Non-linear bi-harmonic Choquard equations

This note studies the fourth-order Choquard equation \begin{document}$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $\end{document} In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.

Fourth-order Schrödinger equations, take into account the role of small fourthorder dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr non-linearity [12,13].

T. SAANOUNI
The exponent s c = 0 is called mass-critical case and corresponds to p * := 1 + α+4 N . The energy-critical case s c = 2 is equivalent to The well-posedness issues for the mass-super-critical and energy sub-critical classical Choquard equation were investigated recently by many authors [7,20,23]. See also [8,4,22], for the fractional Choquard equation.
Recall the conservation laws for the Schrödinger problem (1. The positive (respectively negative) sign of refers to the attractive or defocusing (respectively focusing) case, where a local solution in the energy space is claimed to be global and scatters (respectively blows-up in finite time). It is the purpose of this manuscript to obtain a sharp dichotomy in the mass super-critical and energy sub-critical cases of global well-posedness and scattering versus finite time blow-up of solutions to the fourth-order Choquard problem (1.1), by use of a sharp Gagliardo-Nirenberg type inequality and the existence of ground states. In the scattering part, one uses the concentration-compactnessrigidity method, due to Kenig and Merle [14], which has a deep influence on asymptotic study of Schrödinger problems [5,10].
To the author knowledge, this paper is the first one dealing with scattering of bi-harmonic Choquard equations.
The plan of this paper is as follows. Section two contains some classical estimates needed in the sequel. In the third section a sharp Gagliardo-Nirenberg type inequality is given. The existence of ground states is proved in section four. In section 5, local well-posedness in the energy space is given. A variance identity is established in section six. The existence of global/non global solutions to (1.1) are discussed in section 7. The goal of the last section is to investigate scattering of global solutions.
Here and hereafter C will denote a constant which may vary from line to line and if A and B are non-negative real numbers, A B means that A ≤ CB.
Denote the Lebesgue space L r := L r (R N ) with the standard norm · r := · L r and · := · 2 . Take H 2 := H 2 (R N ) the inhomogeneous Sobolev space endowed with the complete norm If X is an abstract space C T (X) := C([0, T ], X) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X, moreover for an eventual solution to (1.1), T * > 0 denotes it's lifespan.

2.
Preliminary. This section contains some estimates needed in the sequel. Let us start with a Hardy-Littlewood-Sobolev inequality [18].
Lemma 2.5. Let s ∈ (0, 1] and 1 < p, p i , q i < ∞ satisfying 1 Definition 2.6. A couple of real numbers (q, r) is said to be s admissible if Strichartz estimate [11,24] is a classical tool to control solutions to (1.1).
Proposition 2.7. Let N ≥ 2, 0 ≤ s < 2, (q, r) be an admissible pair and (q,r) be −s admissible pair. Then, there exists C := C N,q,q,s such that if u 0 ∈Ḣ s , The time sequences have the pairwise divergence property: The remainder sequence has the following asymptotic smallness property For fixed M and any 0 ≤ α ≤ 2, the asymptotic Pythagorean expansions hold Proof. Taking account of [11], the last equality is the only point to prove. It is sufficient to prove that Q(u) : Assume as a first case that there exists some j for which t j n converges to a finite number, which is supposed to be zero without loss of generality. From the proof of Lemma 5.3 in [11] and the compact embedding H 2 rd → → L q for 2 < q < 2N N −4 , we get W j−1 n → ψ j in L q for 2 < q < 2N N −4 . Write using Lemma 2.1, for r : Since, p < p * , we get 2 < rp < 2N N −4 , which implies that |Q(W j−1 n ) − Q(ψ j )| → 0. Let k = j. Then, |t k n | → ∞. Since p > p * , from Lemma 2.1 and the L p space-time decay estimates of the linear flow associated to (1.1), one gets u n → ψ j in L q for 2 < q < 2N N −4 . As previously, it follows that Q(u n ) → Q(ψ j ). Finally, using the identity one gets W M n → 0 and Q(W M n ) → 0 for M > j. Similarly, we get the second case: for all j, t j n → ∞.
3. Gagliardo-Nirenberg inequality. Denote the real numbers The goal of this section is to prove a sharp Gagliardo-Nirenberg inequality related to the Choquard problem (1.1).
Proof. The proof contains three steps. First, let us start by proving the interpolation inequality (3.1). Taking account of Lemma 2.4 and Corollary 2.2, it follows that Second, one proves the equation (3.2). Denote β : Then, ψ n ψ in H 2 and using Sobolev injections, one gets for a sub-sequence denoted also (ψ n ), In fact, thanks to Lemma 2.1 and Sobolev embedding, This implies that, when n goes to infinity The semi continuity of · H 2 gives max{ ψ , ∆ψ } ≤ 1. Then, because otherwise, one gets the absurdity J(ψ) < β. Thus, ψ satisfies (3.2) because the minimizer satisfies the Euler equation where ψ is given in (3.2). Define, the scaling ψ = φ a,b := aφ(b.), for a, b ∈ R. Then, the equation The proof is closed.

