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Stability analysis for linear heat conduction with memory kernels described by Gamma functions
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 |
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[12] |
T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).
|
[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).
|
[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. |
[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. |
[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. |
[18] |
P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.
|
[19] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).
|
[20] |
F. Rellich, Perturbation Theory of Eigenvalue Problems,, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, (1969).
|
[21] |
J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.
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.
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.
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.
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.
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.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[12] |
T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).
|
[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).
|
[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. |
[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. |
[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. |
[18] |
P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195.
|
[19] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).
|
[20] |
F. Rellich, Perturbation Theory of Eigenvalue Problems,, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, (1969).
|
[21] |
J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.
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.
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.
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.
doi: 10.1007/s002200100589. |
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