Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation

Nguyen, Van Tien; Zaag, Hatem

doi:10.24033/asens.2344

Geodesic

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Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation
[Nombre fini de degrés de liberté du profil raffiné de l'équation semilinéaire de la chaleur]

Nguyen, Van Tien ; Zaag, Hatem

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1241-1282.

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Abstract (VO)
Résumé

We refine the asymptotic behavior of solutions to the semilinear heat equation with Sobolev subcritical power nonlinearity which blow up in some finite time at a blow-up point where the (supposed to be generic) profile holds. In order to obtain this refinement, we have to abandon the explicit profile function as a first order approximation, and take a non explicit function as a first order description of the singular behavior. This non explicit function is in fact a special solution which we construct, obeying some refined prescribed behavior. The construction relies on the reduction of the problem to a finite dimensional one and the use of a topological argument based on index theory to conclude. Surprisingly, the new non explicit profiles which we construct make a family with finite degrees of freedom, namely $\frac{N (N + 1)}{2}$ if $N$ is the dimension of the space.

Nous raffinons le comportement asymptotique des solutions de l'équation semilinéaire de la chaleur avec une non-linéarité sous-critique au sens de Sobolev, qui explosent en temps fini à un point d'explosion avec le profil communément admis comme générique. Pour obtenir ce raffinement, nous devons abandonner le profil explicite comme premier ordre de l'approximation, et prenons à la place une fonction non explicite comme première description du comportement au voisinage de la singularité. Cette fonction non explicite est en fait une solution spécifique que nous construisons, obéissant à un certain comportement prescrit. La construction repose sur la réduction du problème à un problème en dimension finie et l'utilisation d'un argument topologique basé sur la théorie du degré pour conclure. De façon étonnante, on constate que le nouveau profil non explicite produit une famille avec un nombre fini de degrés de liberté, soit $\frac{(N + 1) N}{2}$ si $N$ est la dimension de l'espace.

Publié le : 2017-05-27

DOI : 10.24033/asens.2344

Classification : 35K58, 35K55; 35B40, 35B44.
Keywords: Semilinear heat equations, finite-time blow-up, blow-up profile, stability.
Mots-clés : Équation semilinéaire de la chaleur, explosion en temps fini, profil à l'explosion, stabilité.

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     title = {Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1241--1282},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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     url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2344/}
}

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Nguyen, Van Tien; Zaag, Hatem. Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1241-1282. doi : 10.24033/asens.2344. http://geodesic.mathdoc.fr/articles/10.24033/asens.2344/

Bibliographie
Cité par

Ball, J. M. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., Volume 28 (1977), pp. 473-486 (ISSN: 0033-5606) | MR | Zbl | DOI

Bebernes, J. W.; Bressan, A.; Eberly, D. A description of blowup for the solid fuel ignition model, Indiana Univ. Math. J., Volume 36 (1987), pp. 295-305 (ISSN: 0022-2518) | DOI | MR

Bernoff, A. J.; Bertozzi, A. L.; Witelski, T. P. Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff, J. Statist. Phys., Volume 93 (1998), pp. 725-776 (ISSN: 0022-4715) | MR | Zbl | DOI

Bebernes, J. W.; Kassoy, D. R. A mathematical analysis of blowup for thermal reactions—the spatially nonhomogeneous case, SIAM J. Appl. Math., Volume 40 (1981), pp. 476-484 (ISSN: 0036-1399) | MR | Zbl | DOI

Bricmont, J.; Kupiainen, A. Universality in blow-up for nonlinear heat equations, Nonlinearity, Volume 7 (1994), pp. 539-575 http://stacks.iop.org/0951-7715/7/539 (ISSN: 0951-7715) | MR | Zbl | DOI

Bressan, A. On the asymptotic shape of blow-up, Indiana Univ. Math. J., Volume 39 (1990), pp. 947-960 (ISSN: 0022-2518) | MR | Zbl | DOI

Bressan, A. Stable blow-up patterns, J. Differential Equations, Volume 98 (1992), pp. 57-75 (ISSN: 0022-0396) | MR | Zbl | DOI

