We study global well-posedness below the energy norm of the Cauchy problem for the Klein-Gordon equation in R^n with n ≥ 3. By means of Bourgain's method along with the endpoint Strichartz estimates of Keel and Tao, we prove the H^s-global well-posedness with s < 1 of the Cauchy problem for the Klein-Gordon equation. This we do by establishing a series of nonlinear a priori estimates in the setting of Besov spaces.

The author discusses the asymptotic behavior of the radial minimizer of the p-Ginzburg-Landau functional with a weight in the case p > n ≥ 2. The location of the zeros and the uniqueness of the radial minimizer are derived. Moreover, the W^{1,p} convergence of the radial minimizer of this functional is proved.

The well-posedness of the Cauchy problem for the system {i∂_tu + ∂²_xu = uv + |u|²u, t, x ∈ \mathbb{R}, ∂_tv + ∂_xΗ∂_xv = ∂_x|u|², u(0, x) = u_0(x), v(0, x) = v_0(x), is considered. It is proved that there exists a unique local solution (u(x, t), v(x, t)) ∈ C([0, T);H^s) ×C([0, T);H^{s-\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\mathbb{R}) ×H^{s-\frac{1}{2}} (\mathbb{R})(s ≥ \frac{1}{4}) and the solution depends continuously on the initial data.

We study a free boundary problem of parabolic equations with a positive parameter τ included in the coefficient of the derivative with respect to the time variable t. This problem arises from some reaction-diffusion system. We prove that, if τ is large enough, the solution exists for 0 < t < +∞ while, if τ is small enough, the solution exists only in finite time.

In this paper, Under the assumption that the relaxation time ε is sufficiently small, we prove the existence of the global smooth solution to the Cauchy problem for the Jin-Xin model without any smallness assumption for the initial data. The analysis is based on some a priori estimates which are obtained by the method of characteristic and the maximum principle of first-order quasilinear hyperbolic system.

Global-in-time classical solutions near Maxwellians with small amplitude are constructed for the Vlasov-Poisson system with certain generalized Landau collision operator. The construction of global solution is based on an energy method.