<p>We consider the Cauchy problem for the logarithmic Schrödinger equation and prove uniqueness of weak <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^s({\mathbb {R}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> solutions for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which improves on the previous uniqueness result in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1({\mathbb {R}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The main difficulty for this equation is that the nonlinearity has a singularity at the origin and breaks the local Lipschitz continuity. The proof is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity, based on the localization argument. We also study uniqueness on the torus and uniqueness of the equation perturbed by pure power nonlinearities. To the best of the author’s knowledge, the arguments in this paper are the first to effectively apply dispersive/smoothing estimates of the Schrödinger group to the logarithmic Schrödinger equation.</p>

Uniqueness of solutions for the logarithmic Schrödinger equation

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发表日期：

2026年4月17日

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Masayuki Hayashi

摘要

We consider the Cauchy problem for the logarithmic Schrödinger equation and prove uniqueness of weak $H^s({\mathbb {R}}^d)$ $H^{s} (R^{d})$ solutions for $s\in (0,1)$ $s \in (0, 1)$ , which improves on the previous uniqueness result in $H^1({\mathbb {R}}^d)$ $H^{1} (R^{d})$ . The main difficulty for this equation is that the nonlinearity has a singularity at the origin and breaks the local Lipschitz continuity. The proof is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity, based on the localization argument. We also study uniqueness on the torus and uniqueness of the equation perturbed by pure power nonlinearities. To the best of the author’s knowledge, the arguments in this paper are the first to effectively apply dispersive/smoothing estimates of the Schrödinger group to the logarithmic Schrödinger equation.