On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

Abstract: In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation $\p_t^2u-t^m\Delta u=f(t,x,u)$ with typical discontinuous initial data $(u(0,x), \p_tu(0,x))=(0, \vp(x))$; here $m\in\Bbb N$, $x=(x_1, ..., x_n)$, $n\ge 2$, and $f(t,x,u)$ is $C^{\infty}$ smooth in its arguments. When the initial data $\vp(x)$ is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane ${t=x_1=0}$, it is shown that the local solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^n)$ exists and is $C^{\infty}$ away from the forward cuspidal cone $\Gamma_0=\bigl{(t,x)\colon t>0, |x|^2=\ds\f{4t^{m+2}}{(m+2)^2}\bigr}$ and the characteristic cuspidal wedge $\G_1^{\pm}=\bigl{(t,x)\colon t>0, x_1=\pm \ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$, respectively. On the other hand, for $n=2$ and piecewise smooth initial data $\vp(x)$ singular along the two straight lines ${t=x_1=0}$ and ${t=x_2=0}$, we establish the local existence of a solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^2)\cap C([0, T], H^{\f{m+6}{2(m+2)}-}(\Bbb R^2))$ and show further that $u(t,x)\not\in C^2((0,T]\times\Bbb R^2\setminus(\G_0\cup\G_1^{\pm}\cup\G_2^{\pm}))$ in general due to the degenerate character of the equation under study; here $\G_2^{\pm}=\bigl{(t,x)\colon t>0, x_2=\pm\ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$. This is an essential difference to the well-known result for solutions $v(t,x)\in C^{\infty}(\Bbb R^+\times\Bbb R^2\setminus (\Sigma_0\cup\Sigma_1^{\pm}\cup \Sigma_2^{\pm}))$ to the 2-D semilinear wave equation $\p_t^2v-\Delta v=f(t,x,v)$ with $(v(0,x), \p_tv(0,x))=(0, \vp(x))$, where $\Sigma_0={t=|x|}$, $\Sigma_1^{\pm}={t=\pm x_1}$, and $\Sigma_2^{\pm}={t=\pm x_2}$.