论文标题
在较高空间维度中，无限梯度系统稳定的解决方案的全球行为：无入侵案例
Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case
论文作者
论文摘要
本文与表格的抛物线梯度系统有关\ [u_t = - \ nabla v（u） +δ_xu \ ,, \]，其中空间变量$ x $和状态变量$ u $是多维的，而潜在的$ v $是无限的cocercive。对于这样的系统，研究了$ | x | $的无限解决方案的渐近行为，即稳定的均质平衡，正在研究$ | x | $。根据溶液的渐近能，该溶液的平均生长速度不接近该平衡，这是对这种溶液的全局渐近行为的部分描述。如果此平均速度为零，则渐近能量是非负的，并且时间派生$ u_t $在太空中均匀地达到$ 0 $。如果相反，平均速度为非零，则渐近能量等于$ - \ infty $。在伴侣论文中调用了这一结果，其中描述了径向对称溶液在无穷大时的稳定的全局行为。该证明主要依赖于实验室框架中的能量估计以及以微小的非零速度行驶的框架。
This paper is concerned with parabolic gradient systems of the form \[ u_t = -\nabla V(u) + Δ_x u \,, \] where the space variable $x$ and the state variable $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, the asymptotic behaviour of solutions stable at infinity, that is approaching a stable homogeneous equilibrium as $|x|$ goes to $+\infty$, is investigated. A partial description of the global asymptotic behaviour of such a solution is provided, depending on the mean speed of growth of the spatial domain where the solution is not close to this equilibrium, in relation with the asymptotic energy of the solution. If this mean speed is zero, then the asymptotic energy is nonnegative, and the time derivative $u_t$ goes to $0$ uniformly in space. If conversely the mean speed is nonzero, then the asymptotic energy equals $-\infty$. This result is called upon in a companion paper where the global behaviour of radially symmetric solutions stable at infinity is described. The proof relies mainly on energy estimates in the laboratory frame and in frames travelling at a small nonzero velocity.
