Global Well-posedness for Cubic Nls with Nonlinear Damping

u(0) = u0(x), with given parameters λ ∈ R and σ > 0, the latter describing the strength of the dissipation within our model. We shall consider the physically relevant situation of d 6 3 spatial dimensions and assume that the dissipative nonlinearity is at least of the same order as the cubic one, i.e. p > 3. However, in dimension d = 3, we shall restrict ourselves to 3 6 p 6 5. In other words, we assume that the dissipative effects in our model can be described by nonlinearities, which are at most energy critical in the sense of [12, 34]. In the following, we shall also assume the external potential V to be an anisotropic quadratic confinement, i.e.