Remarks on blowing-up of solutions for some nonlinear Schrödinger equations
The paper is concerned with blowing up properties of nonlinear Schrödinger equations. Two types of nonlinearities are studied: the local interaction \(f(| u|^ 2)u\) and the convolution \((V*| u|^ 2)u\). For the special cases \(f(t^ 2)=t^{p-1}+t^{q-1}\) and \(V(x)=| x|^{-\lambda}+| x|^{-\mu}\), sufficient conditions are given which ensure blow-up of the solution of the Cauchy problem for suitable initial data. For certain exterior-domain problems with Dirichlet or Neumann boundary conditions, similar results are proved for \(f(t^ 2)=t^{p-1}\). Finally, it is shown by variational arguments that the Cauchy problem with \(V(x)=| x|^{-2}\) has a global solution for small initial data.
scientific article; zbMATH DE number 4208622