Stability of the Hartree equation with time-dependent coefficients

• Main Authors: Binhua Feng, Honghong Zhang, Yanjun Zhao
• Format: Article
• Language: English
• Published: SpringerOpen 2017-08-01
• Series: Boundary Value Problems
• Subjects: nonlinear Hartree equation; stability; time-dependent coefficients; Lipschitz continuity
• Online Access: http://link.springer.com/article/10.1186/s13661-017-0854-1

Abstract In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients i ∂ t u + Δ u + α ( t ) 1 | x | u + β ( t ) ( W ∗ | u | 2 ) u = 0 . $$i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{ \vert x \vert }u+\beta(t) \bigl(W \ast \vert u \vert ^{2}\bigr)u=0. $$ We first obtain the Lipschitz continuity of the solution u = u ( α , β ) $u=u(\alpha ,\beta)$ with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients. Our results improve some recent results.