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<p>This paper studies the initial-boundary value problem of a parabolic-elliptic coupled system and the asymptotic behavior towards the rarefaction wave on a half line (0, +∞). The initial-boundary value is imposed as <i>u</i><sub>−</sub> on the boundary and <i>u</i><sub>+</sub> at infinity, where 0 &lt; <i>u</i><sub>−</sub> &lt; <i>u</i><sub>+</sub>. We show that the solution converges to the rarefaction wave at a decay rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O({(1+t)}^{-{1 \over 2}} \, {\rm ln}^{3}(2+t))\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow> <mi mathvariant="normal">ln</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, which is equal to the convergence rate in the whole space in [45]. In our analysis, there are two main difficulties due to the boundary effect. We can no longer handle high-order energy estimates like the Cauchy problem. Instead, we first deal with tangential derivatives, i.e., derivatives with respect to <i>t</i>. Then we take advantage of equations to obtain the energy estimate of derivatives with respect to <i>x</i>. Another difficulty is to improve the decay estimate of the second-order derivatives with respect to <i>x</i>, which is solved by an iteration method. In addition, we need to overcome the growth of nonlinear terms due to large disturbance.</p>

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Unconditional global stability of solutions to a 1D parabolic-elliptic coupled system: The initial-boundary value problem

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Changjiang Zhu,
  • Qiaolong Zhu

摘要

This paper studies the initial-boundary value problem of a parabolic-elliptic coupled system and the asymptotic behavior towards the rarefaction wave on a half line (0, +∞). The initial-boundary value is imposed as u− on the boundary and u+ at infinity, where 0 < u− < u+. We show that the solution converges to the rarefaction wave at a decay rate of \(O({(1+t)}^{-{1 \over 2}} \, {\rm ln}^{3}(2+t))\) O ( ( 1 + t ) − 1 2 ln 3 ( 2 + t ) ) , which is equal to the convergence rate in the whole space in [45]. In our analysis, there are two main difficulties due to the boundary effect. We can no longer handle high-order energy estimates like the Cauchy problem. Instead, we first deal with tangential derivatives, i.e., derivatives with respect to t. Then we take advantage of equations to obtain the energy estimate of derivatives with respect to x. Another difficulty is to improve the decay estimate of the second-order derivatives with respect to x, which is solved by an iteration method. In addition, we need to overcome the growth of nonlinear terms due to large disturbance.