In this paper, we study the existence and the properties of standing waves of the form $u_{\omega}(x,t)=\phi_{\omega}(x)e^{i\omega t},$ where $ x\in \mathbb R^2,$ $t\geq 0$, for a quasilinear Schrödinger equation. Using the minimization method introduced by T. Cazenave and P.L. Lions, we prove a stability theorem for such waves.

For reaction-diffusion equations, equilibrium solutions or traveling wave solutions with thin transition layers are constructed by singular perturbation methods. It is usually difficult to study their stability. This is because the linearized eigenvalue problem has a critical eigenvalue in a small neighborhood of zero, and its location is difficult to determine. The SLEP method is known as one of the most powerful tools to study this critical eigenvalue. To apply this method rigorously, a uniform convergence theorem for the inverse of a differential operator, for instance the inverse Allen-Cahn operator, in some function space plays a crucial role. However, there has been a significant difficulty in the cases of unbounded intervals including those of traveling waves, and no rigorous result was available previously. This paper presents a uniform convergence theorem in a general framework. Our new uniform convergence theorem makes the SLEP method applicable to various kinds of problems including stability of traveling waves.

This work is concerned with instability properties of solutions $u(x,t)=\phi(x-ct)$ of the equation $u_t+(u^p)_x + l H u_{xx}+u_{xxx}=0$ in $\mathbb R$, where $p\in \mathbb N$, $p\geqq 2$, and $H$ is the Hilbert transform. Here, $\phi$ will be a solution of the pseudo-differential equation $\phi''+l H \phi'-c\phi=-\phi^{p}$ solving a certain variational problem. We prove that the set $$ \Omega_{\phi}=\{\phi(\cdot+y) : y\in \mathbb R\;\} $$ is unstable by the flow of the evolution equation above provided $l$ is small, $c>\frac14 l^2$ and $p>5$. Moreover, the trajectories used to exhibit instability are global and uniformly bounded. Finally, we extend these results for a natural generalization of the evolution equation above with general forms of competing dispersion, in particular, we obtain instability results for some Korteweg-de Vries type equations without requiring spectral conditions.

This paper is concerned with the following Cahn-Hilliard equation $ \psi _t = \Delta \mu, $ where $ \mu= -\Delta \psi -\psi +\psi ^3, $ subject on the boundary $\Gamma$ to the following dynamic boundary condition $ \sigma _s \Delta _{||} \psi - \partial _{\nu} \psi + h_s -g_s \psi = \frac{1}{\Gamma _s} \psi _t $ and $ \partial _{\nu} \mu =0, $ and the initial condition $ \psi |_{t=0}= \psi _0. $ This problem was recently proposed by physicists to describe spinodal decomposition of binary mixtures where the effective interaction between the wall (i.e., the boundary $\Gamma$) and two mixture components are short-ranged. The global existence and uniqueness of solutions to this initial boundary value problem with highest-order boundary conditions is proved.

We study the existence, multiplicity and nonexistence of positive radial solutions to boundary value problems for the quasilinear equation $\text{ div} \left ( A(| \nabla u|)\nabla u \right ) + \lambda h(|x|)f(u) =0$ in annular domains under general assumptions on the function $A(u)$. Various possible behaviors of the quotient $\frac{f(u)}{A(u)u}$ at zero and infinity are considered. We shall use fixed point theorems for operators on a Banach space.