Chinese Annals of Mathematics

In the paper [2] the authors defined the singular Darboux transformations and established an explicit formula for constructing unitons from a simply connected Riemann surface M to the group U(N). The formula is obtained as a limit of an infinite consequence of Darboux transformations through some renormalization procedure. In the present paper the authors give a complete proof of the fact that the formula gives a global solution of harmonic maps without singularity.

Under certain conditions, the dynamic equations of membrane shells and the dynamic equations of flexural shells are obtained from dynamic equations of Koiter shells by the method of asymptotic analysis.

The author proves blow up of solutions to the Cauchy problems of certain nonlinear wave equations and, also, estimates the time when the blow up occurs.

Let G be a p-group (p odd prime) and let X=Cay(G,S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Aut(X) of X is isomorphic to the semidirect product G_RAut(G, S), where G_R is the right regular representation of G and Aut(G, S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a couterexample.

The authors consider the existence of singular limit solutions for a family of nonlinear elliptic problems with exponentially dominated nonlinearity and Dirichlet boundary condition and generalize the results of [3].

The authors investigate the global propeties of general autonomous systems on the plane and establish criteria for the nonexistence, existence and uniqueness of limit cycles. As application examples, the limit cycles for some polynomial systems are studied.

This paper gives all the two-dimensional membrane models obtained from formal asymptotic analysis of the three-dimensional geometrically exact nonlinear model of a thin elastic shell made with a Saint Venant-Kirchhoff material. Therefore, the other models can be quoted as flexural nonlinear ones. The author also gives the formal equations solved by the associated stress tensor and points out that only one of those models leads, by linearization, to the "classical" linear limiting membrane model, whose justification has already been established by a convergence theorem.

By means of an equivalent invariant form of boundary conditions, the author get the existence and uniqueness of semi-global C¹ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.

This paper gives a Fan's type minimax theorem, a nearest point theorem and two existence theorems of solutions for a kind of generalized quasi-variational inequalities in H-spaces without any linear structure.

This paper studies the asymptotic stability of traveling wave solutions of nonlinear systems of integral-differential equations. It has been established that linear stablility of traveling waves is equivalent to nonlinear stability and some "nice structure" of the spectrum of an associated operator implies the linear stability. By using the method of variation of parameter, the author defines some complex analytic function, called the Evans function. The zeros of the Evans function corresponds to the eigenvalues of the associated linear operator. By calculating the zeros of the Evans function, the asymptotic stability of the travling wave solutions is established.

The aim of this paper is to discuss the instantaneous shrinking and localization of the support of functions in Y_λ(m,p,q,N) and their applications to some nonlinear parabolic equations including the porous medium equation u_t=Δu^m-u^q, m>0, q>0 and the p-Laplace equation u_t=div(|∇u|^p-2∇u)-u^q, p>1, q>0. In particular, as an application of the results, the necessary and sufficient condition for the solutions of the above p-Laplace equation with nonnegative finite Borel measures as initial conditions to have the instantaneous shrinking property of the support is obtained. This is an answer to an open problem posed by R. Kersner and A. Shishkov.

It is proved that there is no nonconstant harmonic function of finite energy on product of certain Kählerian manifolds with a pole.

The existence and uniqueness of a strong periodic solution of the evolution system describing geophysical flow in bounded domains of R^N(N=3,4) are proven if external forces are periodic in time and sufficiently small.

Let C be a nonempty bounded subset of a p-uniformly convex Banach space X, and T={T(t):t∈S} be a Lipschitzian semigroup on C with , where N_p is the normal structure coefficient of X. Suppose also there exists a nonempty bounded closed convex subset E of C with the following properties: (P₁)x ∈ E implies ω_w(x)⊂ E; (P₂)T is asymptotically regular on E. The authors prove that there exists a z∈E such that T(s)z=z for all s ∈ S. Further, under the similar condition, the existence of fixed points of Lipschitzian semigroups in a uniformly convex Banach space is discussed.