# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022075
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## Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls

*Corresponding author: J. I. Díaz

Dedicated to Juan Luis Vázquez on occasion of his 75th birthday

Received  January 2022 Revised  May 2022 Early access June 2022

Fund Project: The research of A. C. Casal, G. Díaz and J. I. Díaz was partially supported by the project ref. PID2020-112517GB-I00 of the Agencia Estatal de Investigación (Spain)

The main goal of this paper is to show that the blow up phenomenon (the explosion of the ${{\rm{L}}}^{\infty }$-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 ${{\rm{L}}}_{loc}^{1}(0,+\infty : {{\rm{X}}})$, for some functional space ${{\rm{X}}}$) 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 nonlinear variation of constants formula, due to V. M. Alekseev in 1961, to the case of neutral delayed equations (since the control is only in the space ${{\rm{W}}}_{loc}^{-1,q\prime }(0,+\infty : \mathbb{R} )$, 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 a 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 ${{\rm{B}}}_{ {{\rm{R}}}}$ and for a constant as initial datum.

Citation: Alfonso Carlos Casal, Gregorio Díaz, Jesús Ildefonso Díaz, José Manuel Vegas. Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022075
##### References:
 [1] S. Alarcón, G. Díaz and J. M. Rey, Large solutions of elliptic semilinear equations in the borderline case. An exhaustive and intrinsic point of view, Journal of Mathematical Analysis and Applications, 431 (2015), 365-405.  doi: 10.1016/j.jmaa.2015.05.068. [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), 28-36. [3] H. Amann and M. Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition, Acta Math. Univ. Comenian., 66 (1997), 321-328. [4] H. Amann and P. Quittner, Optimal control problems governed by semilinear parabolic equations with low regularity data, Adv. Differ. Equ., 11, (2006) 1–33. [5] J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary equations, Proc. Amer. Math. Soc., 136 (2008), 151-160.  doi: 10.1090/S0002-9939-07-08980-0. [6] K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 9 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009. [7] C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Annals of University of Craiova, Math. Comp. Sci. Ser., 32 (2005), 1-8. [8] C. Bandle and G. Díaz et J. I. Díaz, Solutions d'équations de réaction-diffusion non linéaires explosant au bord parabolique, C. R. Acad Sci Paris, 318 (1994), 455-460. [9] C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 2006, 35–67. doi: 10.4171/RLM/453. [10] P. Baras and L. Cohen, Complete blow-up after Tmax for the solution of a semilinear heat equation, J. Funct. Anal., 71 (1987), 142-174.  doi: 10.1016/0022-1236(87)90020-6. [11] I. Bejenaru, J. I. Díaz and I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamical boundary conditions, Electr. J. Diff. Eqns., (2001), No. 50, 19 pp. [12] H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Mathematical Studies, Amsterdam, 1973. [13] H. Brezis, Th. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_{t}-\Delta u=g(u)$ revisited, Adv. Differ. Equat., 1 (1996), 73-90. [14] L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Commun. Partial Differential Equations, 32 (2007) 1245–1260. doi: 10.1080/03605300600987306. [15] A. C. Casal, J. I. Díaz and J. M. Vegas, Blow-up in some ordinary and partial differential equations with time-delay, Dynam. Systems Appl., 18 (2009), 29-46. [16] A. C. Casal, J. I. Díaz and J. M. Vegas, Controlled explosions of blowing-up trajectories in semilinear problems and a nonlinear variation of constant formula, XXIII Congreso de Ecuaciones Diferenciales y Aplicaciones, XIII Congreso de Matemática Aplicada, Castellón, 9–13 septiembre 2013. e-Proccedings. [17] A. C. Casal, J. I. Díaz and J. M. Vegas, Complete recuperation after the blow up time for semilinear problems, AIMS Procceding, 2015 (2015), 223-229.  doi: 10.3934/proc.2015.0223. [18] Th. Cazenave, Y. Martel and L. Zhao, Solutions blowing up on any given compact set for the energy subcritical wave equation, J. Differential Equations, 268 (2020), 680-706.  doi: 10.1016/j.jde.2019.08.030. [19] J. M. Coron and E. Trélat, Global steady-state controllability of 1-D semilinear heat equations, SIAM J. Control and Optimization, 43 (2004), 549-569.  