Nonlinear Schrödinger equation with singular potential and initial data

Abstract

We consider a nonlinear Schrödinger equation with singular potential and initial data when the nonlinear term is an Lloc∞-function which does not satisfy the Lipschitz condition. To avoid non-Lipshitz nonlinearity we use the cut-off method of regularization and as a framework for existence-uniqueness theorems we employ Colombeau vector space GC1,W2,2 ([0,T),Rn), n≤3. As an example we prove the existence-uniqueness result for nonlinear mapping f(u)=|u|p-1u, p≥1, in the space GC1,W2,2([0,T),Rn), n≤3.