<p>We study the existence of subharmonic and time-periodic solutions with prescribed minimal period for wave equations on a disk with radius <i>R</i> in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. Such issues were posed by Rabinowitz as open problems in the context of finite dimensional Hamiltonian systems. The wave equation has an infinite dimensional Hamiltonian and its energy functional is strongly indefinite, which create considerable challenges. Our contributions are threefold: (i) By a deep analysis of Bessel functions, we obtain new spectral properties and a compact embedding theorem to the wave operator on a disk. (ii) For any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m \in {\mathbb {Z}}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_m = R/m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>m</mi> </msub> <mo>=</mo> <mi>R</mi> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, we study the existence of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-periodic solutions for the focusing problem and show the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$nT_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <msub> <mi>T</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-periodic solutions (called subharmonics) are distinct for different <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \in {\mathbb {Z}}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation>. (iii) Under some suitable restrictions, we prove that the solutions for the defocusing problem have <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> as the minimal period. This is the first result about the minimal period for the solutions and subharmonics of wave equations in multidimensional space.</p>

On Rabinowitz’s minimal period conjecture and subharmonic solutions for the wave equation on a disk

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发表日期：

2026年2月17日

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Jianyi Chen,
Kui Li,
Zhitao Zhang

摘要

We study the existence of subharmonic and time-periodic solutions with prescribed minimal period for wave equations on a disk with radius R in $\mathbb {R}^2$ $R^{2}$ . Such issues were posed by Rabinowitz as open problems in the context of finite dimensional Hamiltonian systems. The wave equation has an infinite dimensional Hamiltonian and its energy functional is strongly indefinite, which create considerable challenges. Our contributions are threefold: (i) By a deep analysis of Bessel functions, we obtain new spectral properties and a compact embedding theorem to the wave operator on a disk. (ii) For any $m \in {\mathbb {Z}}_+$ $m \in Z_{+}$ and $T_m = R/m$ $T_{m} = R / m$ , we study the existence of $T_m$ $T_{m}$ -periodic solutions for the focusing problem and show the $nT_1$ $n T_{1}$ -periodic solutions (called subharmonics) are distinct for different $n \in {\mathbb {Z}}_+$ $n \in Z_{+}$ . (iii) Under some suitable restrictions, we prove that the solutions for the defocusing problem have $T_m$ $T_{m}$ as the minimal period. This is the first result about the minimal period for the solutions and subharmonics of wave equations in multidimensional space.