平面上变系数Allen-Cahn方程的multiple-end解（英文）

Multiple-end solutions of the variable coefficient Allen-Cahn equation on the plane

平面上Allen-Cahn方程具有multiple-end解,进一步推广,变系数Allen-Cahn方程可以构造一类类似的整体解.给定k≥1,可以发现一个解集远离紧集,且它的零点集渐近于2k条直线(称为ends).这些解具有这样的性质:存在θ_0 <…<θ_(2k)=θ_0+2π,j=0,…,2k-1,且θ是(θ_j,θ_(j+1))的紧子集,使得■关于θ一致成立.

The Allen-Cahn equation has multiple-end solutions on the plane,we construct a similar class of entire solutions for the variable coefficient Allen-Cahn equation for further study.Given k≥1,we can find a family of solutions whose zero level sets are,away from a compact set,asymptotic to 2k straight lines (which are called the ends).These solutions have the property that lim r→+∞ u(re^iθ)=(-1)^j uniformly in θ on compact subsets of (θ j,θ j+1) when there exist θ 0<θ 1<…<θ 2k =θ 0+2 π,for j=0,…,2k-1.