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We consider the focusing, mass-supercritical NLS equation augmented with anonlinear damping term. We provide sufficient conditions on the nonlinearityexponents and damping coefficients for finite-time blow-up. In particular,singularities are formed for focusing and dissipative nonlinearities of thesame power, provided that the damping coefficient is sufficiently small. Ourresult thus rigorously proves the non-regularizing effect of nonlinear dampingin the mass-supercritical case, which was suggested by previous numerical andformal results.  We show that, under our assumption, the damping term may be controlled insuch a way that the self-similar blow-up structure for the focusing NLS isapproximately retained even within the dissipative evolution. The nonlineardamping contributes as a forcing term in the equation for the perturbationaround the self-similar profile, that may produce a growth over finite timeintervals. We estimate the error terms through a modulation analysis and acareful control of the time evolution of total momentum and energy functionals.

On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping