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In this paper, we consider the generalized Camassa-Holm equation ut + 2kux -uxxt + au2ux = 2uxuxx-uuxxxUnder substitution ζ = x - ct, some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to infinity on ζ - u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.