Blow-up phenomena for positive solutions of  semilinear diffusion equations in a half-space:  the influence of the dispersion kernel

Matthieu Alfaro; Otared Kavian

doi:10.5802/afst.1718

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

Matthieu Alfaro ¹; Otared Kavian ²

¹ Université de Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël Salem, Saint-Etienne-du-Rouvray, France & BioSP, INRAE, 84914, Avignon, France
² Université Paris-Saclay (site de Versailles), 45 avenue des États-Unis, 78035 Versailles cedex, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1259-1286.

Abstract
Abstract

We consider the semilinear diffusion equation $\partial_{t} u = A u + {| u |}^{α} u$ in the half-space $ℝ_{+}^{N} : = ℝ^{N - 1} \times (0, + \infty)$ , where $A$ is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data $u (0, x) = u_{0} (x)$ which is nonnegative in the half-space $ℝ_{+}^{N}$ , and the Dirichlet boundary condition $u (t, x^{'}, 0) = 0$ for $x^{'} \in ℝ^{N - 1}$ .

We prove that if the symbol of the operator $A$ is of order ${a | ξ |}^{β}$ near the origin $ξ = 0$ , for some $β \in (0, 2]$ and $a > 0$ , then any positive solution of the semilinear diffusion equation blows up in finite time whenever $0 < α \leq β / (N + 1)$ . On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $α > β / (N + 1)$ . Notice that in the case of the half-space, the exponent $β / (N + 1)$ is smaller than the so-called Fujita exponent $β / N$ in $ℝ^{N}$ .

As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of $ℝ^{N}$ , which are odd in the $x_{N}$ direction (and thus sign changing).

On considère l’équation semi linéaire $\partial_{t} u = A u + {| u |}^{α} u$ dans le demi-espace $ℝ_{+}^{N} : = ℝ^{N - 1} \times (0, + \infty)$ , où $A$ est un opérateur linéaire de diffusion, qui peut être le Laplacien « classique », ou le Laplacien fractionnaire, ou un opérateur non local et non régularisant. L’équation est complétée par une donnée initiale $u (0, x) = u_{0} (x)$ positive dans le demi-espace $ℝ_{+}^{N}$ , et la condition de Dirichlet $u (t, x^{'}, 0) = 0$ pour $x^{'} \in ℝ^{N - 1}$ .

On prouve que si le symbole de l’opérateur $A$ est de l’ordre de ${a | ξ |}^{β}$ au voisinage de l’origine $ξ = 0$ , avec $β \in (0, 2]$ et $a > 0$ , alors toute solution strictement positive explose en temps fini dès que $0 < α \leq β / (N + 1)$ . En revanche, on prouve l’existence de solutions strictement positives globales quand $α > β / (N + 1)$ . Notons que, dans le cas considéré du demi-espace $ℝ_{+}^{N}$ , l’exposant $β / (N + 1)$ est inférieur à l’exposant de Fujita $β / N$ dans l’espace $ℝ^{N}$ .

Ceci nous permet également de résoudre la question de l’explosion pour les solutions de l’équation semi linéaire dans tout l’espace, $ℝ^{N}$ , qui sont impaires dans la direction $x_{N}$ (et qui, donc, changent de signe).

Received: 2020-04-18
Accepted: 2020-10-06
Published online: 2022-12-08

DOI: 10.5802/afst.1718

Classification: 35B40, 35B33, 45K05, 47G20
Keywords: Sign changing solutions, half-space, blow-up solutions, global solutions, Fujita exponent, nonlocal diffusion, dispersal tails

Author's affiliations:

Matthieu Alfaro ¹; Otared Kavian ²

¹ Université de Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël Salem, Saint-Etienne-du-Rouvray, France & BioSP, INRAE, 84914, Avignon, France
² Université Paris-Saclay (site de Versailles), 45 avenue des États-Unis, 78035 Versailles cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AFST_2022_6_31_5_1259_0,
     author = {Matthieu Alfaro and Otared Kavian},
     title = {Blow-up phenomena for positive solutions of  semilinear diffusion equations in a half-space:  the influence of the dispersion kernel},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1259--1286},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 31},
     number = {5},
     year = {2022},
     doi = {10.5802/afst.1718},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1718/}
}

