EVOLUTION EQUATIONS AND doi:10.3934/eect.2018003 CONTROL THEORY Volume 7, Number 1, March 2018 pp. 53–60 SELF-SIMILAR SOLUTIONS TO NONLINEAR DIRAC EQUATIONS AND AN APPLICATION TO NONUNIQUENESS Hyungjin Huh Department of Mathematics Chung-Ang University Seoul, 156-756, Korea (Communicated by Thierry Cazenave) Abstract. Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space C([0,T ]; H s (R)) (s< 0) to the system (1) which is L 2 (R) scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order. 1. Introduction. We are interested in the initial value problem for the following nonlinear Dirac equations i(∂ t U 1 + ∂ x U 1 )= U 2 (|U 1 | 2 + |U 2 | 2 ) + 2Re( ¯ U 1 U 2 )U 1 , i(∂ t U 2 − ∂ x U 2 )= U 1 (|U 1 | 2 + |U 2 | 2 ) + 2Re( ¯ U 2 U 1 )U 2 , (1) and i(∂ t U 1 + ∂ x U 1 )= |U 2 | 2 U 1 , i(∂ t U 2 − ∂ x U 2 )= |U 1 | 2 U 2 , (2) with the initial data U j (x, 0) = u j (x). Here U j : R 1+1 → C for j =1, 2 and ¯ U is a complex conjugate of U . The systems (1) and (2) have the charge conservation  R (|U 1 | 2 + |U 2 | 2 )(x, t) dx =  R (|u 1 | 2 + |u 2 | 2 )(x) dx. Another important property of the systems (1), (2) is invariance under the scaling U λ j (x, t)= λU j (λ 2 x, λ 2 t), from which we deduce a scale invariant Sobolev space L 2 (R). We study the initial value problem of (1) and (2) in Sobolev space H s (R). We call H s as sub-critical space for s> 0, critical space for s = 0 and super-critical space for s< 0. The system (1) occurs in the context of a nonlinear refractive index [1] and has been studied in [7] where local existence for H s (s> 1/2) has been proved. The system (2) is called the Thirring model and the associated Cauchy problem has been studied by several authors [2, 4, 6, 8]. The global existence of solutions to the Thirring equations was studied in [4] in terms of Sobolev space H s (s ≥ 1). Low regularity well-posedness was discussed in [2, 6, 8] showing that there exist a 2000 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Nonlinear Dirac equations, L 2 critical problem, nonuniqueness. 53