Fully fractional Schrödinger equations with memory: well-posedness via extension
Abstract
We study the nonlinear Schrödinger equation with memory Les u=μ|u|p−1u for t>0, where Les=(∂t−iΔx)s, 0<s<1, and the solution is prescribed for all t≤0. We identify a Hilbert space of histories and prove local well-posedness for 1<p≤ p*(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↑1. For every fixed Schwartz history and 1<p≤1+4/d, including the classical mass critical exponent, the memory solutions converge in C([0,T];L2∩ Lp+1) to the classical solution with initial value u0(·,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 L2 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 Les as a weighted Dirichlet-to-Neumann map and establish the corresponding extension bounds on L2 for 0<s≤1/2, and from Dom(Les) to L2 for 1/2<s<1.
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BibTeX
@misc{garofalo2026fully,
title = {{Fully fractional Schrödinger equations with memory: well-posedness via extension}},
author = {Nicola Garofalo and Gigliola Staffilani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01826},
eprint = {2609.01826},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01826},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fully fractional Schrödinger equations with memory: well-posedness via extension
AU - Nicola Garofalo
AU - Gigliola Staffilani
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.01826
UR - https://arxiv.org/abs/2609.01826
AB - We study the nonlinear Schrödinger equation with memory Le^s u=μ|u|^p−1u for t>0, where Le^s=(∂_t−iΔ_x)^s, 0<s<1, and the solution is prescribed for all t≤0. We identify a Hilbert space of histories and prove local well-posedness for 1<p≤ p_*(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↑1. For every fixed Schwartz history and 1<p≤1+4/d, including the classical mass critical exponent, the memory solutions converge in C([0,T];L^2∩ L^p+1) to the classical solution with initial value u_0(·,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≤1/2, and from Dom(Le^s) to L^2 for 1/2<s<1.
ER -