We consider the following parabolic system whose nonlinearity has no gradient structure:
in the whole space , where and . We show the existence of initial data such that the corresponding solution to this system blows up in finite time simultaneously in and only at one blowup point , according to the following asymptotic dynamics:
with and . 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 ; and the fact that the case 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.
Blowup solutions for a non-variational semilinear parabolic system] Construction and stability of blowup solutions for a non-variational semilinear parabolic system
T. Ghoul, V. T. Nguyen, H. Zaag]
\subjclassPrimary: 35K50, 35B40; Secondary: 35K55, 35K57.
V. T. Nguyen]Tien.Nguyen@nyu.edu
††thanks: H. Zaag is supported by the ERC Advanced Grant no. 291214, BLOWDISOL and by the ANR project ANAÉ ref. ANR-13-BS01-0010-03.
July 3, 2019
Tej-Eddine Ghoul, Van Tien Nguyen and Hatem Zaag
New York University in Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates.
Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France.
In this paper we are concerned with finite time blowup for the semilinear parabolic system:
in the whole space , where
The local Cauchy problem for (1.1) can be solved in . We denote by the maximal existence time of the classical solution of problem (1.1). If , then the solution blows up in finite time in the sense that
In that case, is called the blowup time of the solution. A point is said to be a blowup point of if is not locally bounded near in the sense that for some sequence as . We say that the blowup is simultaneous if
and that it is non-simultaneous if (1.2) does not hold, i.e. if one of the two components remains bounded on . For the system (1.1), it is easy to see that the blowup is always simultaneous. Indeed, if is uniformly bounded on , then the second equation would yield a uniform bound on . More specifically, we say that and blow up simultaneously at the same point if is a blowup point both for and .
In the case of a single equation, namely when system (1.1) is reduced to the scalar equation
the blowup question for equation (1.3) has been studied intensively by many authors and no list can be exhaustive. Let us sketch the main results for the case of the equation (1.3). Considering a blowup solution to (1.3) and its blowup time, we know from Giga and Kohn [GK87] that
for some positive constant , provided that or with . This result was extended by Giga, Matsui and Sasayama [GMS04] for all without assuming the non-negativity of initial data.
The study of the blow-up behavior of solution (1.3) is done through the introduction of similarity variables:
where may or not be a blow-up point of . From (1.3), we see that solves the new equation in :
uniformly on compact sets , where .
This estimate has been refined until the higher order by Filippas, Kohn and Liu [FK92], [FL93], Herrero and Velázquez [HV92a], [HV93], [Vel92], [Vel93b], [Vel93a]. More precisely, they classified the behavior of for bounded, and showed that one of the following cases occurs (up to replacing by if necessary),
either there exists ,
or there exists an even integer and constant not all zero such that
where the homogeneous multilinear form is non-negative.
From Bricmont and Kupiainen [BK94], Herrero and Velázquez [HV93], we have examples of initial data leading to each of the above mentioned scenarios. Moreover, Herrero and Velázquez [HV92b] proved that the asymptotic behavior (1.6) is generic in the one dimensional case, and they announced the same for the higher dimensional case, but they never published it. Note also that the asymptotic profile described in (1.6) with has been proved to be stable with respect to perturbations in the initial data or the nonlinearity by Merle and Zaag in [MZ97] (see also Fermanian, Merle and Zaag [FMZ00], [FZ00], Nguyen and Zaag [NZ16b] for other proofs of the stability).
As for system (1.1), much less result is known, in particular in the study of the asymptotic behavior of the solution near singularities. As far as we know, the only available results concerning the blowup behavior are due to Andreucci, Herrero and Velázquez [AHV97] and Zaag [Zaa01] where the system (1.1) is considered with .
and both functions and must blow up simultaneously. See also [EH93] for the case of boundary value problems.
In [AHV97], the authors proved that if
then every positive solution of (1.1) exhibits the Type I blowup, namely that there exists some constant such that
where solves the following ODE system
whose solution is explicitly given by
where defined by
The estimate (1.8) has also been proved by Caristi and Mitidieri [CM94] in a ball under assumptions on and different from (1.7). See also Deng [Den96], Fila and Souplet [FS01] for other results relative to estimate (1.8).
The study of blowup solutions for system (1.1) is done through the introduction of the following similarity variables for all ( may or may not be a blowup point):
From (1.1), (or for simplicity) satisfy the following system: for all ,
Assuming (1.8) holds, namely that
and considering a blowup point of , we know from [AHV97] that (remind that we are considering the case when )
either goes to exponentially fast,
or there exists such that after an orthogonal change of space coordinates and up to replacing by if necessary,
where is given by (1.9) and
and the convergence takes place in for any .
In the first case, we have other profiles, some of them are different from those occurring in the scalar case of (1.3), see Theorem 3 and 4 in [AHV97] for more details. Note that the value of given in (1.13) was not precised in [AHV97], but we can justify it by explicit computations as in [AHV97].
