• Previous Article
    Stability analysis for linear heat conduction with memory kernels described by Gamma functions
  • DCDS Home
  • This Issue
  • Next Article
    On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models
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

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.
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

[1]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435

[2]

Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025

[3]

José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43

[4]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[5]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[6]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[7]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[8]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[9]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[10]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[11]

Maria Antonietta Farina, Monica Marras, Giuseppe Viglialoro. On explicit lower bounds and blow-up times in a model of chemotaxis. Conference Publications, 2015, 2015 (special) : 409-417. doi: 10.3934/proc.2015.0409

[12]

Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135

[13]

Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435

[14]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715

[15]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[16]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[17]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

[18]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569

[19]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

[20]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]