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

TL;DR

This paper investigates the existence and singularity structures of low regularity solutions to higher-order degenerate hyperbolic equations, extending previous work on second-order cases to include discontinuous and unbounded initial data.

Contribution

It establishes local existence and detailed singularity structures for low regularity solutions of higher-order degenerate hyperbolic equations with discontinuous initial data.

Findings

Solutions exist in L-infinity spaces and are smooth away from characteristic surfaces.

Singularity structures include cusp and conic surfaces.

Results extend understanding of low regularity solutions in degenerate hyperbolic equations.

Abstract

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the initial data (u,\p_t u)(0,x) belongs to suitable conormal classes. In the present paper, we will study low regularity solutions of higher-order degenerate hyperbolic equations in the category of discontinuous and even unbounded functions. More specifically, we are concerned with the local existence and singularity structure of low regularity solutions of the higher-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\le i\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 its arguments. When the \phi_i and \psi_j are piecewise smooth with respect to the hyperplane \{x_1=0\} at t=0, we show that local solutions u(t,x), v(t,x)\in L^{\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 are homogeneous of degree zero close to x=0, then there exist local solutions u(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 a cuspidal conic surface and l_0=\{(t,x): t\ge 0, |x|=0\} is a ray.