We first consider the initial value problem of nonlinear Schrödinger equation with the effect of dissipation, and prove the existence of global generalized solution and smooth solution as some conditions respectively. Secondly, we disscuss the asymptotic behavior of solution of mixed problem in bounded domain for above equation. Thirdly, we find the “blow up” phenomenon of the solution of mixed problem for equation iu_t = Δu + βf(|u|²)u - i\frac{ϒ(t)}{2}u, \quad x ∈ Ω ⊂ R³, t > 0 i. e. there exists T_0 > 0 such that lim^{t→Γ_0} || ∇u || ²_{L_t(Ω)} = ∞. The main means are a prior estimates on fractional degree Sobolev space, related properties of operator's semigroup and some integral identities.

ln this peper we obtained the Hopf bifurcation theorem for an abstract functional differential equation by the results of [1]. The asymptotic expression of bifurcation formulae and stability condition were given in detail. Applying the result, we considered the Hopf bifurcation problem for a reaction-diffusion equation with time delay.

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0

Let M be a n-dimensional simply connected, complete Riemannian manifold with constant negative curvature. The heat kernel on M is denoted by H^M_t(x, y) = H^M_t(r(x, y)), where r(x, y) = dist(x, y). We have the explicit formula of H^M_t(x, y) for n=2, 3, and the induction formula of H^M_t(x, y) for n ≥ 4^{[-1]}. But the explicit formula is very complicated for n ≥ 4. ln this paper we give some simple and useful global estimates of H^M_t(x, y), and apply these estimates to the problem of eigenvalue.

This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.