In this paper, we consider the following complex-valued semilinear heat equation
in the whole space , where . We aim at constructing for this equation a complex solution , which blows up in finite time and only at one blowup point , with the following estimates for the final profile
Note that the imaginary part is non-zero and that it blows up also at point . Our method relies on two main arguments: the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. Up to our knowledge, this is the first time where the blowup behavior of the imaginary part is derived in multi-dimension.
The profile for the imaginary part of a blowup solution]Profile for the imaginary part of a blowup solution for a complex-valued semilinear Heat Equation G. K. Duong] \subjclassPrimary: 35K55, 35K57 35K50, 35B44; Secondary: 35K50, 35B40. ††thanks: August 4, 2019
Giao Ky Duong 111 G. K. Duong is supported by the project INSPIRE. This project has received funding from the European Unionâs Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665850.
Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France.
August 4, 2019
In this work, we are interested in the following complex-valued semilinear heat equation
where and , , . Though our results hold only when (see Theorem 1.1 below), we keep in the introduction, in order to broaden the discussion.
In particular, when , model (1.1) evidently becomes
We remark that equation (1.2) is rigidly related to the viscous Constantin-Lax-Majda equation with a viscosity term, which is a one dimensional model for the vorticity equation in fluids. The readers can see more in some of the typical works: Constantin, Lax, Majda [CLM85], Guo, Ninomiya and Yanagida in [GNSY13], Okamoto, Sakajo and Wunsch [OSW08], Sakajo in [Sak03a] and [Sak03b], Schochet [Sch86]. The local Cauchy problem for model (1.1) can be solved (locally in time) in if is integer, by using a fixed-point argument. However, when is not integer, the local Cauchy problem has not been sloved yet, up to our knowledge. This probably comes from the discontinuity of on . In addition to that, let us remark that equation (1.1) has the following family of space independent solutions:
If , this makes a finite number of solutions.
If then the set
is countable and dense in the unit circle of .
This latter case (), is somehow intermediate between the case and the case of the twin PDE
which admits the following family of space independent solutions
for any , which turns to be infinite and covers all the unit circle, after rescaling as in (1.4). In fact, equation (1.5) is certainly much easier than equation (1.1). As a mater of fact, it reduces to the scalar case thanks to a modulation technique, as Filippas and Merle did in [FM95].
Since the Cauchy problem for equation (1.1) is already hard when , and given that we are more interested in the asymptotic blowup behavior, rather than the well-posedness issue, we will focus in our paper on the case . In this case, from the Cauchy theory, the solution of equation (1.1) either exists globally or blows up in finite time. Let us recall that the solution blows up in finite time if and only if it exists for all and
If blows up in finite time , a point is called a blowup point if and only if there exists a sequence as such that
The blowup phenomena occur for evolution equations in general, and in semilinear heat equations in particular. Accordingly, an interesting question is to construct for those equations a solution which blows up in finite time and to describe its blowup behavior. These questions are being studied by many authors in the world. Let us recall some blowup results connected to our equation:
The real case: Bricmont and Kupiainen [BK94] constructed a real positive solution to (1.1) for all , which blows up in finite time , only at the origin and they also gave the profile of the solution such that
where the profile is defined as follows
In addition to that, with a different method, Herrero and Velázquez in [HV92] obtained the same result. Later, in [MZ97] Merle and Zaag simplified the proof of [BK94] and proposed the following two-step method (see also the note [MZ96]):
Reduction of the infinite dimensional problem to a finite dimensional one.
Solution of the finite dimensional problem thanks to a topological argument based on Index theory.
We would like to mention that this method has been successful in various situations such as the work of Tayachi and Zaag [TZ15], and also the works of Ghoul, Nguyen and Zaag in [GNZ16b], [GNZ16c], and [GNZ16a]. In those papers, the considered equations were scale invariant; this property was believed to be essential for the construction. Fortunately, with the work of Ebde and Zaag [EZ11] for the following equation
that belief was proved to be wrong.
where . Though the results of [EZ11] and [NZ17] show that the invariance under dilations of the equation in not necessary in the construction method, we might think that the construction of [EZ11] and [NZ17] works because the authors adopt a perturbative method around the pure power case . If this is true with [EZ11], it is not the case for [NZ17]. In order to totally prove that the construction does not need the invariance by dilation, Duong, Nguyen and Zaag considered in [DNZ18] the following equation
for some where and , where we have no invariance under dilation, not even for the main term on the nonlinearity. They were successful in constructing a stable blowup solution for that equation. Following the above mentioned discussion, that work has to be considered as a breakthrough.
