# American Institute of Mathematical Sciences

2015, 2015(special): 223-229. doi: 10.3934/proc.2015.0223

## Complete recuperation after the blow up time for semilinear problems

Received  September 2014 Revised  September 2015 Published  November 2015

We consider explosive solutions $y^{0}(t)$, $t\in \lbrack 0,T_{y^{0}}),$ of some ordinary differential equations \begin{equation*} P(T_{y^{0}}): \begin{array}{lc} \frac{dy}{dt}(t)=f(y(t)),y(0)=y_{0}, & \end{array} \end{equation*} where $f:$ $\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is a locally Lipschitz superlinear function and $d\geq 1$. In this work we analyze the following question of controlability: given $\epsilon >0$, a continuous deformation $y(t)$ de $y^{0}(t)$, built as a solution of the perturbed control problem obtained by replacing $f(y(t))$ by $f(y(t))+u(t),$ for a suitable control $u$, such that $y(t)=y^{0}(t)$ for any $t\in \lbrack 0,T_{y^{0}}-\epsilon ]$ and such that $y(t)$ also blows up in $t=T_{y_{0}}$ but in such a way that $y(t)$ could be extended beyond $T_{y_{0}}$ as a function $y\in L_{loc}^{1}(0,+\infty :\mathbb{R}^{d})$?
Citation: Alfonso C. Casal, Jesús Ildefonso Díaz, José Manuel Vegas. Complete recuperation after the blow up time for semilinear problems. Conference Publications, 2015, 2015 (special) : 223-229. doi: 10.3934/proc.2015.0223
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##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [2] V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary differential equations (Russian),, Vestnik Moskov Univ. Ser. I Mat. Meh., 2 (1961), 28. Google Scholar [3] H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,, North-Holland Mathematical Studies, (1973). Google Scholar [4] A. Casal, J. I. Díaz, and J. M. Vegas, Blow-up in some ordinary and partial differential equations with time-delay., Dynam. Systems Appl. 18(1) (2009), 18 (2009), 29. Google Scholar [5] J. I. Díaz and A. V. Fursikov, A simple proof of the approximate controllability from the interior for nonlinear evolution problems., Applied Mathematics Letters 7(5) (1994), 7 (1994), 85. Google Scholar [6] V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications,, Vols. I and II, (1969). Google Scholar [7] S. Mirica, On differentiability with respect to initial data in the theory of differential equations., Rev. Roumaine des Math. Pures Appl., 48 (2003), 153. Google Scholar
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