Chapman, S. J.; Hunton, B. J.; Ockendon, J. R. Vortices and boundaries, Quart. Appl. Math., Volume 56 (1998), pp. 507-519 (ISSN: 0033-569X) | MR | Zbl | DOI

Côte, R.; Zaag, H. Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., Volume 66 (2013), pp. 1541-1581 (ISSN: 0010-3640) | MR | Zbl | DOI

Ebde, M. A.; Zaag, H. Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term, SeMA J., Volume 55 (2011), pp. 5-21 (ISSN: 1575-9822) | MR | Zbl | DOI

Filippas, S.; Kohn, R. V. Refined asymptotics for the blowup of $u_{t} - Δ u = u^{p}$ , Comm. Pure Appl. Math., Volume 45 (1992), pp. 821-869 (ISSN: 0010-3640) | MR | Zbl | DOI

Fermanian Kammerer, C.; Merle, F.; Zaag, H. Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Ann., Volume 317 (2000), pp. 347-387 (ISSN: 0025-5831) | MR | Zbl | DOI

Fermanian Kammerer, C.; Zaag, H. Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation, Nonlinearity, Volume 13 (2000), pp. 1189-1216 (ISSN: 0951-7715) | MR | Zbl | DOI

Filippas, S.; Liu, W. X. On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 10 (1993), pp. 313-344 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

Fujita, H. On the blowing up of solutions of the Cauchy problem for $u_{t} = Δ u + u^{1 + α}$ , J. Fac. Sci. Univ. Tokyo Sect. I, Volume 13 (1966), pp. 109-124 (ISSN: 0040-8980) | MR | Zbl

Glimm, J.; Jaffe, A., Springer, New York, 1987, 535 pages (ISBN: 0-387-96476-2) | DOI | MR

Giga, Y.; Kohn, R. V. Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., Volume 38 (1985), pp. 297-319 (ISSN: 0010-3640) | MR | Zbl | DOI

Giga, Y.; Kohn, R. V. Characterizing blowup using similarity variables, Indiana Univ. Math. J., Volume 36 (1987), pp. 1-40 (ISSN: 0022-2518) | MR | Zbl | DOI

Giga, Y.; Kohn, R. V. Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., Volume 42 (1989), pp. 845-884 (ISSN: 0010-3640) | MR | Zbl | DOI

Herrero, M. A.; Velázquez, J. J. L. Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, Volume 5 (1992), pp. 973-997 (ISSN: 0893-4983) | MR | Zbl | DOI

Herrero, M. A.; Velázquez, J. J. L. Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 19 (1992), pp. 381-450 (ISSN: 0391-173X) | MR | Zbl | mathdoc-id

Herrero, M. A.; Velázquez, J. J. L. Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 10 (1993), pp. 131-189 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

Kaplan, S. On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., Volume 16 (1963), pp. 305-330 (ISSN: 0010-3640) | MR | Zbl | DOI

Kassoy, D. R.; Poland, J. The thermal explosion confined by a constant temperature boundary. I. The induction period solution, SIAM J. Appl. Math., Volume 39 (1980), pp. 412-430 (ISSN: 0036-1399) | MR | Zbl | DOI

Kassoy, D. R.; Poland, J. The thermal explosion confined by a constant temperature boundary. II. The extremely rapid transient, SIAM J. Appl. Math., Volume 41 (1981), pp. 231-246 (ISSN: 0036-1399) | MR | Zbl | DOI

Levine, H. A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $P u_{t} = - A u + ℱ (u)$ , Arch. Rational Mech. Anal., Volume 51 (1973), pp. 371-386 (ISSN: 0003-9527) | MR | Zbl | DOI

Merle, F.; Raphaël, P.; Rodnianski, I. Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map, C. R. Math. Acad. Sci. Paris, Volume 349 (2011), pp. 279-283 (ISSN: 1631-073X) | MR | Zbl | DOI

Merle, F.; Zaag, H. A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann., Volume 316 (2000), pp. 103-137 (ISSN: 0025-5831) | MR | Zbl | DOI

