Fully fractional Schrödinger equations with memory: well-posedness via extension

Abstract

We study the nonlinear Schrödinger equation with memory $\Le^s u=μ|u|^{p-1}u$ for $t>0$, where $\Le^s=(\partial_t-iΔ_x)^s$, $0<s<1$, and the solution is prescribed for all $t\le 0$. We identify a Hilbert space of histories and prove local well-posedness for $1<p\le p_{\ast }(s)=1+4s/(d+2-2s)$, with global solutions for sufficiently small histories at the critical exponent. We also establish the passage from this memory problem to the classical nonlinear Schrödinger equation as $s\uparrow 1$. For every fixed Schwartz history and $1<p\le 1+4/d$, including the classical mass critical exponent, the memory solutions converge in $C([0,T];L^2\cap L^{p+1})$ to the classical solution with initial value $u_0(\cdot ,0)$, for every finite $T$ below its maximal existence time. Thus the limiting evolution depends on the prescribed past only through its value at time zero. Our construction uses an oscillatory Poisson extension and an exact identity expressing the weighted $L^2$ norm of the lifted initial datum in terms of the history. This identity allows us to apply the boundary estimates of our earlier work to the memory problem. We recover $\Le^s$ as a weighted Dirichlet-to-Neumann map and establish the corresponding extension bounds on $L^2$ for $0<s\le \frac{1}{2}$, and from $\operatorname{Dom}(\Le^s)$ to $L^2$ for $\frac{1}{2}<s<1$.

arXiv commentv3: Title slightly modified. Well-posedness results strengthened. Added convergence of memory solutions with fixed Schwartz histories to the classical nonlinear Schrödinger evolution as $s\uparrow 1$, including at the mass critical exponent