4.
Existence of ground states. For u ∈ H 2 and a, b ∈ R, here and hereafter define the quantities Definition 4.1. We call ground state of (1.1), any solution to which minimizes the problem The following main result of this section follows with variational methods and ensures the existence of ground states. Theorem 4.3. Take N ≥ 2, a couple of real numbers (a, b) ∈ A and p * < p < p * . Then, 1. m := m a,b is nonzero and independent of (a, b); 2. there is a ground state solution to (4.1)-(4.2).
Let us give some intermediate results.
Since µ ≥ 0 and p > p * , one obtains, if b < 0, Arguing as previously, it follows that The next intermediate result is the following.
Thus, when n → ∞, One can express the minimizing problem (4.2), with negative constraint.
Denoting by r the right hand side of the previous equality, it is sufficient to prove that m a,b ≤ r. Take u∈ H 2 such that K a,b (u) < 0. Because lim λ→0 K Q a,b (u λ ) = 0, by the previous Lemma, there exists λ ∈ (0, 1) such that K a,b (u λ ) > 0. With a continuity argument there exists λ 0 ∈ (0, 1) such that K a,b (u λ0 ) = 0, then since λ → H a,b (u λ ) is increasing, we get This closes the proof.
Proof of Theorem 4.3. Let (φ n ) be a minimizing sequence, namely First case a > 0 and b > 0. Denoting λ : So the following sequence is bounded Thus, for any real number β, the following sequence is also bounded Second case a > 0 and −2a Moreover, if b < 0,μ = 2a + (N − 4)b. Then, since µ ≥ 0 and p > p * , we obtain 2a(p − 1) + (α + 4)b > 0. Because K a,b (φ n ) = 0, this implies that • Second step: the limit of (φ n ) is nonzero and m > 0. Taking account of the compact injection in Lemma 2.3, take Assume that φ = 0. Using Corollary 2.2, with the fact that 1 + α N < p < p * , write
Let us end this section with the so-called generalized Pohozaev identity [15].  Remark 5.2. 1. Thanks to the inequality (3.1), the energy is well-defined for 1 + α N ≤ p ≤ p * . So, the condition p ≥ 2 which gives a restriction on the space dimension, seems to be technical; 2. the proof is omitted because it follows as in [22]. 6. Virial type identity. This section is devoted to prove a Virial type identity, which will be useful in order to obtain finite time blow-up of some solutions to the Choquard problem (1.1). Here and hereafter, denote ψ R := R 2 ψ( . R ), R > 0, where ψ ∈ C ∞ 0 (R n ) is a radial function satisfying ψ ≤ 1 and A direct computation gives ψ R ≤ 1, ψ R (r) ≤ r and ∆ψ R ≤ N.  Then, M ψ [u(t)] =< u(t), Γ ψ u(t) > . The main result of this section reads as follows.