Let us mention that a classification of the blowup behavior of (1.2) was made available by many authors such as Herrero and Velázquez in [HV92] and Velázquez in [Vel92], [Vel93a], [Vel93b] (see also Zaag in [Zaa02] for some refinement). More precisely and just to stay in one space dimension for simplicity, it is proven in [HV92] that if a real solution of (1.1), which blows up in finite time and is a given blowup point, then:
for any where is defined in (1.6).
Or, there exist and such that
for any , where .
The complex case: The blowup question for the complex-valued parabolic equations has been studied intensively by many authors, in particular for the Complex Ginzburg Landau (CGL) equation
This is the case of an ealier work of Zaag in [Zaa98] for equation (1.7) when and small enough. Later, Masmoudi and Zaag in [MZ08] generalized the result of [Zaa98] and constructed a blowup solution for (1.7) with such that the solution satisfies the following
As for equation (1.2), there are many works done in dimension one, such as the work of Guo, Ninomiya, Shimojo and Yanagida, who proved in [GNSY13] the following results (see Theorems 1.2, 1.3 and 1.5 in this work):
(A Fourier- based blowup crieterion). We assume that the Fourier transform of initial data of (1.2) is real and positive, then the solution blows up in finite time.
(A simultaneous blowup criterion in dimension one) If the initial data satisfies
Then, the fact that the blowup set is compact implies that blow up simultaneously.
Assume that satisfy
for some constant . Then, the solution of (1.2), with initial data , blows up at time with . Moreover, the real part blows up only at space infinity and remains bounded.
Still for equation (1.2), Nouaili and Zaag constructed in [NZ15] a complex solution which blows up in finite time only at the origin. Moreover, the solution satisfies the following asymptotic behavior
where and the imaginary part satisfies the following astimate for all
for some and small enough. Note that the real and the imaginary parts blow up simultaneously at the origin. Note also that [NZ15] leaves unanswered the question of the derivation of the profile of the imaginary part, and this is precisely our aim in this paper, not only for equation (1.2), but also for equation (1.1) with .
Before stating our result (see Theorem 1.1 below), we would like to mention some classification results by Harada for blowup solutions of (1.2). As a matter of fact, in [Har16], he classified all blowup solutions of (1.2) in dimension one, under some reasonable assumption (see (1.9), (1.10)), as follows (see Theorems 1.4, 1.5 and 1.6 in that work):
Consider a blowup solution of (1.2) in one dimension space with blowup time and blowup point which satisfies
Assume in addition that
where is defined as follows
and is defined by the following change of variables (also called similarity variables):
Then, one of the following cases occurs
where and is defined in (1.11) and is a rescaled version of the Hermite polynomial of order defined as follows:
Besides that, Harada has also given a profile to the solutions in similarity variables:
There exist such that
Furthemore , he also gave the final blowup profiles
The blowup profile of is given by
Then, from the work of Nouaili and Zaag in [NZ15] and Harada in [Har16] for equation (1.2), we derive that the imaginary part also blows up under some conditions, however, none of them was able to give a global profile (i.e. valid uniformly on , and not just on an expanding ball as in (1) and (1)) for the imaginary part. For that reason, our main motivation in this work is to give a sharp description for the profile of the imaginary part. Our work is considered as an improvement of Nouaili and Zaag in [NZ15] in dimension , which is valid not only for , but also for any . In particular, this is the first time we give the profile for the imiginary part when the solution blows up. More precisely, we have the following Theorem:
Theorem 1.1 (Existence of a blowup solution for (1.1) and a sharp discription of its profile).