doi: 10.1137/S036301290342471X. [20] G. Díaz, J. I. Díaz and J. Otero, Construction of the maximal solution of Backus' problem in geodesy and geomagnetism, Studia Geophysica et Geodaetica, 55 (2011), 415-440.  doi: 10.1007/s11200-011-0024-3. [21] G. Díaz and J. I. Díaz, Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing, Discrete and Continuous Dynamical Systems Series S. doi: 10.3934/dcdss. 2021165. [22] G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Analysis, 20 (1993), 97-125.  doi: 10.1016/0362-546X(93)90012-H. [23] J. I. Díaz, D. Gómez-Castro and J. L. Vázquez., The fractional Schrödinger equation with general nonnegative potentials, The weighted space approach, Nonlinear Analysis, 177 (2018), 325-360.  doi: 10.1016/j.na.2018.05.001. [24] J. I. Díaz and J.-L. Lions, Sur la contrôlabilité de problèmes paraboliques avec phenomenes d'explosion, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 173-177.  doi: 10.1016/S0764-4442(98)80083-9. [25] J. I. Díaz and J. L. Lions, On the approximate controllability for some explosive parabolic problems, In: Hoffmann, K. -H., et al. (eds. ), Optimal Control of Partial Differential Equations, (Chemnitz, 1998), Internat. Ser. Numer. Math., vol. 133 (1993), Birkhäuser, Basel, 115–132. [26] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré Anal. Non Linèaire, 17 (2000), 583–616. doi: 10.1016/s0294-1449(00)00117-7. [27] M. Fila and J. Filo, Blow-up on the boundary: A survey, In Singularities and Differential Equations, Banach Center Publications, volume 33 (1996). Institute of Mathematics, Polish Academy of Sciences, Warszawa, 67–77. [28] M. Fila and P. Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition, in Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume, (Joachim Escher and Gieri Simonett eds. ), Progress in Nonlinear Differential Equations and Their Applications, Vol. 35 (1999), Birkhauser, 251–272. [29] Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electronic Journal of Qualitative Theory of Differential Equations, 2016, Paper No. 70, 17 pp. doi: 10.14232/ejqtde. 2016.1.70. [30] V. A. Galaktionov and J. L. Vázquez, A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach, Progress in Nonlinear Differential Equations and Their Applications Vol. 56. Birkhauser, Boston, 2003. doi: 10.1007/978-1-4612-2050-3. [31] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18460-4. [32] M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 222-229.  doi: 10.14492/hokmj/1381413677. [33] M. Kirane, E. Nabana and S. I. Pokhozhaev, The absence of solutions of elliptic systems with dynamic boundary conditions, Differential Equations, 38 (2002), 808-815.  doi: 10.1023/A:1020358228313. [34] V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications, Vols. I and II, Academic Press, New York, 1969. [35] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equations with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.  doi: 10.1016/0022-0396(74)90018-7. [36] J. -L. Lions, Contrôle Des Systêmes Distribués Singuliers, Gauthier-Villars, Bordas, Paris. 1983. [37] J. López Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Eq., 92 (1991), 384-401.  doi: 10.1016/0022-0396(91)90056-F. [38] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.  doi: 10.1002/cpa.3160450303. [39] A. Porretta and E. Zuazua, Null controllability of viscous Hamilton-Jacobi equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 301-333.  doi: 10.1016/j.anihpc.2011.11.002. [40] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser, Berlin, 2007. [41] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka Journal of Mathematics, 12 (1975), 45–51. [42] J. L. Vázquez, The mathematical theories of diffusion, Nonlinear and Fractional Diffusion, Lecture Notes in Mathematics, 2186. Fondazione CIME/CIME Foundation Subseries. Springer, Cham; Fondazione C.I.M.E., Florence, 2017. [43] J. L. Vázquez and E. Vitillaro, On the Laplace equation with dynamical boundary conditions of reactive diffusive type, J. Math. Anal. Appl., 354 (2009), 674-688.  doi: 10.1016/j.jmaa.2009.01.023. [44] N. Yamazaki, A class of nonlinear evolution equations governed by time-dependent operators of subdifferential type, Hokkaido University Preprint Series in Mathematics, 696 (2005), 1–16.

