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This paper is a continuation of our previous work [21], where we haveestablished that, for the second-order degenerate hyperbolic equation(\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutionsu(t,x) exist when the initial data (u,\p_t u)(0,x) belongs to suitable conormalclasses. In the present paper, we will study low regularity solutions ofhigher-order degenerate hyperbolic equations in the category of discontinuousand even unbounded functions. More specifically, we are concerned with thelocal existence and singularity structure of low regularity solutions of thehigher-order degenerate hyperbolic equations \p_t(\p_t^2-t^m\Delta_x)u=f(t,x,u)and (\p_t^2-t^{m_1}\Delta_x)(\p_t^2-t^{m_2}\Delta_x)v=f(t,x,v) in\R_+\times\R^n with discontinuous initial data \p_t^iu(0,x)=\phi_i(x) (0\lei\le 2) and \p_t^jv(0,x)=\psi_j(x) (0\le j\le 3), respectively; here m, m_1,m_2\in\N, m_1\neq m_2, x\in\R^n, n\ge 2, and f is C^\infty smooth in itsarguments. When the \phi_i and \psi_j are piecewise smooth with respect to thehyperplane \{x_1=0\} at t=0, we show that local solutions u(t,x), v(t,x)\inL^{\infty}((0,T)\times\R^n) exist which are C^\infty away from \G_0\cup\G_m^\pm and \G_{m_1}^\pm\cup\G_{m_2}^\pm in [0,T]\times\R^n, respectively;here \G_0=\{(t,x): t\ge 0, x_1=0\} and the \Gamma_k^\pm = \{(t,x): t\ge 0,x_1=\pm \f{2t^{(k+2)/2}}{k+2}\} are two characteristic surfaces forming a cusp.When the \phi_i and \psi_j belong to C_0^\infty(\R^n\setminus\{0\}) and arehomogeneous of degree zero close to x=0, then there exist local solutionsu(t,x), v(t,x)\in L_{loc}^\infty((0,T]\times\R^n) which are C^\infty away from\G_m\cup l_0 and \G_{m_1}\cup\G_{m_2} in [0,T]\times\R^n, respectively; here\Gamma_k=\{(t,x): t\ge 0, |x|^2=\f{4t^{k+2}}{(k+2)^2}\} (k=m, m_1, m_2) is acuspidal conic surface and l_0=\{(t,x): t\ge 0, |x|=0\} is a ray.

The existence and singularity structure of low regularity solutions of higher-order degenerate hyperbolic equations