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August  2015, 35(8): 3585-3626. doi: 10.3934/dcds.2015.35.3585

On the blow-up results for a class of strongly perturbed semilinear heat equations

Van Tien Nguyen 1,

1. 

Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France

Received  May 2014 Revised  December 2014 Published  February 2015

We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.
Keywords: blow-up profile, lower order perturbations., semilinear heat equation, asymptotic behavior, Finite-time blow-up.
Mathematics Subject Classification: Primary: 35K58, 35K55; Secondary: 35B40, 35B4.
Citation: Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
References:
[1]

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

[2]

T. Cazenave and P. L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations, 9 (1984), 955-978. doi: 10.1080/03605308408820353.

[3]

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

[4]

S. Filippas and W. X. Liu, On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 313-344.

[5]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[6]

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.

[7]

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.

[8]

Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514. doi: 10.1512/iumj.2004.53.2401.

[9]

M. A. Hamza and H. Zaag, Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case, J. Hyperbolic Differ. Equ., 9 (2012), 195-221. doi: 10.1142/S0219891612500063.

[10]

M. A. Hamza and H. Zaag, A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations, Nonlinearity, 25 (2012), 2759-2773. doi: 10.1088/0951-7715/25/9/2759.

[11]

M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 131-189.

[12]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[14]

F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u + \| u \|^{p-1}u$, Duke Math. J., 86 (1997), 143-195. doi: 10.1215/S0012-7094-97-08605-1.

[15]

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.

[16]

V. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, preprint, arXiv:1406.5233.

[17]

N. Nouaili and H. Zaag, A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up, Trans. Amer. Math. Soc., 362 (2010), 3391-3434. doi: 10.1090/S0002-9947-10-04902-0.

[18]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203.

[19]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Blow-up, global existence and steady states.

[20]

F. Rellich, Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York, 1969.

[21]

J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596. doi: 10.1080/03605309208820896.

[22]

J. J. L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., 338 (1993), 441-464. doi: 10.1090/S0002-9947-1993-1134760-2.

[23]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[24]

H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549. doi: 10.1007/s002200100589.

show all references

References:
[1]

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

[2]

T. Cazenave and P. L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations, 9 (1984), 955-978. doi: 10.1080/03605308408820353.

[3]

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

[4]

S. Filippas and W. X. Liu, On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 313-344.

[5]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[6]

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.

[7]

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.

[8]

Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514. doi: 10.1512/iumj.2004.53.2401.

[9]

M. A. Hamza and H. Zaag, Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case, J. Hyperbolic Differ. Equ., 9 (2012), 195-221. doi: 10.1142/S0219891612500063.

[10]

M. A. Hamza and H. Zaag, A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations, Nonlinearity, 25 (2012), 2759-2773. doi: 10.1088/0951-7715/25/9/2759.

[11]

M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 131-189.

[12]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[14]

F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u + \| u \|^{p-1}u$, Duke Math. J., 86 (1997), 143-195. doi: 10.1215/S0012-7094-97-08605-1.

[15]

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.

[16]

V. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, preprint, arXiv:1406.5233.

[17]

N. Nouaili and H. Zaag, A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up, Trans. Amer. Math. Soc., 362 (2010), 3391-3434. doi: 10.1090/S0002-9947-10-04902-0.

[18]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203.

[19]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Blow-up, global existence and steady states.

[20]

F. Rellich, Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York, 1969.

[21]

J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596. doi: 10.1080/03605309208820896.

[22]

J. J. L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., 338 (1993), 441-464. doi: 10.1090/S0002-9947-1993-1134760-2.

[23]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[24]

H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549. doi: 10.1007/s002200100589.

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