show all references

##### References:
 [1] S. Alarcón, G. Díaz and J. M. Rey, Large solutions of elliptic semilinear equations in the borderline case. An exhaustive and intrinsic point of view, Journal of Mathematical Analysis and Applications, 431 (2015), 365-405.  doi: 10.1016/j.jmaa.2015.05.068. [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), 28-36. [3] H. Amann and M. Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition, Acta Math. Univ. Comenian., 66 (1997), 321-328. [4] H. Amann and P. Quittner, Optimal control problems governed by semilinear parabolic equations with low regularity data, Adv. Differ. Equ., 11, (2006) 1–33. [5] J. M. Arrieta, On boundedness of solutions of reaction-diffusion equations with nonlinear boundary equations, Proc. Amer. Math. Soc., 136 (2008), 151-160.  doi: 10.1090/S0002-9939-07-08980-0. [6] K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 9 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009. [7] C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Annals of University of Craiova, Math. Comp. Sci. Ser., 32 (2005), 1-8. [8] C. Bandle and G. Díaz et J. I. Díaz, Solutions d'équations de réaction-diffusion non linéaires explosant au bord parabolique, C. R. Acad Sci Paris, 318 (1994), 455-460. [9] C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 2006, 35–67. doi: 10.4171/RLM/453. [10] P. Baras and L. Cohen, Complete blow-up after Tmax for the solution of a semilinear heat equation, J. Funct. Anal., 71 (1987), 142-174.  doi: 10.1016/0022-1236(87)90020-6. [11] I. Bejenaru, J. I. Díaz and I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamical boundary conditions, Electr. J. Diff. Eqns., (2001), No. 50, 19 pp. [12] H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Mathematical Studies, Amsterdam, 1973. [13] H. Brezis, Th. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_{t}-\Delta u=g(u)$ revisited, Adv. Differ. Equat., 1 (1996), 73-90. [14] L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Commun. Partial Differential Equations, 32 (2007) 1245–1260. doi: 10.1080/03605300600987306. [15] A. C. Casal, J. I. Díaz and J. M. Vegas, Blow-up in some ordinary and partial differential equations with time-delay, Dynam. Systems Appl., 18 (2009), 29-46. [16] A. C. Casal, J. I. Díaz and J. M. Vegas, Controlled explosions of blowing-up trajectories in semilinear problems and a nonlinear variation of constant formula, XXIII Congreso de Ecuaciones Diferenciales y Aplicaciones, XIII Congreso de Matemática Aplicada, Castellón, 9–13 septiembre 2013. e-Proccedings. [17] A. C. Casal, J. I. Díaz and J. M. Vegas, Complete recuperation after the blow up time for semilinear problems, AIMS Procceding, 2015 (2015), 223-229.  doi: 10.3934/proc.2015.0223. [18] Th. Cazenave, Y. Martel and L. Zhao, Solutions blowing up on any given compact set for the energy subcritical wave equation, J. Differential Equations, 268 (2020), 680-706.  doi: 10.1016/j.jde.2019.08.030. [19] J. M. Coron and E. Trélat, Global steady-state controllability of 1-D semilinear heat equations, SIAM J. Control and Optimization, 43 (2004), 549-569.  doi: 10.1137/S036301290342471X. [20] G. Díaz, J. I. Díaz and J. Otero, Construction of the maximal solution of Backus' problem in geodesy and geomagnetism, Studia Geophysica et Geodaetica, 55 (2011), 415-440.  doi: 10.1007/s11200-011-0024-3. [21] G. Díaz and J. I. Díaz, Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing, Discrete and Continuous Dynamical Systems Series S. doi: 10.3934/dcdss. 2021165. [22] G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Analysis, 20 (1993), 97-125.  doi: 10.1016/0362-546X(93)90012-H. [23] J. I. Díaz, D. Gómez-Castro and J. L. Vázquez., The fractional Schrödinger equation with general nonnegative potentials, The weighted space approach, Nonlinear Analysis, 177 (2018), 325-360.  doi: 10.1016/j.na.2018.05.001. [24] J. I. Díaz and J.