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
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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"> title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/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"> title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303808-si163.gif"> class="mathContainer hidden">class="mathCode">s > max { 1 + 1 p , 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 ∈ [ 1 , + ∞ ] ). 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"> title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/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"> title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303808-si44.gif"> class="mathContainer hidden">class="mathCode">s = 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.