Blowup and ill-posedness results for a Dirac equation without gauge invariance

We consider the Cauchy problem for 
 a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$, 
 with a power type, non gauge invariant nonlinearity $\sim|u|^{p}$. 
 We prove several ill-posedness and blowup results for 
 both large and small $H^{s}$ data. In particular we prove that: 
 for (essentially arbitrary) large data in $H^{\frac n2+}(\mathbb{R} ^n)$ 
 the solution blows up in a 
 finite time; 
 for suitable large $H^{s}(\mathbb{R} ^n)$ data and 
 $s< \frac{n}{2}-\frac{1}{p-1}$ no weak solution exist; 
 when $1< p <1+\frac1n$ 
 (or $1< p <1+\frac2n$ in $n=1,2,3$), there exist arbitrarily small 
 initial data data for which the solution blows up in a finite time.


(Communicated by Thierry Cazenave)
Abstract. We consider the Cauchy problem for a nonlinear Dirac equation on R n , n ≥ 1, with a power type, non gauge invariant nonlinearity ∼ |u| p . We prove several ill-posedness and blowup results for both large and small H s data. In particular we prove that: for (essentially arbitrary) large data in H n 2 + (R n ) the solution blows up in a finite time; for suitable large H s (R n ) data and s < n 2 − 1 p−1 no weak solution exist; when 1 < p < 1 + 1 n (or 1 < p < 1 + 2 n in n = 1, 2, 3), there exist arbitrarily small initial data data for which the solution blows up in a finite time.
1. Introduction. We consider the Cauchy problem for the Dirac equation where u = u(t, x) : R × R n → C N is an unknown function, N = 2 [ n+1 2 ] , [x] is the integer part of x ∈ R, u 0 is a given function, and D = n j=1 α j ∂ j . Here, ∂ j are the partial derivatives, β and α j are Dirac matrices which satisfy β * = β, β 2 = I N , βα j + α j β = 0 for j = 1, . . . , n, α * j = α j for j = 1, . . . , n, α j α k + α k α j = 2δ jk I N for j, k = 1, . . . , n, δ jk is the Kronecker delta, and I N the identity matrix in M N (C). Throughout this paper, we assume that the function F =     The work of the second author was supported by JSPS KAKENHI Grant numbers 26887017, JP16K17624.

PIERO D'ANCONA AND MAMORU OKAMOTO
Let p > 1. The function F satisfies the estimates The most important examples of nonlinearities satisfying (A) are |u| p−1 u, u, βu p−1 2 βu, |u| p . In these cases, equation (1) is invariant under the scaling for σ > 0. Thus we see that the scale invariant Sobolev exponent is s c := n 2 − 1 p−1 . We summarize the results for (1) with F (u) = u, βu βu. For n = 1, Candy [4] proved global (in time) well-posedness in L 2 (R). Pecher [14] proved local wellposedness in H s (R 2 ) with s > 3/4. Escobedo and Vega [6] proved local wellposedness in H s (R 3 ) with s > 1; they also considered more general nonlinearities. Machihara et. al. [12] proved small data scattering in H 1 (R 3 ) with some additional regularity in the angular variables. Note that H 1 (R 3 ) is the scale critical space in the 3D case. Recently, Bejenaru and Herr [1], [2] and Bournaveas and Candy [3] obtained small data scattering in the scale critical Sobolev spaces for n = 2, 3. These results rely on the null structure of the nonlinearity u, βu βu, hence it is not clear if they can be extended to more general nonlinearities.
In this paper, we prove several blowup and ill-posedness results for (1) in the case when the nonlinear term is not gauge invariant. Before stating our results, we define our notion of weak solution: Here we denote by t ψ the transpose of ψ. Writing for brevity we shall consider for simplicity data and nonlinearities which are scalar multiples of the vector e 1 ; it is clear that several extensions are possible.
In our first result, we prove that for suitable large data the life span of any weak solution is finite, and satisfies an explicit bound.

