Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping: $$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
Formally, one might have
$$\frac{\partial^{2k+1}u(0,t)}{\partial \delta^{2k+1}}=\int_0^t S(t-\tau)|S(\tau)v_0|^{2k}S(\tau)v_0 d\tau=:u_{2k+1}$$ where $S(t)=e^{it\Delta}$ (free Schrodinger propogator)
Let $(X, \|\cdot\|_{X}$ be some Banach space of functions on $\mathbb R^N$. Suppose that $\delta u_0\mapsto u(\delta)$ is of class $C^{2k+1}$ at origin in X. Then
My question: Can we say that it is necessary that following estimates hold: $$\sup_{t\in [0, T]} \left\| u_{2k+1}\right\|_{X}\leq C \|v_0\|_{X}^{2k+1}?$$ If so, how?
Motivation: (i) to understand the idea that how to show that the solution map (of NLS) $u_0\to u(t)$ is not $C^{2k+1}$ at origin in $X$ (let say some Banach space, e.g., $H^{s}$ or some other function space..) (ii) this ideas have been used by many authors, see e.g., (a) see section 6 in this paper (b) see Section 5 in this paper
