Using scaling to simulate Finite-time Linda El Alaoui LAGA Universit´e Paris 13

Hatem Zaag CNRS LAGA Universit´e Paris 13

November 27, 2010

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Singularities in ODEs

Finite-time blow-up occurs already in Ordinary Differential Equations (ODEs). For example, the maximal solution of the equation u0 (t) = u(t)2 with u(0) = a ≥ 0, is given by

a . 1 − at As we can see, if a = 0, then the solution is identically zero, and it happens that it is defined on the whole interval [0, ∞). We say that it is a global solution. If a > 0, then the solution exists on the time interval t ∈ [0, a1 ), and “blows-up” at time T (a) = a1 , in the sense that lim u(t) = +∞. u(t) =

t→ a1

Other examples of ODEs leading to blow-up exist in the literature. For more information, see Galaktionov and V´ azquez [3].

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Singularities in PDEs

When dealing with Partial Differential Equations (PDEs), we still encounter the blow-up phenomenon, with maximal solutions existing on a finite time interval. This is the case for the semilinear heat equation ∂t u = ∆u + |u|p−1 u,

(1)

∂tt u = ∆u + |u|p−1 u,

(2)

or the semilinear wave equation

where p > 1, u = u(x, t) with x ∈ RN and t ∈ [0, T ). Though a huge literature is devoted to the first equation (see [3], Khenissy et al. [5] and the references therein), the results about the second equation are more recent (see the review paper by Merle and Zaag [7] 1

and the references therein). Blow-up may be encountered in more realistic situations such as the model of Keller-Segel for chemotaxis, where amoeba concentrate in some localized region of the Petri dish, in finite time. We say in that case that their density blows up in finite time (see Horstmann [4]).

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Blow-up profile

For some PDEs such as (1), when the solutions blows up, we can prove the existence of blow-up profile which is (generically) given by the fact   1 x − x0 − p−1 u(x, t) ∼ (T − t) as t → T, (3) f √ T −t where f (z) = p − 1 + b(p)|z|2


−

1 p−1

with b(p) > 0,

if T is the blow-up time and x0 is the blow-up point. For more details, see [5].

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Numerical simulation and scaling invariance

From (3), we see that the solution becomes large as t approaches T , in a ball of radius √ T − t centered on x0 , which shrinks to {x0 } as t goes to T . Therefore, we understand that simulating a blow-up solution is not an easy task (see Baruch, Fibich and Gavish [1]). Some PDEs are invariant under a rescaling property, defined in the case of (1) by √ 1 λ 7→ uλ (ξ, τ ) = λ p−1 u( λξ, λτ ). This property allows to make a zoom of the solution when (x, t) is close to the singularity (x0 , T ), still keeping the same equation. Numerically speaking, we can make a meshrefinement technique based on the scaling invariance and try to find the blow-up behavior of the solution, as already tested by Berger and Kohn in [2].

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The subject

The aim of this work is to confirm numerically some new blow-up results. The candidate will have to gain knowledge with the results of [8] and references therein. In addition he will have to program with Matlab the mesh-refinement method to simulate various PDEs leading to blow-up. The candidate will consider in particular the semilinear wave equation (2) presented in [8] and/or the following Complex Ginzburg-Landau equation, (see Masmoudi and Zaag [6]). ∂t u = (1 + iβ)∆u + ( + iδ)|u|p−1 u − γu. Contact: Linda El Alaoui: [email protected] Hatem ZAAG: [email protected] 2

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References [1] G. Baruch, G. Fibich, and N. Gavich. Singular standing-ring solutions of nonlinear partial differential equations. Physica D, 239(1968–1983), 2010. [2] M. Berger and R. V. Kohn. A rescaling algorithm for the numerical calculation of blowing-up solutions. Comm. Pure Appl. Math., 41(6):841–863, 1988. [3] V. A. Galaktionov and J. L. V´ azquez. The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst., 8(2):399–433, 2002. Current developments in partial differential equations (Temuco, 1999). [4] D. Horstmann. From 1970 until present : the keller-segel model in chemotaxis and its consequences. Jahresber. Deutsch. Math. Verein., pages 103–165, 2003. [5] S. Khenissy, Y Rebai, and H. Zaag. Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 2010. to appear, doi:10.1016/j.anihpc.2010.09.006. [6] N Masmoudi and H Zaag. Blow-up profile for the complex Ginzburg-Landau equation. J. Funct. Anal., 2008. to appear. [7] F. Merle and H. Zaag. Estimations uniformes `a l’explosion pour les ´equations de la ´ chaleur non lin´eaires et applications. In S´eminaire sur les Equations aux D´eriv´ees ´ Partielles, 1996–1997, pages Exp. No. XIX, 10. Ecole Polytech., Palaiseau, 1997. [8] F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear wave equa´ tion in one space dimension. In S´eminaire sur les Equations aux D´eriv´ees Partielles, ´ 2009–2010, pages Exp. No. 11, 10p. Ecole Polytech., Palaiseau, 2010.

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