We study a scalar, bi-stable reaction-diffusion-convection equation in $\mathbb{R}^N$. With a hyperbolic scaling, we reduce it to a singularly perturbed equation. In the singular limit, solutions converge to functions which take on only two different values in bulk regions. We also derive an equation describing the motion of the interface between the two bulk regions. To the lowest order, the normal speed $s(\nu)$ of the interface depends only on the unit normal vector $\nu$, where the wave speed $s(\nu)$ is determined by nonlinear reaction and convection terms. When the convection term is even and the reaction term is odd in their argument, the wave speed $s(\nu)$ vanishes identically for all directions $\nu$. In this situation, a parabolic scaling reduces the equation to another singularly perturbed equation for which we also establish convergence of solutions in the singular limit. The singular limit dynamics is governed by an anisotropic mean curvature flow, in which the anisotropy comes from the convection term. Our method of proof for convergence consists of constructing approximate solutions of any order and using the comparison principle.

This paper is devoted to the study of the integrodifferential equation $$ u'(t) =Au (t) +\int^t_0 a(t-s) A_1 u(s) ds +f (t) , \quad t\ge 0, $$ where $A$ is a Hille-Yosida operator in a Banach space $X$, $A_1 \in {\mathcal L} (D (A);$ $ X)$ and $a$ has bounded variation. Existence, uniqueness and estimates of strict and weak solutions are proved by extrapolation methods and the Miller scheme. Applications are given to the Cauchy-Dirichlet problem for the integrodifferential wave equation $$ w_{tt} (t,x) =w_{xx} (t,x)+\int^t_0 a (t-s) w_{xx} (s,x) ds +f(t,x), \quad t\ge 0, \quad x\in [0, \ell]. $$

We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: $ih\frac{\partial\psi}{\partial t} =\left(\frac{h}{i}\nabla-A(x)\right)^2\psi+W(x)\psi-f(|\psi|^2)\psi$ on ${{\mathbb R}}\times{\Omega}$. $\Omega\subset{{\mathbb R}}^N$ is a domain which may be bounded or unbounded. For $h>0$ small we obtain the existence of multi-bump bound states $\psi_h (x,t)=e^{-iEt/h}u_h(x)$ where $u_h$ concentrates simultaneously at possibly degenerate, non-isolated local minima of $W$ as $h\to0$. We require that $W\geq E$ and allow the possibility that $\{x\in\Omega:W(x)=E\}\not=\emptyset$. Moreover, we describe the asymptotic behavior of $u_h$ as $h\to\, 0$.

We consider a nonlinear Schrödinger equation with a nonlinearity of the form $V(x)g(u)$. Assuming that $V(x)$ behaves like $|x|^{-b}$ at infinity and $g(s)$ like $|s|^p$ around $0$, we prove the existence and orbital stability of travelling waves if $1 < p < 1+(4-2b)/N$.