Construction and stability of blowup solutions for a non-variational semilinear parabolic system

Ghoul, Tej-Eddine; Nguyen, Van Tien; Zaag, Hatem

doi:10.1016/j.anihpc.2018.01.003

Ghoul, Tej-Eddine ; Nguyen, Van Tien ; Zaag, Hatem

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1577-1630.

Résumé
Abstract

Nous considérons le système parabolique suivant :

{\begin{matrix} \partial_{t} u = Δ u + | v |^{p - 1} v, & \partial_{t} v = μ Δ v + | u |^{q - 1} u, \\ u (\cdot, 0) = u_{0}, & v (\cdot, 0) = v_{0}, \end{matrix}

dans tout l'espace

R^{N}

, où

p, q > 1

μ > 0

. Nous prouvons l'existence d'une donnée initiale telle que la solution associée explose en temps fini

T (u_{0}, v_{0})

simultanément en u et v, et en un unique point a. Plus présicément, nous prouvons que cette solution a le comportement asymptotique suivant :

{\begin{matrix} u (x, t) \sim Γ {[(T - t) (1 + \frac{b | x - a |^{2}}{(T - t) | \log (T - t) |})]}^{- \frac{(p + 1)}{p q - 1}}, \\ v (x, t) \sim γ {[(T - t) (1 + \frac{b | x - a |^{2}}{(T - t) | \log (T - t) |})]}^{- \frac{(q + 1)}{p q - 1}}, \end{matrix}

avec

b = b (p, q, μ) > 0

(Γ, γ) = (Γ (p, q), γ (p, q))

. La construction de cette solution découle de la réduction du problème à un problème de dimension finie et un argument topologique basé sur la théorie de l'index pour conclure. Deux difficultés majeures se sont posées lors de la preuve :

– d'une part, le linearisé autour du profil n'est pas auto-adjoint, même pas pour $μ = 1$ ;
– d'autre part, lorsque $μ \neq 1$ , cela brise toute symmétrie dans le problème.

Dans la dernière section, grâce à une interprétation géométrique des paramètres du problème, nous prouvons la stabilité du comportement asymptotique des solutions construites par rapport à des perturbations dans les données initiales.

We consider the following parabolic system whose nonlinearity has no gradient structure:

{\begin{matrix} \partial_{t} u = Δ u + | v |^{p - 1} v, & \partial_{t} v = μ Δ v + | u |^{q - 1} u, \\ u (\cdot, 0) = u_{0}, & v (\cdot, 0) = v_{0}, \end{matrix}

in the whole space

R^{N}

, where

p, q > 1

and

μ > 0

. We show the existence of initial data such that the corresponding solution to this system blows up in finite time

T (u_{0}, v_{0})

simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

{\begin{matrix} u (x, t) \sim Γ {[(T - t) (1 + \frac{b | x - a |^{2}}{(T - t) | \log (T - t) |})]}^{- \frac{(p + 1)}{p q - 1}}, \\ v (x, t) \sim γ {[(T - t) (1 + \frac{b | x - a |^{2}}{(T - t) | \log (T - t) |})]}^{- \frac{(q + 1)}{p q - 1}}, \end{matrix}

with

b = b (p, q, μ) > 0

and

(Γ, γ) = (Γ (p, q), γ (p, q))

. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case

μ = 1

; and the fact that the case

μ \neq 1

breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

MR Zbl

DOI : 10.1016/j.anihpc.2018.01.003

Classification : 35K50, 35B40, 35K55, 35K57
Mots clés : Blowup solution, Blowup profile, Stability, Semilinear parabolic system

@article{AIHPC_2018__35_6_1577_0,
     author = {Ghoul, Tej-Eddine and Nguyen, Van Tien and Zaag, Hatem},
     title = {Construction and stability of blowup solutions for a non-variational semilinear parabolic system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1577--1630},
     publisher = {Elsevier},
     volume = {35},
     number = {6},
     year = {2018},
     doi = {10.1016/j.anihpc.2018.01.003},
     mrnumber = {3846237},
     zbl = {1394.35222},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.003/}
}

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Ghoul, Tej-Eddine; Nguyen, Van Tien; Zaag, Hatem. Construction and stability of blowup solutions for a non-variational semilinear parabolic system. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1577-1630. doi : 10.1016/j.anihpc.2018.01.003. http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.003/

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