Beside the results already cited, let us mention to the work by Zaag [Zaa01] where the author obtained a Liouville theorem for system (1.1) that improves the results of [AHV97]. Based on this theorem, he was able to derive sharp estimates of asymptotic behaviors as well as a localization property for blowup solutions of (1.1). For other aspects of system (1.1), especially concerning the blowup set, see Friedman and Giga [FG87], Mahmoudi, Souplet and Tayachi [MST15], Souplet [Sou09].
In this paper, we want to study the profile of the solution of (1.1) near blowup, and the stability of such behavior with respect to perturbations in initial data. More precisely, we prove the following result.
Theorem 1.1 (Existence of a blow-up solution for system (1.1) with the description of its profile).
Consider . There exists such that system (1.1) has a solution defined on such that:
and blow up in finite time simultaneously at one blowup point and only there.
There holds that
with given by (1.9) and
for all , with
The initial data for which system (1.1) has a solution blowing up in finite time at only one blowup point and verifying (1.14) is given by formula (4.2), which is expressed in the original variables as follows:
where is given by (1.9), and are positive constants fixed sufficiently large, and are parameters in our proof, and with and on .
Note that in the setting (1.10), the terms and turn to be exponentially small. Therefore, a perturbation of our method works although we need in addition some parabolic regularity results in order to handle the nonlinear gradient terms (see [EZ11] and [TZ16] for such parabolic regularity techniques). For simplicity, we only give the proof when the nonlinear terms are exactly given by and .
Our method can be naturally extended to the system of equations of the form
where and for . Up to a complication in parameters, we suspect that our analysis yields the existence of a solution for (1.17) which blows up in finite time only at one blowup point and satisfies the asymptotic behavior: for ,
where , is given by
As a consequence of our techniques, we show the stability of the constructed solution with respect to perturbations in initial data. More precisely, we have the following result.
Theorem 1.8 (Stability of the blowup profile (1.15)).
Let be the initial data of system (1.1) such that the corresponding solution blows up in finite time at only one blowup point and satisfies (1.14) with and . Then, there exists a neighborhood of in such that for any , system (1.1) has a unique solution with initial data which blows up in finite time at only one blowup point . Moreover, parts and of Theorem 1.1 are satisfied, and
as in .
With the stability result, we expect that the blowup profile (1.15) is generic, i.e. there exists an open, everywhere dense set of initial data whose corresponding solution to (1.1) either converges to the steady state (1.9) or blows up in finite time at a single point, according the asymptotic behavior (1.14). In particular, we suspect that a numerical simulation of (1.1) should lead to the profile (1.15). Up to our knowledge, the only available proof for the genericity is given by Herrero and Velázquez [HV92b] for the case of equation (1.3) in one-dimensional case. As in [HV92b], a first step towards the genericity of the profile (1.15) is to classify all possible asymptotic behaviors of the blowup solution of (1.1) which was established in [AHV97] (see also [Zaa01]) in the case when .
Let us now give the main idea of the proof of Theorem 1.1. Our proof uses some ideas developed by Merle and Zaag [MZ97] and Bricmont and Kupiainen [BK94] for the equation (1.3). This kind of method has been proved to be successful for various situations including parabolic and hyperbolic equations. For the parabolic equations, we would like to mention the work by Masmoudi and Zaag [MZ08] (see also the earlier work by Zaag [Zaa98]) for the complex Ginzburg-Landau equation with no gradient structure,
where , , satisfying
There are also the works by Nguyen and Zaag [NZ16a] for a logarithmically perturbed equation of (1.3) (see also Ebde and Zaag [EZ11] for a weakly perturbed version of (1.3)), by Nouaili and Zaag [NZ15] for a non-variational complex-valued semilinear heat equation, or the recent work by Tayachi and Zaag [TZ16] for the nonlinear heat equation with a critical power nonlinear gradient term,
When , this equation is reduced to
which is studied in [GNZ16]. There are also the cases for the construction of multi-solitons for the semilinear wave equation in one space dimension by Côte and Zaag [CZ13], for the wave maps by Raphaël and Rodnianski [RR12], for the Schrödinger maps by Merle, Raphaël and Rodnianski [MRR11], for the critical harmonic heat flow by Schweyer [Sch12] and for the two-dimensional Keller-Segel equation by Raphaël and Schweyer [RS14], Ghoul and Masmoudi [GM16].
