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Preface
Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term
1. | Université de Sfax, Faculté des Sciences de Sfax, Département de Mathematiques, BP 1171, Sfax 3000, Tunisia |
2. | Université Sorbonne Paris Nord, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France |
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work [
References:
[1] |
B. Abdelhedi and H. Zaag,
Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term, J. Differential Equations, 272 (2021), 1-45.
doi: 10.1016/j.jde.2020.09.020. |
[2] |
J. M. Ball,
Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser.(2), 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[3] |
M. Berger and R. V. Kohn,
A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.
doi: 10.1002/cpa.3160410606. |
[4] |
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. |
[5] |
M. Chipot and F. B. Weissler,
Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.
doi: 10.1137/0520060. |
[6] |
G. K. Duong and H. Zaag,
Profile of touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci., 29 (2019), 1279-1348.
doi: 10.1142/S0218202519500222. |
[7] |
S. Filippas and R. V. Kohn,
Refined asymptotics for the blow-up of $u_t-\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[8] |
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.
doi: 10.1016/S0294-1449(16)30215-3. |
[9] |
H. Fujita,
On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
|
[10] |
V. A. Galaktionov and J. L. Vázquez,
Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.
doi: 10.1137/0524071. |
[11] |
V. A. Galaktionov and J. L. Vázquez,
Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations, 127 (1996), 1-40.
doi: 10.1006/jdeq.1996.0059. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
M. A. Herrero and J. J. L. Velázquez,
Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.
|
[16] |
M. A. Herrero and J. J. L. Velázquez,
Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450.
|
[17] |
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.
doi: 10.1016/S0294-1449(16)30217-7. |
[18] |
M. A. Herrero and J. J. L. Velázquez,
Comportement générique au voisinage d'un point d'explosion pour des solutions d'équations paraboliques unidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 201-203.
|
[19] |
F. Merle,
Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.
doi: 10.1002/cpa.3160450303. |
[20] |
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.
|
[21] |
F. Merle and H. Zaag,
Stabilité du profil à l'explosion pour les équations du type $u_ t = \Delta u+\vert u\vert ^ {p-1}u$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 345-350.
|
[22] |
V. T. Nguyen,
Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time, Phys. D, 339 (2017), 49-65.
doi: 10.1016/j.physd.2016.09.002. |
[23] |
V. T. Nguyen and H. Zaag,
Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method, Anal. PDE, 9 (2016), 229-257.
doi: 10.2140/apde.2016.9.229. |
[24] |
P. Quittner and P. Souplet, Superlinear parabolic Problems. Blow-up, Global Existence and Steady States, Second Edition. Birkhäuser Advanced Texts, 2019.
doi: 10.1007/978-3-030-18222-9. |
[25] |
P. Souplet, S. Tayachi and F. B. Weissler,
Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45 (1996), 655-682.
doi: 10.1512/iumj.1996.45.1197. |
[26] |
S. Tayachi and H. Zaag,
Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Trans. Amer. Math. Soc., 371 (2019), 5899-5972.
doi: 10.1090/tran/7631. |
[27] |
S. Tayachi and H. Zaag, Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term, Actes du Colloque EDP-Normandie, Le Havre, 21–22, octobre 2015. |
[28] |
F. B. Weissler,
Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224.
doi: 10.1016/0022-0396(84)90081-0. |
[29] |
H. Zaag,
Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 581-622.
doi: 10.1016/S0294-1449(98)80002-4. |
show all references
References:
[1] |
B. Abdelhedi and H. Zaag,
Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term, J. Differential Equations, 272 (2021), 1-45.
doi: 10.1016/j.jde.2020.09.020. |
[2] |
J. M. Ball,
Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser.(2), 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[3] |
M. Berger and R. V. Kohn,
A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.
doi: 10.1002/cpa.3160410606. |
[4] |
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. |
[5] |
M. Chipot and F. B. Weissler,
Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.
doi: 10.1137/0520060. |
[6] |
G. K. Duong and H. Zaag,
Profile of touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci., 29 (2019), 1279-1348.
doi: 10.1142/S0218202519500222. |
[7] |
S. Filippas and R. V. Kohn,
Refined asymptotics for the blow-up of $u_t-\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[8] |
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.
doi: 10.1016/S0294-1449(16)30215-3. |
[9] |
H. Fujita,
On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
|
[10] |
V. A. Galaktionov and J. L. Vázquez,
Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.
doi: 10.1137/0524071. |
[11] |
V. A. Galaktionov and J. L. Vázquez,
Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations, 127 (1996), 1-40.
doi: 10.1006/jdeq.1996.0059. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
M. A. Herrero and J. J. L. Velázquez,
Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.
|
[16] |
M. A. Herrero and J. J. L. Velázquez,
Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450.
|
[17] |
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.
doi: 10.1016/S0294-1449(16)30217-7. |
[18] |
M. A. Herrero and J. J. L. Velázquez,
Comportement générique au voisinage d'un point d'explosion pour des solutions d'équations paraboliques unidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 201-203.
|
[19] |
F. Merle,
Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.
doi: 10.1002/cpa.3160450303. |
[20] |
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.
|
[21] |
F. Merle and H. Zaag,
Stabilité du profil à l'explosion pour les équations du type $u_ t = \Delta u+\vert u\vert ^ {p-1}u$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 345-350.
|
[22] |
V. T. Nguyen,
Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time, Phys. D, 339 (2017), 49-65.
doi: 10.1016/j.physd.2016.09.002. |
[23] |
V. T. Nguyen and H. Zaag,
Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method, Anal. PDE, 9 (2016), 229-257.
doi: 10.2140/apde.2016.9.229. |
[24] |
P. Quittner and P. Souplet, Superlinear parabolic Problems. Blow-up, Global Existence and Steady States, Second Edition. Birkhäuser Advanced Texts, 2019.
doi: 10.1007/978-3-030-18222-9. |
[25] |
P. Souplet, S. Tayachi and F. B. Weissler,
Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45 (1996), 655-682.
doi: 10.1512/iumj.1996.45.1197. |
[26] |
S. Tayachi and H. Zaag,
Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Trans. Amer. Math. Soc., 371 (2019), 5899-5972.
doi: 10.1090/tran/7631. |
[27] |
S. Tayachi and H. Zaag, Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term, Actes du Colloque EDP-Normandie, Le Havre, 21–22, octobre 2015. |
[28] |
F. B. Weissler,
Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224.
doi: 10.1016/0022-0396(84)90081-0. |
[29] |
H. Zaag,
Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 581-622.
doi: 10.1016/S0294-1449(98)80002-4. |
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