For each and , there exists such that for all there exist initial data such that equation (1.1) has a unique solution for all satisfying the following:
The solution blows up in finite time only at the origin. Moreover, it satisfies the following estimates
where is defined in (1.6) and is defined as follows
There exists a complex function such that as uniformly on compact sets of and we have the following asymptotic expansions:
The initial data is given exactly as follows
with , are positive constants fixed large enough, are parametes we fine tune in our proof, and .
We deduce from that blows up only at . In particular, note both the and blow up. However, the blowup speed of is softer than because of the quantity .
Nouaili and Zaag constructed a blowup solution of (1.2) with a less explicit behavior for the imaginary part (see (1.8)). Here, we do better and we obtain the profile the the imaginary part in (1.16) and we also describe the asymptotics of the solution in the neighborhood of the blowup point in (1.19). In fact, this refined behavior comes from a more involved formal approach (see Section 2 below), and more parameters to be fine tuned in initial data (see Definition 3.3 where we need more parameters than in Nouaili and Zaag [NZ15], namely ). Note also that our profile estimates in (1.15) and (1.16) are better than the estimates (1) and (1) by Harada (), in the sense that we have a uniform estimate for whole space , and not just for all for some . Another point: our result hold in space dimensions, unlike the work of Harada in [Har16], which holds only in one space dimension.
As in the case treated by Nouaili and Zaag [NZ15], we suspect this behavior in Theorem 1.1 to be unstable. This is due to the fact that the number of parameters in the initial data we consider below in Definition 3.3 is higher than the dimension of the blowup parameters which is ( for the blowup points and for the blowup time).
Besides that, we can use the technique of Merle [Mer92] to construct a solution which blows up at arbitrary given points. More precisely, we have the following Corollary:
Corollary 1.7 (Blowing up at distinct points).
This paper is organized as follows:
Acknowledgement: I would like to send a huge thank to Professor Hatem ZAAG, my PhD advisor at Paris 13. He led my first steps of the study. Not only did he introduced me to the subject, he also gave me valuable indications on the reductions of a mathematics paper. I have no anymore words to describe this wondeful. Beside that, I also thank my family who encouraged me in my mathematical stidies.
2 Derivation of the profile (formal approach)
In this section, we aim at giveing a formal approach to our problem which helps us to explain how we derive the profile of solution of (1.1) given in Theorem (1.1), as well the asymptotics of the solution.
2.1 Modeling the problem
In this part, we will give definitions and special symbols important for our work and explain how the functions arise as blowup profiles for equation (1.1) as stated in (1.15) and (1.16). Our aim in this section is to give solid (though formal) hints for the existence of a solution to equation (1.1) such that
with and being respectively the real and the imaginary part of and for all Let us introduce the similarity-variables:
Thanks to (2.2), we derive the system satisfied by for all and as follows:
Then note that studying the asymptotics of as is equivalent to studying the asymptotics of in long time. We are first interested in the set of constant solutions of (2.5), denoted by
Introducing our goal because to get
From (2.5), we deduce that satisfy the following system
It is important to study the linear operator and the asymptotics of as which will appear as quadratic.
The properties of :
We observe that the operator plays an important role in our analysis. It is not really difficult to find an analysis space such that is self-adjoint. Indeed, is self-adjoint in , where is the weighted space associated with the weight defined by
and the spectrum set of
Moreover, we can find eigenfunctions which correspond to each eigenvalue :
The one space dimensional case: the eigenfunction corresponding to the eigenvalue is , the rescaled Hermite polynomial given in (1.12). In particular, we have the following orthogonality property:
The higher dimensional case: , the eigenspace , corresponding to the eigenvalue is defined as follows:
As a matter of fact, so we can represent an arbitrary function as follows
where: is the projection of on for any which is defined as follows:
The asymptotic of : The following asymptotics hold:
as (see Lemma A.1 below).
2.2 Inner expansion
In this part, we study the asymptotics of the solution in Moreover, for simplicity we suppose that , and we recall that we aim at constructing a solution of (2.6) such that . Note first that the spectrum of contains two positive eigenvalues , a neutral eigenvalue and all the other ones are strictly negative. So, in the representation of the solution in it is reasonable to think that the part corresponding to the negative spectrum is easily controlled. Imposing a symmetry condition on the solution with respect of , it is reasonable to look for a solution of the form: