Scholarlink

<p>We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data. By employing <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> estimates for the linear problem and a fractional Leibniz rule in suitable homogeneous Besov spaces, we show the existence of the solution for initial data that may not belong to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> at the spatial infinity in general. The main novelty of our result is to construct the solution for initial data having the decay like <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(r&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. A crucial point in our argument is to control the derivative loss from the high frequency part by appropriately choosing the function space.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Existence of solutions to the semilinear damped wave equation with non-\(L^2\) slowly decaying data: polynomial nonlinearity case

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年2月22日

查看全文
  • Masahiro Ikeda,
  • Takahisa Inui,
  • Yuta Wakasugi

摘要

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data. By employing \(L^{p}\) L p – \(L^{q}\) L q estimates for the linear problem and a fractional Leibniz rule in suitable homogeneous Besov spaces, we show the existence of the solution for initial data that may not belong to \(L^{2}\) L 2 at the spatial infinity in general. The main novelty of our result is to construct the solution for initial data having the decay like \(L^r\) L r with \(r>2\) r > 2 . A crucial point in our argument is to control the derivative loss from the high frequency part by appropriately choosing the function space.