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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1088/0951-7715/7/2/011.  Google Scholar

[2]

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

[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.  doi: 10.1002/cpa.3160450703.  Google Scholar

[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.   Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1512/iumj.2004.53.2401.  Google Scholar

[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.  doi: 10.1142/S0219891612500063.  Google Scholar

[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.  doi: 10.1088/0951-7715/25/9/2759.  Google Scholar

[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.   Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar

[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, (1968).   Google Scholar

[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.  doi: 10.1215/S0012-7094-97-08605-1.  Google Scholar

[15]

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

[16]

V. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations,, preprint, ().   Google Scholar

[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.  doi: 10.1090/S0002-9947-10-04902-0.  Google Scholar

[18]

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

[19]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

[20]

F. Rellich, Perturbation Theory of Eigenvalue Problems,, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, (1969).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

show all references

References:
[1]

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

[2]

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

[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.  doi: 10.1002/cpa.3160450703.  Google Scholar

[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.   Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1512/iumj.2004.53.2401.  Google Scholar

[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.  doi: 10.1142/S0219891612500063.  Google Scholar

[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.  doi: 10.1088/0951-7715/25/9/2759.  Google Scholar

[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.   Google Scholar

[12]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar

[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, (1968).   Google Scholar

[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.  doi: 10.1215/S0012-7094-97-08605-1.  Google Scholar

[15]

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

[16]

V. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations,, preprint, ().   Google Scholar

[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.  doi: 10.1090/S0002-9947-10-04902-0.  Google Scholar

[18]

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

[19]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

[20]

F. Rellich, Perturbation Theory of Eigenvalue Problems,, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, (1969).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

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