Masmoudi, N.; Zaag, H. Blow-up profile for the complex Ginzburg-Landau equation, J. Funct. Anal., Volume 255 (2008), pp. 1613-1666 (ISSN: 0022-1236) | MR | Zbl | DOI

Merle, F.; Zaag, H. Reconnection of vortex with the boundary and finite time quenching, Nonlinearity, Volume 10 (1997), pp. 1497-1550 (ISSN: 0951-7715) | MR | Zbl | DOI

Merle, F.; Zaag, H. Stability of the blow-up profile for equations of the type $u_{t} = Δ u + {| u |}^{p - 1} u$ , Duke Math. J., Volume 86 (1997), pp. 143-195 (ISSN: 0012-7094) | MR | Zbl | DOI

Merle, F.; Zaag, H. Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., Volume 51 (1998), pp. 139-196 (ISSN: 0010-3640) | MR | Zbl | 3.0.CO;2-C class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

Merle, F.; Zaag, H. Refined uniform estimates at blow-up and applications for nonlinear heat equations, Geom. Funct. Anal., Volume 8 (1998), pp. 1043-1085 (ISSN: 1016-443X) | MR | Zbl | DOI

Nouaili, N.; Zaag, H. Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation, Comm. Partial Differential Equations, Volume 40 (2015), pp. 1197-1217 (ISSN: 0360-5302) | DOI | MR

Nguyen, V. T.; Zaag, H. Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 16 (2016), pp. 1275-1314 | MR | DOI

Raphaël, P.; Rodnianski, I. Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. IHÉS, Volume 115 (2012), pp. 1-122 (ISSN: 0073-8301) | MR | Zbl | mathdoc-id | DOI

Raphaël, P.; Schweyer, R. Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Comm. Pure Appl. Math., Volume 66 (2013), pp. 414-480 (ISSN: 0010-3640) | MR | Zbl | DOI

Schweyer, R. Type II blow-up for the four dimensional energy critical semi linear heat equation, J. Funct. Anal., Volume 263 (2012), pp. 3922-3983 (ISSN: 0022-1236) | MR | Zbl | DOI

Segur, H.; Kruskal, M. D. Nonexistence of small-amplitude breather solutions in $ϕ^{4}$ theory, Phys. Rev. Lett., Volume 58 (1987), pp. 747-750 (ISSN: 0031-9007) | DOI | MR

Soner, H. M.; Souganidis, P. E. Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Comm. Partial Differential Equations, Volume 18 (1993), pp. 859-894 (ISSN: 0360-5302) | MR | Zbl | DOI

Velázquez, J. J. L. Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations, Volume 17 (1992), pp. 1567-1596 (ISSN: 0360-5302) | MR | Zbl | DOI

Velázquez, J. J. L. Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., Volume 338 (1993), pp. 441-464 (ISSN: 0002-9947) | MR | Zbl | DOI

Velázquez, J. J. L. Estimates on the $(n - 1)$ -dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J., Volume 42 (1993), pp. 445-476 (ISSN: 0022-2518) | MR | Zbl | DOI

Ward, M. J., ICIAM 95 (Hamburg, 1995) (Math. Res.), Volume 87, Akademie Verlag, Berlin, 1996, pp. 435-462 | MR | Zbl

Weissler, F. B. Single point blow-up for a semilinear initial value problem, J. Differential Equations, Volume 55 (1984), pp. 204-224 (ISSN: 0022-0396) | MR | Zbl | DOI

Zaag, H. On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 19 (2002), pp. 505-542 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

Zaag, H. One-dimensional behavior of singular $N$ -dimensional solutions of semilinear heat equations, Comm. Math. Phys., Volume 225 (2002), pp. 523-549 (ISSN: 0010-3616) | MR | Zbl | DOI

Zaag, H. Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J., Volume 133 (2006), pp. 499-525 (ISSN: 0012-7094) | MR | Zbl | DOI

Zaag, H. Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 15 (1998), pp. 581-622 (ISSN: 0294-1449) | MR | Zbl | mathdoc-id | DOI

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