On semilinear Tricomi equations in one space dimension

TL;DR

This paper investigates the global existence and finite-time blowup of small data solutions to the 1-D semilinear Tricomi equation, identifying a critical exponent p=5 that determines the solution's behavior.

Contribution

It establishes the critical exponent for 1-D semilinear Tricomi equations and develops a weighted Strichartz inequality for the degenerate linear equation.

Findings

Global solutions exist for p>5 with small initial data.

Solutions blow up in finite time for 1<p<5.

A new weighted Strichartz inequality is proved for the degenerate equation.

Abstract

For 1-D semilinear Tricomi equation $\partial_t^2 u-t\partial_x^2u=|u|^p$ with initial data $(u(0,x), \partial_t u(0,x))$ $=(u_0(x), u_1(x))$, where $t\ge 0$, $x\in\mathbb{R}$, $p>1$, and $u_i\in C_0^\infty(\mathbb{R})$ ($i=0,1$), we shall prove that there exists a critical exponent $p_{\rm crit}=5$ such that the small data weak solution $u$ exists globally when $p>p_{\rm crit}$; on the other hand, the weak solution $u$, in general, blows up in finite time when $1<p<p_{\rm crit}$. We specially point out that for 1-D semilinear wave equation $\partial_t^2 v-\partial_x^2v=|v|^p$, the weak solution $v$ will generally blow up in finite time for any $p>1$. By this paper and \cite{HWYin1}-\cite{HWYin3}, we have given a systematic study on the blowup or global existence of small data solution $u$ to the equation $\partial_t^2 u-t\Delta u=|u|^p$ for all space dimensions. One of the main ingredients in the paper is to establish a crucial weighted Strichartz-type inequality for 1-D linear degenerate equation $\partial_t^2 w-t\partial_x^2 w=F(t,x)$ with $(w(0,x), \partial_t w(0,x))=(0,0)$, i.e., an inequality with the weight $(\frac{4}{9}t^3-|x|^2)^{\alpha}$ between the solution $w$ and the function $F$ is derived for some real numbers $\alpha$.