One may think that the method used in [MZ97] and [BK94] should work the same for system (1.1) perhaps with some technical complications. This is not the case, since the fact that breaks any symmetry in the problem, and makes the diffusion operator associated to (1.1) not self-adjoint. In other words, the method we present here is not based on a simple perturbation of the equation (1.3) treated in [MZ97] and [BK94]. More precisely, our proof relies on the understanding of the dynamics of the selfsimilar version (1.11) around the profile (1.15). In the setting (1.10), constructing a solution for (1.1) satisfying (1.14) is equivalent to construct a solution for (1.11) such that
Satisfying such a property is guaranteed by a condition that belongs to some set which shrinks to as (see Definition 4.1 below for an example). Since the linearization of system (1.11) around the profile gives positive modes,
zero modes, and an infinite dimensional negative part (see Lemma 3.2 and Remark 3.3), we can use the method of [MZ97] and [BK94] which relies on two arguments:
- The use of the bounding effect of the heat kernel (see Proposition 5.3) to reduce the problem of the control of in to the control of its positive modes. Note that the linearized operator around the profile, that is defined in (3.4) and (3.5), is not self-adjoint. This is one of the major difficulties arising in this paper.
- The control of the positive modes thanks to a topological argument based on the index theory.
In addition to the difficulties concerning the linearized operator mentioned above, we also deal with the number of parameters in the problem (, and ) leading to actual complications in the analysis. According to the general framework of [MZ97], some crucial modifications are needed. In particular, we have to overcome the following challenges:
Finding the profile is not obvious, in particular in determining the values of given by (1.13), which is crucial in many algebraic identities in the rigorous analysis. See Section 2 for a formal analysis to justify such a profile. We emphasize that the formal approach actually gives us an appreciated profile to be linearized around (see (2.8) and (2.9)).
Defining the shrinking set (see Definition 4.1) to trap the solution. Note that our definition of is different from that of [MZ97]. Here, we follow the idea of [MZ08] to find out such an appreciated definition for . In particular, it comes from many relations in our proof, one of them is related to the dynamics of the linearized problem stated in Proposition 5.3.
A good understanding of the dynamics of the linearized operator of equation (3.3) around the appreciated profile given in (2.8) and (2.9) is needed, according to the definition of the shrinking set . Because the behavior of the potential defined in (3.6) inside and outside the blowup region is different, the effect of the linearized operator is therefore considered accordingly to this region. Outside the blowup region, the linear operator behaves as one with fully negative spectrum, which greatly simplifies the analysis in this region (see Section 5.2.4). Inside the blowup region, the potential is considered as a perturbation of the effect of , therefore, a good study of the spectral properties of is needed. Note that the linear operator is not diagonal, but it is diagonalized (see Lemma 3.2). Using this diagonalization, we then define the projection on subspaces of the spectrum of (see Lemma 3.4).
For the proof of single blowup point (part of Theorem 1.1), we use part and an extended result of [GK89] that is called no blow-up under some threshold criterion for parabolic inequalities (see Proposition 4.7). The derivation of the final profile (part of Theorem 1.1) follows from part by using the same argument as [Zaa98] and [Mer92].
The rest of the paper is organized as follows:
- In Section 2, we first explain formally how we obtain the profile and give a suggestion for an appreciated profile to be linearized around.
- In Section 3, we give a formulation of the problem in order to justify the formal argument. We also give the spectral properties of the linear operator as well as the definition of the projection on eigenspaces of .
- In Section 4, we give all the argument of the proof of Theorem 1.1 assuming technical results, which are left to the next section.
- Section 5 is central in our analysis. It is devoted to the study of the dynamics of the linearized problem. In particular, we prove Proposition 5.3 from which we reduce the problem to a finite dimensional one.
- In Section 6, we give the proof of Theorem 1.8. Since its proof is a consequence of the existence proof (part of Theorem 1.1), thanks to a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we only explain the main ideas of the proof there.
2 A formal analysis.
In this section, we give a formal analysis leading to the asymptotic behaviors described in (1.14) by means of matching asymptotic. For simplicity, we shall look for , a positive solution of (1.1) in one dimensional case. By the translation invariant in space, we assume that blows up in finite time at the origin, and write instead of for short. From the transformation (1.10), the behavior (1.14) is equivalent to showing that
We use here the method of [MZ08] treated for the complex Ginzburg-Landau equation, which was slightly adapted from the method of Berger and Kohn [BK88] for equation (1.3). Following the approach of [MZ08], we try to search formally for system (1.11) a regular solution of the form
and for ,
Solving system (2.3) equipped with data at zero
for some integration constant , and is given by (1.9). Since we want to be regular, we impose the condition
Using the definition of given in (1.9), one can simplify this system and obtain
Let us now expand in power of , namely
Injecting these forms into (2.4) and expanding and in powers of , we obtain at the order ,
A straightforward computation gives , hence,
For the terms of order in the expansion of and , we have
Multiplying the second equation by , then combining with the first equation, we find that the coefficients of and disappear leading to
In conclusion, we obtain the following profile for :
with given in (1.16).
3 Formulation of the problem.
In this section, we give a formulation for the proof of Theorem 1.1. We will only give the proof in one dimensional case () for simplicity, but the proof remains the same for higher dimensions . We want to prove the existence of suitable initial data so that the corresponding solution of system (1.1) blows up in finite time only at one point and verifies (1.14). From translation invariance of equation (1.1), we may assume that . Through the transformation (1.10), we want to find and such that the solution of system (1.11) with initial data satisfies