In this paper, the existence and uniqueness of the time-periodic solutions to the Ginzburg-Landau-BBM equations are proved by using a priori estimates and Leray-Schauder fixed point theorem.

We consider a semilinear wave equation of the form u_tt(x, t) - Δu(x, t) = - m(x, t)u_t(x, t) + ∇Φ(x) ⋅ ∇u(x, t ) + b(x)|u(x, t)|^{p-2}u(x, t) where p > 2. We show, under suitable conditions on m, Φ, b, that weak solutions break down in finite time if the initial energy is negative. This result improves an earlier one by the author [1].

We prove the refined ABP maximum principle, comparison principle, and related existence and uniqueness theorem for the positive solutions of the Dirichlet problems of second order fully nonlinear elliptic equations on arbitrary bounded domains.

In this paper, we study the spectral asymptotic behavior for a class of Schrödinger operators on 1-dimensional fractal domains. We have obtained, if the potential function is locally constant, the exact second term of the spectral asymptotics. In general, we give a sharp estimate for the second term of the spectral asymptotics.

In this paper, the global existence of a unique smooth solution for the Landan-Lifshitz-Maxwell equations of the ferromagnetic spin chain in n(1 ≤ n ≤ 2) dimensions is established by using a coupled priori estimates in Sobolev spaces.

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).

ln this paper, we prove Moser-Trüdinger inequality in any two dimensional manifolds. Let (M,g_M,) be a two dimensional manifold without boundary and (g, g_N) with boundary, we shall prove the following three inequalities: u∈H¹(M), \sup\limits_{and ||u||_{H¹(M)}}=1∫_M^{e^{4\pi u²}<+∞} u∈H¹(M), \sup\limits_{∫_M u=0, and} ∫_M|∇u|²=1∫_M^{e^{4\pi u²}<+∞} u∈H¹_0(N), \sup\limits_{and ∫_M|∇u²|=1∫_M^{e^{4\pi u²}<+∞} Moreover, we shall show that there exist of extremal functions which at tain the above three inequalities.