一维无界空间维度下梯度系统双稳态解的全局行为
Global behaviour of bistable solutions for gradient systems in one unbounded spatial dimension
AI总结：
本文研究了实数轴上多维抛物梯度系统的双稳态解，在势能强制性等一般假设下，利用纯变分方法证明其全局渐近行为表现为向两端传播的波叠堆及中间缓慢分离的稳态轮廓模式，将 Fife 和 McLeod 的经典理论从标量情形推广至系统情形。
摘要：
本文研究形如 \[ u_t=-\ abla V (u) + u_{xx}\,, \] 的抛物梯度系统，其中空间域为整个实数轴，状态变量 $u$ 是多维的，势能 $V$ 在无穷远处是强制的。对于此类系统，在关于势能的一般性假设下，描述了每个双稳态解（即在空间两端均接近稳定齐次平衡态的解）的渐近行为。每个此类解在空间极左处趋近于向左传播的双稳态波叠堆族，在空间极右处趋近于向右传播的双稳态波叠堆族，而在中间则是趋向于稳定齐次平衡态的同宿或异宿稳态解的轮廓模式，且这些轮廓彼此缓慢分离。通过将结果扩展至系统情形，本研究进一步推进了 Fife 和 McLeod 在七十年代末发起的关于双稳态解全局渐近行为的研究计划。在缺乏极值原理的情况下，论证纯粹采用变分方法，并引用了配套论文中的已有结果。
英文摘要：
This paper is concerned with parabolic gradient systems of the form \[ u_t=-\nabla V (u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of every bistable solution (that is, every solution close at both ends of space to stable homogeneous equilibria) is described. Every such solution approaches, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right in space a stacked family of bistable fronts travelling to the right, and in between a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. This result pushes one step further the program initiated in the late seventies by Fife and McLeod about the global asymptotic behaviour of bistable solutions, by extending their results to the case of systems. In the absence of maximum principle, the arguments are purely variational, and call upon previous results obtained in companion papers.
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