Construction of two-bubble blow-up solutions for the mass-critical gKdV equations

📝 Abstract

For the mass-critical generalized Korteweg-de Vries equation,

$$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$

We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.

💡 Analysis

For the mass-critical generalized Korteweg-de Vries equation,

$$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$

We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.

📄 Content

1. Introduction 1.1. Main result. In this article, we consider the dynamics of two-bubbles for the mass-critical generalized Korteweg-de Vries (gKdV) equation,

x u + u 5 = 0, (t, x) ∈ [0, ∞) × R.

(1.1)

The mass M (u) and energy E(u) are formally conserved by the flow of (1.1) where

Recall that, the Cauchy problem for equation (1.1) is locally well-posed in the energy space H 1 (R) (see Kenig-Ponce-Vega [17,18]). More precisely, for any initial data u 0 ∈ H 1 (R), there exists a unique (in a certain class) maximal solution u of (1.1) in C [0, T ); H 1 (R) . For this Cauchy problem, the following blow-up criterion holds: T < ∞ =⇒ ∥∇u(t)∥ L 2 = ∞ as t ↑ T.

(1.2) For such H 1 solution u, the mass M (u) and energy E(u) are conserved on [0, T ). Recall also that, if u is a solution of (1.1) then for any λ > 0, the function

is also a solution of (1.1). In addition, this scaling symmetry keeps the L 2 -norm invariant so that the problem is mass-critical.

We recall the family of solitary wave solutions of (1.1). Let Q(x) = 3sech 2 (2x) 1 4 be the unique (up to translation) positive H 1 solution of the equation

Then, for any (λ 0 , x 0 ) ∈ (0, ∞) × R, the function

is a solution of (1.1) called solitary wave or soliton. Following a variational argument [47], the conservation laws and the blow-up criterion (1.2) imply that any initial data u 0 ∈ H 1 (R) with subcritical mass, i.e. satisfying ∥u 0 ∥ L 2 < ∥Q∥ L 2 , generates a global and bounded solution in H 1 (R). To go beyond the threshold mass ∥Q∥ L 2 , it is natural to restrict to solutions with small supercritical mass, i.e. satisfying ∥Q∥ L 2 ≤ ∥u 0 ∥ L 2 < (1 + δ)∥Q∥ L 2 , for 0 < δ ≪ 1.

(1.

The study of singularity formation for such case was first developed in a series of works by Martel-Merle [28,29,30,31] and Merle [39] via rigidity Liouville-type property, monotonicity formula and localized Virial estimate. Later on, Martel-Merle-Raphaël [33,34,35] revisited the blow-up analysis for (1.1), giving a comprehensive description of the flow near the soliton and thereby completing the results in the above-mentioned works. See Section 1.2 for more discussion.

For the multi-bubble dynamics of (1.1), it is known from Combet-Martel [5] that finite time blow-up solutions exist with an arbitrary number of bubbles and any choice of signs. In this article, we construct the first example of infinite time blowup solution of (1.1) related to the strong interactions of two bubbles with opposite signs. Our main result is formulated as follows.

Theorem 1.1. There exists a global-in-time solution u ∈ C [0, ∞); H 1 of (1.1) that decomposes asymptotically into a sum of two bubbles at infinity:

Here, the position parameters (x 1 , x 2 ) satisfy

x 2 (t) -x 1 (t) = (2 + o(1)) log 5 6 t, as t ↑ ∞. In addition, the scaling and position parameters (λ(t), x 1 (t)) satisfy (λ(t), x 1 (t)) = (1 + o(1)) 1 log 1 6 t , t log 1 3 t , as t ↑ ∞.

In particular, the blow-up rate for u(t) is

t, as t ↑ ∞. Remark 1.2. We mention here that, Theorem 1.1 deal with strong interactions in the sense that the blow-up dynamics of each bubble is perturbed at the main order by the presence of the another bubble (see Section 1.4 for more discussion).

Remark 1.3. The choice of signs and the number of bubbles are related to the solvability of a linear differential system. The source term of this system originates from the interaction between solitons and certain non-localized profiles (see Lemma 2.7). Indeed, the system for two bubbles with the same sign is not solvable, and the multi-bubble case is far more involved than the two bubble case. This is the main reason why we restrict ourselves to the current case.

Exit: The solution eventually exits any small neighborhood of the solitons.

Blow-up: The solution blows up stably in finite time with rate (T -t) -1 . Soliton: : The solution is global and locally converges to a soliton.

Later on, building on the work [33], Martel-Merle-Nakanishi-Raphaël [32] showed that there exists a local C 1 co-dimension one manifold included in A which separates the stable blow-up behavior from solutions that eventually exit the soliton neighborhood. In particular, the solutions on the manifold are global in time and converge in a local norm to a soliton.

On the other hand, Martel-Merle-Raphaël [34] established the existence and description of the minimal mass blow-up solution, a key step for the complete description of the flow around the soliton. Then, the sharp asymptotics in both time and space variables were derived by Combet-Martel [4] for any order derivative of this solution. Building on these sharp properties and the techniques developed mainly in [33,34,35], Combet-Martel [5] constructed finite time multi-bubble blowup solutions with an arbitrary number of bubbles. Such solutions are also related to the strong interaction of bubbles as can be seen from the presence of the tail to the left of the minimal mass blow-up solution (see [5,Section 1.3] f

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