-L. Lions, Sur la contrôlabilité de problèmes paraboliques avec phenomenes d'explosion, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 173-177.  doi: 10.1016/S0764-4442(98)80083-9. [25] J. I. Díaz and J. L. Lions, On the approximate controllability for some explosive parabolic problems, In: Hoffmann, K. -H., et al. (eds. ), Optimal Control of Partial Differential Equations, (Chemnitz, 1998), Internat. Ser. Numer. Math., vol. 133 (1993), Birkhäuser, Basel, 115–132. [26] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré Anal. Non Linèaire, 17 (2000), 583–616. doi: 10.1016/s0294-1449(00)00117-7. [27] M. Fila and J. Filo, Blow-up on the boundary: A survey, In Singularities and Differential Equations, Banach Center Publications, volume 33 (1996). Institute of Mathematics, Polish Academy of Sciences, Warszawa, 67–77. [28] M. Fila and P. Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition, in Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume, (Joachim Escher and Gieri Simonett eds. ), Progress in Nonlinear Differential Equations and Their Applications, Vol. 35 (1999), Birkhauser, 251–272. [29] Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electronic Journal of Qualitative Theory of Differential Equations, 2016, Paper No. 70, 17 pp. doi: 10.14232/ejqtde. 2016.1.70. [30] V. A. Galaktionov and J. L. Vázquez, A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach, Progress in Nonlinear Differential Equations and Their Applications Vol. 56. Birkhauser, Boston, 2003. doi: 10.1007/978-1-4612-2050-3. [31] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18460-4. [32] M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 222-229.  doi: 10.14492/hokmj/1381413677. [33] M. Kirane, E. Nabana and S. I. Pokhozhaev, The absence of solutions of elliptic systems with dynamic boundary conditions, Differential Equations, 38 (2002), 808-815.  doi: 10.1023/A:1020358228313. [34] V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications, Vols. I and II, Academic Press, New York, 1969. [35] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equations with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.  doi: 10.1016/0022-0396(74)90018-7. [36] J. -L. Lions, Contrôle Des Systêmes Distribués Singuliers, Gauthier-Villars, Bordas, Paris. 1983. [37] J. López Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Eq., 92 (1991), 384-401.  doi: 10.1016/0022-0396(91)90056-F. [38] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.  doi: 10.1002/cpa.3160450303. [39] A. Porretta and E. Zuazua, Null controllability of viscous Hamilton-Jacobi equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 301-333.  doi: 10.1016/j.anihpc.2011.11.002. [40] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser, Berlin, 2007. [41] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka Journal of Mathematics, 12 (1975), 45–51. [42] J. L. Vázquez, The mathematical theories of diffusion, Nonlinear and Fractional Diffusion, Lecture Notes in Mathematics, 2186. Fondazione CIME/CIME Foundation Subseries. Springer, Cham; Fondazione C.I.M.E., Florence, 2017. [43] J. L. Vázquez and E. Vitillaro, On the Laplace equation with dynamical boundary conditions of reactive diffusive type, J. Math. Anal. Appl., 354 (2009), 674-688.  doi: 10.1016/j.jmaa.2009.01.023. [44] N. Yamazaki, A class of nonlinear evolution equations governed by time-dependent operators of subdifferential type, Hokkaido University Preprint Series in Mathematics, 696 (2005), 1–16.
Illustrative example of the control $\alpha(t)$ and the effective bang-bang control if $f(s)=s^{p},p>2$

The blowing up solution without control $u^{0}(t)$ and the controlled solution $u^{\alpha}(t)$ defined in the whole $[0,+\infty[$

Spatial profile of the subsolution $\underline{ {{\rm{U}}}}$ and time profiles of the solution at $r= {{\rm{R}}}$ for some values of $p$
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