This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.

The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83-99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing's instability of the corresponding reaction-diffusion system.

No abstract

The human complement system is an important early host defense against infection. Entamoeba histolytica activates the complement system but is resistant to killing by complement C5b-9 complexes deposited on the membrane surface. Our aim was to identify components of the amebic plasma membrane that mediate resistance to human complement C5b-9 by screening for neutralizing monoclonal antibodies. A monoclonal antibody was identified that abrogated amebic resistance to C5b-9, and the mAb was shown to recognize the parasite's galactosespecific adhesin. The purified adhesin bound to C8 and C9 and conferred C5b-9 resistance to sensitive ameba upon reconstitution; these activities of the adhesin were inhibited by the antiadhesin mAb. The E. histolytica adhesin shared sequence similarities and antigenic cross-reactivity with CD59, a membrane inhibitor of C5b-9 in human blood cells, suggesting both molecular mimicry and shared complement-inhibitory functions. (J. Clin. Invest. 1992. 90:1131-1137

This paper is dealing with entire solutions of a bistable reactiondiffusion equation with Nagumo type nonlinearity, so called the Allen-Cahn equation. Here the entire solutions are meant by the solutions defined for all (x; t) 2 R £ R. In this article we first show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of x-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. We also show the existence of an entire solution emanating from the unstable standing pulse solution and converges to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle we prove a rather general result on the existence of an unstable set of an unstable equilibrium to apply to the present case.