Denote the localized Virial
Theorem 6.1. Let N ≥ 2, 0 < α < N such that α ≥ N − 8, 2 ≤ p ≤ p * and u ∈ C T (H 2 rd ) be a solution of (1.1). Then, on [0, T ), for any R > 0 and 1 Proof. Taking account of the equation (1.1), one gets where [X, Y ] := XY − Y X denotes the commutator of X and Y . According to computation done in [1], one has Using computations in [22], it follows that Take 1 2 < µ < min{2, N 2 }. Taking account of (2.1) and (2.2), write 7. Global/non global existence of solutions. In this section, we prove a sharp criteria of finite time blow-up/global existence of solutions to the Choquard problem (1.1) in the focusing regime. In this section one takes = −1. Here and hereafter, denote, for u ∈ H 2 , the scale invariant quantities The main result of this section reads.
Theorem 7.1. Let N ≥ 2, 0 < α < N such that α > N − 8, 0 < s c < 2, φ be a ground state solution to (4.1) and a maximal solution u ∈ C T * (H 2 rd ) of (1.1). Suppose that ME[u] < 1. Remark 7.2. 1. The unnatural condition p < 3 which seems to be technical is due to a lack of a Virial identity similar to the NLS case; 2. the radial condition is required for the Virial identity in the first case and is assumed for simplicity in the second case; 3. scattering is proved in the next section; 4. the proof of next auxiliary result is omitted because it follows like in [22].
Proof. Using the properties of ψ, write Integrating the previous inequality, one obtains for some t * > 0, Then, u cannot be global. Hence T * < ∞.
We are ready to prove Theorem 7.1. Assume that (7.1)-(7.2) are satisfied and take η > 0 satisfying Then, thanks to (7.2), one gets With Theorem 6.1, for O R (1) → 0 uniformly in time, and using Young inequality via the fact that p < 3, one gets The proof is a consequence of Lemma 7.5. By the Strichartz estimate Hölder and Hardy-Littlewood-Sobolev inequalities, one gets for (q, r) := (2p, 2N p N p−4 ) and w := u − v, d(ũ,ṽ) ≤C (I α * |u| p )|u| p−2 u − (I α * |v| p )|v| p−2 v L q (I,L r ) Thanks to the Sobolev injection in the previous remark, yields Taking δ > 0 small enough, it follows that φ u0 is a contraction of X δ,M . Then, the fixed point principle gives the result.

Now, by previous computation
Taking t > 0 large enough such that u S((t,∞)) << 1, then a partition of [0, t) ⊂ ∪I j with sup j u S(Ij ) << 1, this implies that Thus, when t → ∞, With the same way, we prove that when t → ∞, Finally when t → ∞, Taking account of Pohozaev identity, one gets ∆φ 2 = B A φ 2 . Then, Since p > p * , B > 2, E(u) is conserved implies that ∆u(t) is bounded. The claim follows by Proposition 8.2. Now, for each δ > 0, define the set . By contradiction, assume that  Proof. Arguing as in the proof of Proposition 8.2, one can solve for large t > 0, the integral equation Indeed, taking t > 0 such that e i.∆ 2 ψ S(t,∞) < δ, where δ is given in Proposition 8.2, there exist v ∈ C((t, ∞), H 2 ) a solution to (1.1) such that v S(t,∞) ≤ 2δ and ∞) ) → 0. This implies that M (v) = ψ 2 . Since p > p * , from Lemma 2.1 and the L p spacetime decay estimates of the linear flow associated to (1.1), one gets Then, E(v) = lim t→∞ E(v(t)) = ∆ψ 2 . This implies that Then, by Lemma 7.3, v is global, which concludes the proof. Proof. There exists a sequence of solutions u n to (1.1) with H 2 data u n,0 (rescaled to satisfy u n = 1) such that ∆u n,0 < φ Now, since B > 2, there exists C δ > 0 such that F (x) > C δ x 2 for 0 < x < 1 − δ.
Then, on R, The claim follows by the previous inequality via (8.1).

8.3.
Proof of scattering. Thanks to Proposition 8.7, the critical solution u c constructed in Proposition 8.6 satisfies the hypotheses in Proposition 8.9. Therefore, to complete the proof of Theorem 7.1, we apply Proposition 8.9 to u c and find that u c,0 = 0, which contradicts the fact that u c S(R) = ∞. This contradiction shows that (8.2) is false. Thus, by Proposition 8.4, H 2 scattering holds.