<p>In this paper, we prove that the linear term and the nonlinear term of the integral form solution for the Kawahara-type equation in Duhamel’s formula belongs to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C\left( [0,T];H^{s}(\mathbb {R})\right) (s>\frac{1}{4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C\left( [0,T];H^{r+s}(\mathbb {R})\right) (r<\min \left\{ 2,2s-\frac{1}{2}\right\} ,s>\frac{1}{4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>H</mi> <mrow> <mi>r</mi> <mo>+</mo> <mi>s</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo><</mo> <mo movablelimits="true">min</mo> <mfenced close="}" open="{"> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mi>s</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> <mo>,</mo> <mi>s</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, respectively. Further, the local well-posedness result is obtained in the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^{s}(\mathbb {R})(s>\frac{1}{4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we show that the corresponding solution converges to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> uniformly, as <i>t</i> tends to zero for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_{0}\in H^{s}(\mathbb {R})(s>\frac{1}{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Nonlinear smoothing and uniform convergence of the Kawahara-type equation

期刊论文
发表日期：

2026年1月20日

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Weimin Wang,
Jing Wang,
Menghui Li,
Zhaoli Yin

摘要

In this paper, we prove that the linear term and the nonlinear term of the integral form solution for the Kawahara-type equation in Duhamel’s formula belongs to $C\left( [0,T];H^{s}(\mathbb {R})\right) (s>\frac{1}{4})$ $C ([0, T]; H^{s} (R)) (s > \frac{1}{4})$ and $C\left( [0,T];H^{r+s}(\mathbb {R})\right) (r<\min \left\{ 2,2s-\frac{1}{2}\right\} ,s>\frac{1}{4})$ $C ([0, T]; H^{r + s} (R)) (r < min \{2, 2 s - \frac{1}{2}\}, s > \frac{1}{4})$ , respectively. Further, the local well-posedness result is obtained in the space $H^{s}(\mathbb {R})(s>\frac{1}{4})$ $H^{s} (R) (s > \frac{1}{4})$ . Moreover, we show that the corresponding solution converges to $u_{0}$ $u_{0}$ uniformly, as t tends to zero for $u_{0}\in H^{s}(\mathbb {R})(s>\frac{1}{3})$ $u_{0} \in H^{s} (R) (s > \frac{1}{3})$ .