Cubic nonlinear Dirac equation in a quarter plane

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• 作者：I.P. Naumkin ^{ivannaumkinkaikin@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear Dirac equation ; Initial&ndash ; boundary value problem ; Thirring model ; Critical nonlinearity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1633-1664
• 全文大小：521 K

文摘

We study the initial–boundary value problem (IBV) for the cubic nonlinear Dirac equation in one space dimension where ψ=ψ(t,x)∈C²$\psi =\psi (t,x)\in {\mathbf{C}}^2$ is a two-spinor field, α, β   are hermitian (2×2)$(2\times 2)$-matrices satisfying β²=α²=I${\beta }^2={\alpha }^2=I$, αβ+βα=0$\alpha \beta +\beta \alpha =0$, 〈⋅,⋅〉$〈\cdot ,\cdot 〉$ denotes the C²${\mathbf{C}}^2$-scalar product. We prove the global in time existence of solutions of IBV problem for cubic Dirac equations with inhomogeneous Dirichlet boundary conditions. We obtain the sharp time decay of solutions in the uniform norm.