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- W2018152140 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^mDelta_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^mDelta_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_+timesR^n with discontinuous initial data p_t^iu(0,x)=phi_i(x) (0le ile 2) and p_t^jv(0,x)=psi_j(x) (0le jle 3), respectively; here m, m_1, m_2inN, m_1neq m_2, xinR^n, nge 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)timesR^n) exist which are C^infty away from G_0cup G_m^pm and G_{m_1}^pmcupG_{m_2}^pm in [0,T]timesR^n, respectively; here G_0={(t,x): tge 0, x_1=0} and the Gamma_k^pm = {(t,x): tge 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^nsetminus{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]timesR^n) which are C^infty away from G_mcup l_0 and G_{m_1}cupG_{m_2} in [0,T]timesR^n, respectively; here Gamma_k={(t,x): tge 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): tge 0, |x|=0} is a ray." @default.
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- W2018152140 date "2013-07-13" @default.
- W2018152140 modified "2023-09-27" @default.
- W2018152140 title "The existence and singularity structure of low regularity solutions of higher-order degenerate hyperbolic equations" @default.
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