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<p>In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for the perturbed inhomogeneous wave equation in the exterior domain, which yields <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(t^{-\frac{1}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation> decay inside the cone. In addition, two new weighted <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> product estimates are established to produce <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(t^{-\frac{1}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation> decay close to the cone. We then show that the existence lifespan <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(T_\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> for the quasilinear wave equations with general quadratic nonlinearity satisfies <Equation ID="Equ57"> <EquationSource Format="TEX">\(\begin{aligned} \varepsilon ^2T_{\varepsilon }\ln ^3T_{\varepsilon }=A, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>ε</mi> <mn>2</mn> </msup> <msub> <mi>T</mi> <mi>ε</mi> </msub> <msup> <mo>ln</mo> <mn>3</mn> </msup> <msub> <mi>T</mi> <mi>ε</mi> </msub> <mo>=</mo> <mi>A</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for some fixed positive constant <i>A</i>, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem.</p>

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Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in \(2{\textbf{D}}\)

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • Ning-An Lai,
  • Cui Ren,
  • Wei Xu

摘要

In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for the perturbed inhomogeneous wave equation in the exterior domain, which yields \(t^{-\frac{1}{2}}\) t - 1 2 decay inside the cone. In addition, two new weighted \(L^2\) L 2 product estimates are established to produce \(t^{-\frac{1}{2}}\) t - 1 2 decay close to the cone. We then show that the existence lifespan \(T_\varepsilon \) T ε for the quasilinear wave equations with general quadratic nonlinearity satisfies \(\begin{aligned} \varepsilon ^2T_{\varepsilon }\ln ^3T_{\varepsilon }=A, \end{aligned}\) ε 2 T ε ln 3 T ε = A , for some fixed positive constant A, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem.