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

TL;DR

This paper investigates the local existence and singularity structure of solutions to semilinear generalized Tricomi equations with discontinuous initial data, revealing cusp singularities and differences from classical wave equations.

Contribution

It establishes the existence and regularity properties of solutions with low regularity initial data, identifying cusp singularities and their geometric structures.

Findings

Solutions exist and are smooth away from cusp singularities.

Cusp singularities occur along specific geometric cones and wedges.

Differences from classical wave equations in singularity structure.

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 $\Sigma_2^{\pm}={t=\pm x_2}$.