Mass concentration phenomenon for a non-elliptic Schrodinger equation
- Mass concentration phenomenon for a non-elliptic Schrodinger equation
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- For elliptic Schrodinger equations with L2-critical nonlinearity, it was known that there exist a finite time blow up solution. Bourgain proved a
mass concentration phenomenon for L2-critical nonlinear Schrodinger equations in spatial dimension 2, applying bilinear extension estimates for the paraboloid. This result was extended to higher-dimensional cases by Begout
For a non-elliptic hypersurface, e.g. saddle surface in R3, there are also results of bilinear extension estimates. From this, Rogers and Vargas could
investigate phenomena similar to the elliptic cases.
Recently, the result of Begout and Vargas could be extended to the case that a mixed norm of solution to Schrodinger equation blows up, see . We extend the result of  to the case with a mixed norm condition similar to .
The main theorem is the following: If a mixed norm of the solution to a non-elliptic nonlinear Schrodinger equation blows up in finite time, then
there exists a small constant \varepsilon such that the L2-norm of the solution (i.e. mass of the solution) on some spatial region cannot be smaller than \varepsilon.
We need two primary tools, Strichartz estimates and bilinear estimates for the saddle surface. Firstly, Strichartz estimates for Schrodinger equation plays an important role to control the nonlinear part of the given equation.
Secondly, a mixed norm version of bilinear extension estimate for the saddle surface implies a refined Strichartz estimate, which lets us detect the mass concentrating region. Unlike the elliptic case, we need a rectangular Whitney type decomposition, from which we have a proper lower bound condition to satisfy a hypothesis of bilinear estimate. In fact, this rectangular decomposition
arose from the two parameter scaling invariance of our non-elliptic Schrodinger equation.
The proof of the main theorem basically follows from the argument of Bourgain. We begin with decomposition of the time interval such that on
each subinterval a mixed norm of the solution is a given small constant. Then we follow the steps: Controlling the inhomogeneous part, Decomposing the initial data,
Figuring out the concentrating region,
Determining the size of windows.
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