Global behaviour of radially symmetric solutions stable at infinity for gradient systems
Global behaviour of radially symmetric solutions stable at infinity for gradient systems
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梯度系统在无穷远处稳定的径向对称解的全局行为
DOI：
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发表时间：
2017
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通讯作者：
E. Risler
中科院分区：
文献类型：
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作者：
E. Risler
本文研究了一类方程组[ u_t = -u_t]的径向对称解
其中空间变量x和状态参数u是多维的，势V在无穷远处是强制的。对于这样的系统，在一般假设的潜力，渐近行为的解决方案“稳定在无穷远”，即接近一个空间齐次平衡时，|X| $ approaches $+infty$，是一个很好的例子。证明了每一个这样的解决方案，接近一个堆叠家庭的径向对称波前旅行到无穷远。这种行为是类似于一个无界空间维度的梯度系统，在配套文件中所描述的一个numerical解决方案。人们预期（但不幸的是，在这个阶段没有证明），在这些行进的前沿后面，解的行为再次表现为一维情况（即，时间导数接近于零，解接近于稳态解的模式）。
This paper is concerned with radially symmetric solutions of systems of the form [ u_t = -
abla V(u) + Delta_x u ] where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions "stable at infinity", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).