Moving boundary i.b.v.p.s for Schrodinger equations: Solution bounds and related results

作者：Flavin James N^*

来源：Journal of Mathematical Analysis and Applications, 2010, 371(2): 552-563.

DOI：10.1016/j.jmaa.2010.05.047

• Solution bounds
• Moving boundary problems
• Schrodinger equations
• Asymptotically sharp bounds
• Pointwise bounds

The paper is concerned with i.b.v.p.s for Schrodinger equations, linear and nonlinear, in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the L(2) norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region. The application of the methodology to i b.v p.s for other p.d.e.s is discussed briefly.

• 出版日期2010-11-15