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

Complete recuperation after the blow up time for semilinear problems

1. 

Departamento de Matemática Aplicada, E.T.S. Arquitectura. Universidad Politécnica de Madrid, Av. Juan de Herrera 4, Madrid 28040, Spain

2. 

IMI, Universidad Complutense de Madrid and CUNEF, 28040 Madrid, Spain, Spain

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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H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,, North-Holland Mathematical Studies, (1973).   Google Scholar

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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

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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

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V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications,, Vols. I and II, (1969).   Google Scholar

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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

show all references

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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