Nonexistence of global solutions for a class of nonlinear Schrödinger equations (Q2747818)

The author considers an initial boundary value problem for the equation NEWLINE\[NEWLINE\{i\partial_t-\Delta\} u=\alpha f(u)+ kuNEWLINE\]NEWLINE defined on a bounded domain \(\Omega\) where \(f(u)\) satisfies \(f(u)\geq| u|^{1+ p}\), \(p> 0\), and \(u\) is required to satisfy a homogeneous Neumann condition on \(\partial\Omega\) the boundary of \(\Omega\). It is proved that either the \(L^1\)-norm of the solution blows up or that solutions lose smoothness enough to justify calculations in finite time. It is remarked that a nonlinear term of the form a \(\alpha f(u)+ ku\) is not the usual form and is one which is often referred to as nonclassical. In this paper the cases when \(k\neq 0\) and \(k=0\) are studied. In the final section corresponding results are stated for a similar initial boundary value problem associated with the equation \(\{i\partial_t+ \Delta\}u=\alpha f(u)+ ku\).