Wave-breaking and persistence properties in weighted \(L^p\) spaces for a Camassa-Holm type equation with quadratic and cubic nonlinearities (Q6593178)

This paper considers the following Camassa-Holm type equation \N\[\Nu_t-u{txx}+3u^2u_x=2u_xu_{xx}+uu_{xxx}.\tag{1}\N\]\NThe authors studied such an equation from two main viewpoints: formation of singularities and persistence properties.\N\NThe first blow up of this equation is wave-breaking, that is, whenever its \(x-\)derivative becomes unbounded. This is explored in Section 3 and the authors give conditions for the appearance of breaking waves in Theorem 3.1.\N\NIn Section 4 the authors study persistence properties of the solutions of (1). To this end, they use the machinery developed by \textit{L. Brandolese} [Int. Math. Res. Not. 2012, No. 22, 5161--5181 (2012; Zbl 1256.35108)].\N\NA consequence of their results is the following: if the initial datum has enough decaying, then this property is inherited to the solution emanating from the corresponding Cauchy problem. In particular, initial data with asymptotic peakon like behavior will lead to solutions that, for fixed \(t\) and large values of \(|x|\) will have a behavior like \(e^{-|x|}\).\N\NIn close note, equation (1) is a particular case of a larger family of equations whose persistence properties of solutions was considered in [\textit{I. L. Freire}, J. Evol. Equ. 24, No. 1, Paper No. 6, 28 p. (2024; Zbl 1533.35030)] using the same techniques from \textit{L. Brandolese} [Int. Math. Res. Not. 2012, No. 22, 5161--5181 (2012; Zbl 1256.35108)]. As a result, persistence properties for solutions of (1) can also be obtained applying the results reported in [loc. cit.].