Remark 1.
A byproduct of standard proofs of local existence in H s (R n ) is the blowup alternative: if T is the maximal existence time of the solution, then either T = ∞, or T < ∞ and lim t T u(t, ·) H s = ∞. In cases when the blowup alternative holds, the previous result yields blowup of solutions for large data.
Since H s (R n ) for s > n/2 is a Banach algebra, local in time well-posedness with blowup alternative holds in H s (R n ) for n/2 < s < p. Thus a local strong solution u(t), continuous in time with values in H s (R n ), exists on some maximal time interval [0, T (λ)), and if T (λ) < ∞ the H s (R n ) norm of the solution goes to infinity as t T (λ). A similar result holds in the case n = 3, p ≥ 3 for which is valid on R 3 provided s > 1 ( [12]). In both results, we can remove the constraint s < p if the nonlinearity F is smooth. Note that these solutions are also weak solutions in the sense of Definition 1.1 (see Appendix A at the end of the paper for a quick summary of local existence). For large data in H s with s > n 2 we prove the following general blowup result: For the solutions obtained via Stricharz estimates in the 3D case, we obtain a similar result provided the data satisfy the additional constraint (2). Note that the condition 3 2 − 1 p−1 < s < p is compatible with (2) provided k < 1 p−1 : Corollary 2. Let n = 3, p ≥ 3, λ > 0, µ ∈ C\{0}, F (u) = µ|u| p e 1 and u 0 (x) = λf (x)e 1 ≡ 0. Assume f belongs to H s (R n ) for some 3 2 − 1 p−1 < s < 3 2 and satisfies (2) (for some k < 3 2 − s). Let u(t) be the strong solution of (1) in H s (R n ) on the for λ ≥ λ 0 , and we have lim In the regularity range s < n 2 − 1 p−1 we can give a rather strong ill-posedness result: More precisely, one can construct initial data in H s (R n ) for which no weak solution exists.
The previous results do not include the case of small initial data. However, if p is sufficiently close to 1 we can prove by the same methods nonexistence of global weak solutions. Note that in the following Theorem no condition is imposed on the size of f .
For instance, as before, we can choose z = µ so that condition (6) becomes or we can choose z = −i µ, in which case condition (6) becomes Finally, in the low dimensional cases n = 1, 2, 3 we can construct small blowup solution for a larger range of p. Note that the initial data (7) belong to any H s (R n ) space with s > 0 thanks to the assumption k > n 2 , and this ensures the local existence of a strong solution in H s (R n ) for s > n 2 .
where χ ∈ C ∞ 0 ([0, ∞)) with 0 ≤ χ ≤ 1, χ(r) = 1 for 0 ≤ r ≤ 1, and χ(r) = 0 for r ≥ 2. Let T ( ) be the maximal lifespan of the local strong solution of Problem (1), continuous in time with values in H s (R n ). Then there exists ε 0 and C > 0 such that Blowup results for the Schödinger equation without gauge invariance have been proved by Ikeda and Wakasugi [8], Ikeda and Inui [9], [10] and Oh [13]. They used so-called test function method, see [17], [18], which is also used in this paper. Ikeda and Inui [9], [10] showed that the maximal existence time T (λ) for suitable initial data (satisfying conditions similar to (3)) is bounded by The different rate obtained here is of course due to the fact that in the Dirac equation the Laplacian is replaced by a first order operator; Blowup for the wave equation in absence of gauge invariance, u = |u| p , is a classical and well studied problem (see for instance [7], [11], [16]). In this case the sharp blowup range of p is known; note however that one can exploit some positivity properties of the fundamental solution or its averages, which are ruled out for the Dirac equation. It is also clear that blowup results for (1) can not be proved by reduction to the wave equation case, since in the process we obtain a derivative nonlinearity.

Proof. By the definition of a weak solution with the test function ψ
we get x)(1, . . . , 1)α j u(t, x)dx.

PIERO D'ANCONA AND MAMORU OKAMOTO
Multiplying by z and taking the real part we have (µz)I(R) + λ (izJ(R)) where we used [q]p ≥ [q] + 1 in the last inequality, which follows from [q] ≥ q − 1.

2.2.
Proof of Corollaries 1 and 2. Corollary 2 follows by a direct application of Theorem 1.2, hence we focus on the proof of Corollary 1.
By assumption, there exists a point x such that f (x) is not a positive multiple of iµ, and by translation invariance of the problem we can suppose x = 0. Now we need the following elementary fact: If w is not of the form α · iµ for some α > 0, then there exists z ∈ C such that (µz) > 0 and (izw) > 0.
Proof. By assumption, iw and µ are contained in an open half-plane H of C with 0 ∈ ∂H, and any such half-plane can be written H = {v ∈ C : (vz) > 0} for some z.
By continuity of f , we have (izf (x)) ≥ C 0 > 0 for x in a nbd of 0. Thus the assumptions of Theorem 1.2 are satisfied with k = 0 and we obtain that the life span T (λ) is finite for all λ ≥ λ 0 and satisfies the bound in the statement. The final claim concerning the growth of the H s (R n ) norm follows from the blowup alternative.
2.3. Proof of Theorem 1.3. It is not restrictive to assume s > − n 2 since the result for this case includes the result for lower values of s. Let z be such that (µz) > 0, and let u 0 = λf (x)e 1 with f of the form (4), λ > 0 and 1 p−1 < k < n 2 − s (so that f ∈ H s as noticed above). In Section 2.1 we proved that (izJ(R)) ≥ c 0 R n−k for R small. Now assume there exists a weak solution u defined on some maximal time interval [0, T (λ)) with T (λ) > 0; by Lemma 2.1 we can write for R small. Letting R → 0 we obtain λ = 0 since k > 1 p−1 , and this concludes the proof. 2.5. Proof of Proposition 1. By the local existence result in H s (R n ), s > n 2 , we know that there exists 0 > 0 such that T ( ) > 16 for 0 < < 0 . We also note that, for R ≥ 8, Applying Lemma 2.1 with λ = we get ≤ CR k− 1 p−1 for all 8 ≤ R < T ( ) and 0 < < 0 . Choosing R = T ( )/2 and noticing that by assumption k − 1 p−1 = −δ < 0, we obtain the required bound on T ( ). The claim on the norm of the solution follows as usual from the blowup alternative.
Appendix A. Well-posedness. For the convenience of the reader, we briefly recall here the arguments for the local well posedness of the Cauchy problem (1). As usual, we rewrite (1) in integral form: x) be the right hand side of this equality. Our goal is to prove that there exists a unique fixed point of U in a suitable complete metric space X T . We denote by · L p T K the norm of the space L p ([0, T ); K), where K is a Banach space on R n , for example K = H s (R n ) or L q (R n ).
Proof. By the Sobolev embedding theorem, we have for ε > 0. Interpolating it with (5), we obtain the desired bound.
Here, we assume that F satisfy the following fractional chain rule: Let 0 ≤ s < p.
Proposition 2. Let n = 3, p ≥ 3, and s ∈ ( 3 2 − 1 p−1 , p) Assume that F satisfies (A) and (B). Then, (1) is local in time well-posed in H s (R 3 ). Moreover, u is a weak solution to (1) in the sense of Definition 1.1 Furthermore, the following blowup alternative holds: if T is the maximal existence time of the solution, then either T = ∞ or T < ∞ and lim t T u(t, ·) H s = ∞.
Proof of Proposition 2. We only consider the case 3 2 − 1 p−1 < s ≤ 3 2 since the proof for the case s > 3 2 follows by a similar but simpler argument, based on Sobolev embedding instead of Strichartz estimates. Pick γ > p − 1 ≥ 2 with γ < 2 3−2s , which is possible because of the assumption. Form Lemma A.1, the linear part is estimated as follows: Set where u X T := u L ∞ T H s + u L γ T L ∞ . We note that X T is a complete metric space endowed with the metric Indeed, let {v m } be a sequence on X T converging to v in L ∞ ([0, T ); L 2 (R 3 )). Since )) * are separable Banach space, where γ denotes the Hölder conjugate of γ, the weak- * compactness implies that there exist v 0 ∈ X T and a subsequence {v mj } of {v m } such that {v mj } converges to v 0 * -weakly in L ∞ ([0, T ]; H s (R 3 )) and L γ ([0, T ]; L ∞ (R 3 )). By the uniqueness of the limit in S ( Now we can show that U u0 maps X T into X T . Lemma A.1 and (B) imply If we take T so small that CT 1− p−1 γ M p−1 < 1/4, we get U u0 [u] ∈ X T . Next, we show that U u0 is a contraction on X T . This follows by computations similar to the above ones: The previous estimate are sufficient to prove the existence of a unique fixed point which is the required strong solution, and the blowup alternative is proved in the usual way (see e.g. [5]). Finally, a standard approximation argument shows that this solution is also a weak solution in the sense of Definition 1.1.

Remark 2.
For the case s ≤ 3/2 it is not necessary to introduce the distance d but one could estimate directly in the X T norm, thanks to the condition p ≥ 3. Indeed, roughly speaking, we can proceed as follows. If s < p − 1, then, (B) yields