TY  - JOUR
AU  - Matthieu Alfaro
AU  - Otared Kavian
TI  - Blow-up phenomena for positive solutions of  semilinear diffusion equations in a half-space:  the influence of the dispersion kernel
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2022
SP  - 1259
EP  - 1286
VL  - 31
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1718/
DO  - 10.5802/afst.1718
LA  - en
ID  - AFST_2022_6_31_5_1259_0
ER  -

%0 Journal Article
%A Matthieu Alfaro
%A Otared Kavian
%T Blow-up phenomena for positive solutions of  semilinear diffusion equations in a half-space:  the influence of the dispersion kernel
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2022
%P 1259-1286
%V 31
%N 5
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1718/
%R 10.5802/afst.1718
%G en
%F AFST_2022_6_31_5_1259_0

Matthieu Alfaro; Otared Kavian. Blow-up phenomena for positive solutions of  semilinear diffusion equations in a half-space:  the influence of the dispersion kernel. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1259-1286. doi : 10.5802/afst.1718. https://afst.centre-mersenne.org/articles/10.5802/afst.1718/

References
Cited by

[1] Matthieu Alfaro Fujita blow up phenomena and hair trigger effect: the role of dispersal tails, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 6, pp. 1309-1327 | DOI | MR | Zbl

[2] Matthieu Alfaro; Jérôme Coville Propagation phenomena in monostable integro-differential equations: acceleration or not?, J. Differ. Equations, Volume 263 (2017) no. 9, pp. 5727-5758 | DOI | MR | Zbl

[3] Thierry Cazenave; Alain Haraux Introduction aux Problèmes d’Évolution Semi-Linéaires, Mathématiques & Applications (Paris), Éditions Ellipses, 1991

[4] Emmanuel Chasseigne; Manuela Chaves; Julio D. Rossi Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., Volume 86 (2006) no. 3, pp. 271-291 | DOI | MR | Zbl

[5] Jérôme Coville; Louis Dupaigne On a non-local equation arising in population dynamics, Proc. R. Soc. Edinb., Sect. A, Math., Volume 137 (2007) no. 4, pp. 727-755 | DOI | MR | Zbl

[6] Richard Durrett Probability: theory and examples, Duxbury Press, 1996, xiii+503 pages | MR

[7] Hiroshi Fujita 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 | MR | Zbl

[8] Jorge García-Melián; Fernando Quirós Fujita exponents for evolution problems with nonlocal diffusion, J. Evol. Equ., Volume 10 (2010) no. 1, pp. 147-161 | DOI | MR | Zbl

[9] Jimmy Garnier Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., Volume 43 (2011) no. 4, pp. 1955-1974 | DOI | MR | Zbl

[10] Kantaro Hayakawa On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., Volume 49 (1973), pp. 503-505 | MR | Zbl

[11] Otared Kavian Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 4 (1987) no. 5, pp. 423-452 | DOI | Numdam | MR | Zbl

[12] Kusuo Kobayashi; Tunekiti Sirao; Hiroshi Tanaka On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, Volume 29 (1977) no. 3, pp. 407-424 | MR | Zbl

[13] Jan Medlock; Mark Kot Spreading disease: integro-differential equations old and new, Math. Biosci., Volume 184 (2003) no. 2, pp. 201-222 | DOI | MR | Zbl

[14] Sadao Sugitani On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., Volume 12 (1975), pp. 45-51 | MR | Zbl

[15] Fred B. Weissler Existence and nonexistence of global solutions for a semilinear heat equation, Isr. J. Math., Volume 38 (1981) no. 1-2, pp. 29-40 | DOI | MR | Zbl

Cited by Sources: