June  2017, 22(4): 1425-1434. doi: 10.3934/dcdsb.2017068

Feedback controllability for blowup points of semilinear heat equations

School of Mathematics & Statistics, Northeast Normal University, Changchun 130024, China

* Corresponding author: Ping Lin

Received  April 2016 Revised  August 2016 Published  February 2017

Fund Project: The author is supported by NSF of China under grant 11471070

This paper studies a controllability problem for blowup points of two classes of semilinear heat equations.Our goal to act controls on the systems we studied is to make the corresponding solutions blow upat given points. This differs with the controllability problem of equations with the property of blowup in the references, where the purpose of using controls is to prevent blowupby controls. We obtain the feedback controls for our controllability problem of blowup points.

Citation: Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068
References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[2]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.  Google Scholar

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J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136 American Mathematical Society, Providence, RI, 2007.  Google Scholar

[4]

J. M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl., 92 (2009), 528-545.  doi: 10.1016/j.matpur.2009.05.015.  Google Scholar

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J. M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.  Google Scholar

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A. DoubovaE. Fernández-CaraM. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.  Google Scholar

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E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[8]

S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of $u_t-\triangle u=u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.  doi: 10.1002/cpa.3160450703.  Google Scholar

[9]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[10]

H. Fujita, On the blowing-up of solutions of the Cauchy problem for $u_t=Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[11]

Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.  doi: 10.1512/iumj.1987.36.36001.  Google Scholar

[12]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.  doi: 10.1002/cpa.3160420607.  Google Scholar

[13]

M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.   Google Scholar

[14]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4.  Google Scholar

[15]

S. KhenissyY. Rébaï and H. Zaag, Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation, Ann. Inst. H. Poincaré, 28 (2011), 1-26.  doi: 10.1016/j.anihpc.2010.09.006.  Google Scholar

[16]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.  Google Scholar

[17]

P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.  Google Scholar

[18]

Z. Ling and Z. Wang, Global existence and finite time blowup for a nonlocal parabolic system, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 371-383.   Google Scholar

[19]

F. Merle and H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann., 316 (2000), 103-137.  doi: 10.1007/s002080050006.  Google Scholar

[20]

Q. TaoH. Gao and Y. Yang, Controllability results for weakly blowingup reaction-diffusion system, Electron. J. Qual. Theory Differ. Equ., 11 (2012), 1-19.   Google Scholar

[21]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh A, 136 (2006), 415-430.  doi: 10.1017/S0308210500004637.  Google Scholar

[22]

C. Wang and S. Zheng, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20 (2007), 1343-1359.  doi: 10.1088/0951-7715/20/6/002.  Google Scholar

show all references

References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[2]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.  Google Scholar

[3]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136 American Mathematical Society, Providence, RI, 2007.  Google Scholar

[4]

J. M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl., 92 (2009), 528-545.  doi: 10.1016/j.matpur.2009.05.015.  Google Scholar

[5]

J. M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.  Google Scholar

[6]

A. DoubovaE. Fernández-CaraM. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.  Google Scholar

[7]

E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. NonLinéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[8]

S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of $u_t-\triangle u=u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.  doi: 10.1002/cpa.3160450703.  Google Scholar

[9]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[10]

H. Fujita, On the blowing-up of solutions of the Cauchy problem for $u_t=Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[11]

Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.  doi: 10.1512/iumj.1987.36.36001.  Google Scholar

[12]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.  doi: 10.1002/cpa.3160420607.  Google Scholar

[13]

M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.   Google Scholar

[14]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4.  Google Scholar

[15]

S. KhenissyY. Rébaï and H. Zaag, Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation, Ann. Inst. H. Poincaré, 28 (2011), 1-26.  doi: 10.1016/j.anihpc.2010.09.006.  Google Scholar

[16]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.  Google Scholar

[17]

P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.  Google Scholar

[18]

Z. Ling and Z. Wang, Global existence and finite time blowup for a nonlocal parabolic system, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 371-383.   Google Scholar

[19]

F. Merle and H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann., 316 (2000), 103-137.  doi: 10.1007/s002080050006.  Google Scholar

[20]

Q. TaoH. Gao and Y. Yang, Controllability results for weakly blowingup reaction-diffusion system, Electron. J. Qual. Theory Differ. Equ., 11 (2012), 1-19.   Google Scholar

[21]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh A, 136 (2006), 415-430.  doi: 10.1017/S0308210500004637.  Google Scholar

[22]

C. Wang and S. Zheng, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20 (2007), 1343-1359.  doi: 10.1088/0951-7715/20/6/002.  Google Scholar

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