the $1$-D exterior mixed problem of the equation $u_{tt}-u_{xx} + u_{t} = \vert u\vert^{p}$ . A new lower bound $p = 8/3$ on the nonlinear term and the better decay estimates of solutions can be derived through the new method in [6].

We prove certain pointwise upper bounds in terms of distance to the boundary for subsolutions of equations generalizing (0.1) below. This type of estimates originate from work of Joseph B. Keller and are of importance to the recent study of so called large solutions of such equations initiated by Catherine Bandle and Moshe Marcus.

We study positive $C^1(\bar{\Omega})$ solutions to classes of boundary value problems of the form \begin{eqnarray*} -\Delta_{p} u & = & g(\lambda,u)\mbox{ in } \Omega \\ u & = & 0 \mbox{ on } \partial \Omega, \end{eqnarray*} where $ \Delta_{p} $ denotes the p-Laplacian operator defined by $$ \Delta_{p} z:= \mbox{div}(|\nabla z|^{p-2}\nabla z);\, p > 1, $$ where $ \lambda > 0$ is a parameter and $ \Omega $ is a bounded domain in $ R^{N} $; $ N \geq 2 $ with $\partial \Omega$ of class $ C^{2}$ and connected. (If $N=1$, we assume that $\Omega$ is a bounded open interval.) In particular, we establish existence and multiplicity results for classes of nondecreasing, p-sublinear functions $g(\lambda,\cdot)$ belonging to $C^1([0,\infty))$. Our results also extend to classes of p-Laplacian systems. Our proofs are based on comparison methods.