A blowup solution of a complex semi-linear Heat equation with an irrational power
In this paper, we consider the following semi-linear complex heat equation
in with an arbitrary power . In particular, can be non integer and even irrational, unlike our previous work [Duoed], dedicated to the integer case. We construct for this equation a complex solution , which blows up in finite time and only at one blowup point Moreover, we also describe the asymptotics of the solution by the following final profiles:
In addition to that, since we also have and as the blowup in the imaginary part shows a new phenomenon unkown for the standard heat equation in the real case: a non constant sign near the singularity, with the existence of a vanishing surface for the imaginary part, shrinking to the origin. In our work, we have succeeded to extend for any power where the non linear term is not continuous if is not integer. In particular, the solution which we have constructed has a positive real part. We study our equation as a system of the real part and the imaginary part and . Our work 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.
G. K. Duong] \subjclassPrimary: 35K50, 35B40; Secondary: 35K55, 35K57. ††thanks: July 3, 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.
July 3, 2019
1.1 Ealier work
In this work, we are interested in the following complex-valued semilinear heat equation
where and , , .
Typically, when , model (1.1) becomes the following
This model is connected to the viscous Constantin-Lax-Majda equation with a viscosity term, which is a one dimensional model for the vorticity equation in fluids. For more details, the readers are addressed to the following works: Constantin, Lax, Majda [CLM85], Guo, Ninomiya and Yanagida in [GNSY13], Okamoto, Sakajo and Wunsch [OSW08], Sakajo in [Sak03a] and [Sak03b], Schochet [Sch86]. In [Duoed], we treated the case . Indeed, handling the nonlinear term in this case is much easier. In the present paper, we do better, and give a proof which holds also in the case . The local Cauchy problem for model (1.1) can be solved in when is integer, thanks to a fixed-point argument. However, if is not an integer number, then, the local Cauchy problem has not been solved yet, up to our knowledge. In my point of view, this probably comes from the discontinuity of on and this challenge is also one of the main difficulties of the paper. As a matter of fact, we solve the Cauchy problem in Appendix A for data with for some . Accordingly, a maximal solution may be global in time or may exist only for for some In that case, we have to options:
Either as .
Or as .
In this paper, we are interested in the case which is referred to as finite-time blow-up in the sequel.
A blowup solution is called Type I if
Otherwise, the solution is called Type II.
In addtion to that, is called the bolwup time of and a point is called a blowup point if and only if there exists a sequence as such that
In our work, we are interested in constructing a blowup solution of (1.1) which is Type I. Let us quickly mention some typical works for this situation (for more details, see the introduction of [Duoed], treated the integer case).
For the real case: Bricmont and Kupiainen [BK94] constructed a real positive solution to the following equation
which blows up in finite time , only at the origin and they have derived the profile of the solution such that
where the profile is defined as follows
In addition to that, in [HV92], Herrero and Velázquez derived the same result with a different method. Particularly, in [MZ97b], Merle and Zaag gave a proof which is simpler than the one in [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.
Moreover, they also proved the stability of the blowup profile for (1.3). In addition to that, we would like to mention that this method has been successful in various situations such as the work of Ebde and Zaag [EZ11], Tayachi and Zaag [TZ15], and also the works of Ghoul, Nguyen and Zaag in [GNZ16a], [GNZ16b] (with a gradient term) and [GNZ16c]. We would like to mention also the work of Nguyen and Zaag in [NZ17], who considered the following quasi-critical double source equation
and also the work of Duong, Nguyen and Zaag in [DNZ18], who considered the following non scale invariant equation
For the complex case: The blowup problem 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 (1.5) when and small enough. Later, Masmoudi and Zaag in [MZ08] generalized the result of [Zaa98] and constructed a blowup solution for (1.5) with a super critical condition Recently, Nouaili and Zaag in [NZar] has constructed a blowup solution for (1.5), in the critical case where and ).
