We establish an interaction Morawetz estimate for the magnetic Schrödinger equation for $n\geq 3$ under certain smallness conditions on the gauge potentials, but with almost optimal decay. As an application, we prove global wellposedness and scattering in $H^{1}$ for the cubic defocusing magnetic Schrödinger equation for $n=3$.

In this work, we show existence and non-existence results of coexistence states for a Lotka-Volterra symbiotic model with self and cross-diffusion in one species. We study the behavior of the set of positive solutions when the cross-diffusion or the self-diffusion parameter is large.

We consider the Navier-Stokes equations with the Coriolis force (NSC). We establish an improved dispersive estimate for the linear group associated with (NSC), and derive the two-dimensional dispersion effect of the Coriolis forces. As applications, we prove the unique existence of global in time mild solutions for the initial value problem, and the unique existence of time periodic solutions to (NSC).

We obtain the local uniqueness of positive solution for equation with inverse square terms in a smooth bounded domain. This is done by analyzing the Green function and the blow up profile of positive solution of some critical exponent problems with Hardy potential in a smooth bounded domain. We also determine the critical constant of the blow up behavior. This extends the result of Grossi [13] and Cerqueti [5] and Cao-Peng [3] to singular operators.

There exist a delay functional $d$ on an open set $U \subset C([-2,0],$ $\mathbb R)$ and a parameter $\alpha\gt 0$ so that the equation $$ x'(t)=-\alpha\,x(t-d(x_t)) $$ has a solution which is homoclinic to zero, the zero equilibrium is hyperbolic with 2-dimensional unstable manifold, and the stable manifold and unstable manifold intersect transversely along the homoclinic flowline. We prove that close to the homoclinic loop there exists a kind of complicated motion, which involves partially unstable behavior. This does not require any relation between the growth rate in the unstable space and the decay rate in the leading stable plane of the linearized semiflow.

In this paper, we give a comprehensive account of several recent results on the stability of nontrivial soliton structures for some well-known non periodic dispersive models. We will focus on the simpler case of the generalized Korteweg-de Vries equations, covering the classical stability results by Bona, Souganidis, and Strauss until the results by Martel and Merle and our recent collaborations with Miguel Alejo and Luis Vega.