In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N=1, the global existence of the weak solution will be established by the bootstrap argument.

In this paper, the two-layer quasigeostrophic flow model under stochastic wind forcing is considered. It is shown that when the layer depth or density difference across the layers tends to zero, the dynamics on both layers synchronizes to an averaged geophysical flow model.

Let u∈C(R,H^1) be the solution to the initial value problem for a 2D semilinear Schrödinger equation with exponential type nonlinearity, given in [1]. We prove that the L^r norms of u decay as t→±∞, provided that r > 2.

In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized (G'/G)-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.

In the study of the regularity criteria for Leray weak solutions to three-dimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity \tilde{u} satisfies \tilde{u}∈L^2(0,T;BMO(R^3)) or \tilde{u}∈ L^{2/1+r}(0,T; \dot{B}^r_{∞,∞}(R^3)) for 0 < r < 1 are considered.

In this paperwe give a newproof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation for the solution and use this spectral representation to deduce regularity results. By exploiting analyticity properties of the terms within the spectral representation, we are able to give simple estimates for the size of the derivatives of the solutions and interpret them in terms of Gevrey classes.