On the Cauchy problem for a class of shallow water wave equations with (k + 1)-order nonlinearities

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• 作者：Lei Zhang ^{lzhang890701@163.com" class="auth_mail" title="E-mail the corresponding author} ; Bin Liu ; ^{binliu@mail.hust.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Well-posedness ; Shallow water wave equations ; (k+1)(k+1)-order nonlinearities ; Blow-up criteria ; Gevrey regularity ; Analyticity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：151-185
• 全文大小：654 K

文摘

This paper considers the Cauchy problem for a class of shallow water wave equations with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=3f152b5e77764a11dd537db48f3acc57" title="Click to view the MathML source">(k+1)class="mathContainer hidden">class="mathCode">$(k+1)$-order nonlinearities in the Besov spaces which involves the Camassa–Holm, the Degasperis–Procesi and the Novikov equations as special cases. Firstly, by means of the transport equation and the Littlewood–Paley theory, we obtain the local well-posedness of the equations in the nonhomogeneous Besov space class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si167.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=a80511f6935e96c12d84eb841b62f005">class="imgLazyJSB inlineImage" height="18" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303808-si167.gif">class="mathContainer hidden">class="mathCode">$B_{p,r}^s$ (class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si163.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=7dea1e7a1383ab163b8f509306c18d0c">class="imgLazyJSB inlineImage" height="22" width="133" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303808-si163.gif">class="mathContainer hidden">class="mathCode">$s>\max \{1+\frac{1}{p},\frac{3}{2}\}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si156.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=cfc0915db45e1ae5e3cf59643fd87846" title="Click to view the MathML source">p,r∈[1,+∞]class="mathContainer hidden">class="mathCode">$p,r\in [1,+\infty ]$). Secondly, we consider the local well-posedness in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si6.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=81affb084a614f9d5663b9eb3a1f20ad">class="imgLazyJSB inlineImage" height="18" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303808-si6.gif">class="mathContainer hidden">class="mathCode">$B_{2,r}^s$ with the critical index class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303808&_mathId=si44.gif&_user=111111111&_pii=S0022247X16303808&_rdoc=1&_issn=0022247X&md5=d93724f7ae91a85a87281ee078c8be05">class="imgLazyJSB inlineImage" height="20" width="40" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303808-si44.gif">class="mathContainer hidden">class="mathCode">$s=\frac{3}{2}$, and show that the solutions continuously depend on the initial data. Thirdly, the blow-up criteria and the conservative property for the strong solutions are derived. Finally, with the help of a new Ovsyannikov theorem, we investigate the Gevrey regularity and analyticity of the solutions. Moreover, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping.