This paper considers the Cauchy problem for a class of shallow water wave equations with
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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
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3808&_mathId=si163.gif&_user=111111111&_pii=S0022247X1630380 8&_rdoc=1&_issn=0022247X&md5=7dea1e7a1383ab163b8f509306c18d0c"> 3808-si163.gif"> 3808-si163.gif"> s > max { 1 + 1 p , 3 2 } and
3808&_mathId=si156.gif&_user=111111111&_pii=S0022247X1630380 8&_rdoc=1&_issn=0022247X&md5=cfc0915db45e1ae5e3cf59643fd87846" title="Click to view the MathML source">p,r∈[1,+∞] p , r ∈ [ 1 , + ∞ ] ). Secondly, we consider the local well-posedness in
3808&_mathId=si6.gif&_user=111111111&_pii=S0022247X1630380 8&_rdoc=1&_issn=0022247X&md5=81affb084a614f9d5663b9eb3a1f20ad"> 3808-si6.gif"> 3808-si6.gif"> B 2 , r s with the critical index
3808&_mathId=si44.gif&_user=111111111&_pii=S0022247X1630380 8&_rdoc=1&_issn=0022247X&md5=d93724f7ae91a85a87281ee078c8be05"> 3808-si44.gif"> 3808-si44.gif"> 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.