Paper detail

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

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^mΔ_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^mΔ_x)u=f(t,x,u) and (\p_t^2-t^{m_1}Δ_x)(\p_t^2-t^{m_2}Δ_x)v=f(t,x,v) in \R_+\times\R^n with discontinuous initial data \p_t^iu(0,x)=ϕ_i(x) (0\le i\le 2) and \p_t^jv(0,x)=ψ_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 ϕ_i and ψ_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 Γ_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 ϕ_i and ψ_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 Γ_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.