Complete recuperation after the blow up time for semilinear problems (Q260754)

The authors consider blowing-up solutions \(y^0(t)\), \(t\in[0,T_{y^0})\), of the problem NEWLINE\[NEWLINEy'=f(y) \text{ in } \mathbb R^d, \quad y(0)=y_0,NEWLINE\]NEWLINE where \(d\geq 1\) and \(f:\mathbb R^d\to\mathbb R^d\) is a locally Lipschitz function near the infinity. The following problem of controllability is analyzed in the paper. Given \(\varepsilon>0\), find a continuous deformation \(y\) of \(y^0\), built as a solution of the perturbed control problem obtained by replacing \(f(y)\) by \(f(y)+u\), for a suitable control \(u\), such that \(y(t)=y^0(t)\) for any \(t\in[0,T_{y^0}-\varepsilon]\) and such \(y\) also blows up at \(T_{y^0}\), but in such a way that \(y\) could be extended beyond \(T_{y^0}\) as a function \(y\in L^1_{\mathrm{loc}}(0,\infty)\).