Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls

TL;DR

This paper demonstrates that the blow-up phenomenon in certain evolution problems can be controlled using global time-dependent controls, ensuring solutions remain well-defined after the blow-up time, with applications to semilinear elliptic equations.

Contribution

It introduces a method to control blow-up in evolution problems via global controls, extending Alekseev's variation of constants to neutral delayed equations and applying it to boundary blow-up in semilinear elliptic problems.

Findings

Blow-up can be controlled with suitable global controls.

Boundary blow-up occurs only on the domain boundary under certain conditions.

The control method applies to both ODEs and semilinear PDEs.

Abstract

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 $\alpha (t)$ ($i.e.$ 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.