We study a spatial susceptible-infected-susceptible(SIS) model in heterogeneous environments with vary advective rate. We establish the asymptotic stability of the unique disease-free equilibrium(DFE) when R0 > 1 and the existence of the endemic equilibrium when R0 < 1. Here R0 is the basic reproduction number. We also discuss the effect of diffusion on the stability of the DFE.

In this paper, we study the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2uh’(|u|^2)\Delta h(|u|^2)+F(|u|^2)u\mp A[h(|u|^2]^{2^*-1}h’(|u|^2)u,\ x\in \mathbb{R}^N, \ t>0,\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, $h(s)\geq 0$ and $h’(s)\geq 0$ for $s\geq 0$, $N\geq 3$, $A>0$. Besides obtaining sufficient conditions on the blowup in finite time and global existence of the solution, we get Morawetz type estimates and weighted spacetime bounds for the global solution based on pseudoconformal conservation law, we also establish a sharp threshold which gives the watershed for initial data $u_0$ making the solution blow up.

In this paper, we consider Cauchy problem of a quasilinear Schrodinger equation which has general form containing potential term, power type nonlinearity and Hartree type nonlinearity. The space dimension is arbitrary, that is, it is larger than or equals to one. First, we establish the local wellposedness of the solution and discuss the condition on the global existence of the solution. Next, we establish some conservation laws such as mass conservation law, energy conservation law, pseudoconformal conservation law of the solution. Based on these conservation laws, we give Morawetz type estimates, spacetime bounds for the global solution. Last, we take two ideas to establish scattering theory for the global solution in different functional spaces. The first idea is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel's formula in order to keep each term independent, another one is that we factitiously let a continuous function be the sum of two piecewise functions and choose different admissible pairs in Strichartz estimates for the terms containing these functions.