论文标题
受控边界爆炸：爆破后的动态，用于与全球控制的某些半线性问题
Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls
论文作者
论文摘要
本文的主要目的是证明，几种类别的进化问题的解决方案的爆炸现象（$ \ rl^{\ infty} $ - norm爆炸可以通过适当的方式来控制几类进化问题的解决方案（$ n of依赖于$ n的元素）（$ homptect）的元素（$ homptent）是适当的（即$ a）爆炸时间后，对于某些功能空间$ \ rx $）$ \ rl_ {loc}^{1}（0，+\ infty：\ rx）$）。我们首先考虑具有超线性术语的普通微分方程的情况，并表明受控爆炸属性通过使用延迟控制（通过问题解决方案构建，并通过V.M. Alekseev的{\ em em非线性变化，是由于1961年的Alekseev在1961年的{\ em中度延迟式延迟IS}（hofe the Contorations iS Control is Control is Control iS}）（因为$ \ rw_ {loc}^{ - 1，q \ prime}（0，+\ infty：\ rr）$，对于某些$ q> 1 $）$。$。现象。我们证明，在强迫和吸收术语之间的适当平衡中，吹动仅发生在空间域的边界上，这被认为是球$ \ rb _ {\ rr} $，并且作为初始数据的常数。
The main goal of this paper is to show that the blow up phenomenon (the explosion of the $ \rL^{\infty }$-norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls $α(t)$ ($i.e.$ only dependent on time) in such a way that the corresponding solution be well defined (as element of $\rL_{loc}^{1}(0,+\infty :\rX)$, for some functional space $\rX$) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the {\em nonlinear variation of constants formula}, due to V.M. Alekseev in 1961, to the case of {\em neutral delayed equations} (since the control is only in the space $\rW_{loc}^{-1,q\prime }(0,+\infty :\RR )$, for some $q>1$)$.$ We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves the forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball $\rB_{\rR}$ and for a constant as initial datum.