In addtiion to that, there are many works for equation (1.1) or (1.2), such as the work of Nouaili and Zaag in [NZ15a] for equation (1.2), who constructed a complex solution which blows up in finite time only at the origin. Note that the real and the imaginary parts blow up simultaneously. Note also that [NZ15a] 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 . We would like to mention also some classification results, proven by Harada in [Har16], for blowup solutions of (1.2) which satisfy some reasonable assumptions. In particular, in that works, we are able to derive a sharp blowup profile for the imaginary part of the solution. In 2018, in [Duoed], we handled equation (1.1) when is an integer.
1.2 Statement of the result
Although, in [Duoed], we believe we made an important achievement, we acknowledge that we left unanswered the case where and . From the limitation of the above works, it motivates us to study model (1.1) in general even for irrational . The following theorem is considered as a generalization of [Duoed] for all .
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 on satisfying the following:
The solution blows up in finite time only at the origin and on . Moreover, it satisfies the following
where is defined in (1.4) and is defined as follows
There exists a complex function in such that as uniformly on compact sets of and we have the following asymptotic expansions:
We remark that the condition on the parameter comes from the definition of the set (see in item of Definition 3.1), Proposition 4.1 and Lemma B.3. Indeed, this condition ensures that the projections of the quadratic term on the negative and outer parts are smaller than the conditions in . Then, we can conclude (4.6) and (4.8) by using Lemma B.3 and definition of .
We can show that the constructed solution in the above Theorem satisfies the following asymptotics:
as , (see (3.37) and (3.38)). Therefore, we deduce that blows up at time only at . Note that, the real and imaginary parts simultaneously blow up. Moreover, from item of Theorem 1.1, the blowup speed of is softer than because of the quantity (see (1.9) and (1.10)).
Remark 1.4 (A strong singularity of the imaginary part).
We observe from (1.10) and (1.12) that there is a strong sigularity at the neighborhood of as when close to we have which becomes large and positive as , however, we always have as Thus the imaginary part has no constant sign near the singularity. In particular, if is near , there exists in and as such that at time vanishes on some surface close to the sphere of center and radius . Therefore, we don’t have as . This non constant property for the imaginary part is very surprising to us. In the frame work of semilinear heat equation, such a property can be encountered for phase invariant complex equations, such as the Complex Ginzburg-Landau (CGL) equation (see Zaag in [Zaa98], Masmoudi and Zaag in [MZ08], Nouaili-Zaag [NZar]). As for complex parabolic equation with no phase invariance, this is the first time such a sign change in available, up to our knowledge. We would like to mention that such a sign change near the singularity was already observed for the semilinear wave equation non characteristic blowup point (see Merle and Zaag in [MZ12a], [MZ12b]) and Côte and Zaag in [CZ13].
In Theorem (1.1), the initial data is given exactly as follows
with , are positive constants fixed large enough, are parametes we fine tune in our proof, and and for all and is given in (3.32) and related to the final profile given in item of Theorem 1.1. Note that when we took in [Duoed] a simpler expression for initial data, not in involving the final profile nor the term in . In particular, adding this term in our idea to ensure that the real part of the solution straps positive.
As in the case treated by Nouaili and Zaag [NZ15a], and we also mentioned we suspect the 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 3.4 (see also Remark 1.6 above) 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:
1.3 The strategy of the proof
From the singularity of the nonlinear term () when , we can not apply the techniques we used in [Duoed] when (also used in [MZ97b], [NZ15b], …). We need to modify this method. We see that, although our nonlinear term in not continuous in general, it is continuous in the following half plane
Relying on this property, our problem will be derived by using the techniques which were used in [Duoed] and the fine control of the positivity of the real part. We treat this challenge by relying on the ideas of the work of Merle and Zaag in [MZ97a] (or the work of Ghoul, Nguyen and Zaag in [GNZ16b] with inherited ideas from [MZ97a]) for the construction of the initial data. We define a shrinking set (see in Definition 3.1) which allows a very fine control of the positivity of the real part. More precisely, it is procceed to control our solution on three regions and which are given in subsection 3.2 and which we recall here:
- called the intermediate region, i.e : In this region, we will control our solution by control the rescaled function of (see more (3.20)) to approach (see in (3.25)), by using a classical parabolic estimates. Roughly speaking, we control our solution as a perturbation of the final profiles for given in (1.9) and (1.10).
