SMALL DATA BLOW-UP OF SOLUTIONS TO NONLINEAR SCHRODINGER EQUATIONS WITHOUT GAUGE INVARIANCE IN L2

摘要

In this article we study the Cauchy problem of the nonlinear Schrodinger equations without gauge invariance @@@ i partial derivative(t)u + Delta u = lambda(vertical bar u vertical bar(p1) + vertical bar v vertical bar(p2)), (t, x) is an element of [0, T) x R-n, @@@ i partial derivative(t)v + Delta v = lambda(vertical bar u vertical bar(p2) + vertical bar v vertical bar(p1)), (t, x) is an element of [0, T) x R-n, @@@ where 1 <( )p(1,) p(2) < 1 + 4/n and lambda is an element of C\{0}. We first prove the existence of a local solution with initial data in L-2( n). Then under a suitable condition on the initial data, we show that the L-2-norm of the solution must blow up in finite time although the initial data are arbitrarily small. As a by-product, we also obtain an upper bound of the maximal existence time of the solution.

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