- called the regular region, i.e : In this region, we control the solution as a perturbation of initial data (). Indeed, will be chosen small by the end of the proof.
Fixing some constants involved in the definition , we can prove that our problem will be solved by the control of the solution in . Moreover, we prove via a priori estimates in the different regions that the control is reduced to the control of a finite dimensional component of the solution. Finally, we may apply the techniques in [Duoed] to get our conclusion.
We will organize our paper as follows:
- In Section 2: We give a formal approach to explain how the profiles we have in Theorem 1.1 appear naturally. Moreover, we also approach our problem through two independant directions: Inner expansion and Outer expansion, in order to show that our profiles are reasonable.
2 Derivation of the profile (formal approach)
In this section, we would like to give a formal approach to our problem which explains how we derive the profiles for the solution of equation (1.1) given in Theorem (1.1), as well the asymptotics of the solution. In particular, we would like to mention that the main difference between the case and resides in the way we handle the nonlinear term . For that reason, we will give a lot of care for the estimates involving the nonlinear term, and go quickly while giving estimates related to other terms, kindly refering the reader to [Duoed] where the case was treated.
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 the solution of equation (1.1) as stated in (1.6) and (1.7). Our aim in this section is to give solid (though formal) hints for the existence of a solution to equation (1.1) such that
noting that in this case, we already have a difficulty to properly define the nonlinear term as a continuous term. In order to overcome this difficulty, we will restrict ourselves to the case where
Our main challenge in this work will be to show that (2.2) is propagated by the flow, at least for the initial data we are suggesting (see Definition 3.4 below). Therefore, under the condition (2.2), by using equation (1.1), we deduce that solve:
where and for all we have
with and is defined as follows:
Note that, in the case where , we had the following simple expressions for
Of course, both expressions (2.4) and (2.6) coincide when . In fact, we will follow our strategy in [Duoed] for and focus mainly on how we handle the nonlinear terms, since we have a different expression when
Let us introduce the similarity-variables for as follows:
By using (2.3), we obtain a 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 (2.8), denoted by
Introducing our goal because to get
From (2.8), 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 easy to find an analysis space such that is self-adjoint. Indeed, is self-adjoint in , where is the weighted space associated to 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 as follows
In particular, we have the following orthogonality property:
The higher dimensional case: , the eigenspace , corresponding to the eigenvalue is defined as follows:
Accordingly, we can represent an arbitrary function as follows
where: is the projection of on for any which is defined as follows:
The asymptotics of : The following asymptotics hold:
as Note that although we have here the expressions of the nonlinear terms which are different from the case (see (2.4) and (2.6)), the expressions coincide, since we have in all case (see Lemma B.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 (2.9) 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:
From the assumption that , we see that as . We see also that we can understand the asymptotics of the solution in from the study of the asymptotics of We now project equations (2.9) on and Using the asymptotics of in (2.19) and (2.20), we get the following ODEs for
as . Similarly as in [Duoed], where we have we obtain the following asymptotics of
2.3 Outer expansion
As for the inner expansion, we here assume that . We see that asymptotics (2.27) and (2.28) can not give us a shape, since they hold uniformly on compact sets (where we only see the constant solutio ) and not in larger sets. Fortunately, we observe from (2.27) and (2.28) that the profile may be based on the following variable:
This motivates us to look for solutions of the form:
Note that, our purpose is to construct a solution where the real part is positive. So, it is reasonnable to assume that and for all . Besides that, we also assume that are smooth and have bounded derivatives. From the definitions of given in (2.4), we have the following