On semilinear Tricomi equations in one space dimension

For 1-D semilinear Tricomi equation ${\mathrm{\partial }}_t^{2}u-t{\mathrm{\partial }}_x^{2}u=\mathrm{\mid }u{\mathrm{\mid }}^p$ with initial data $(u(0,x),{\mathrm{\partial }}_{t}u(0,x))$ $=(u_0(x),u_1(x))$, where $t\ge 0$, $x\in \mathbb{R}$, $p>1$, and $u_i\in C_0^{\mathrm{\infty }}(\mathbb{R})$ ($i=0,1$), we shall prove that there exists a critical exponent $p_{\rm crit}=5$suchthatthesmalldataweaksolution$u$existsgloballywhen$p>p_{\rm crit}$;ontheotherhand,theweaksolution$u$, in general, blows up in finite time when $1<p<p_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$. We specially point out that for 1-D semilinear wave equation ${\mathrm{\partial }}_t^{2}v-{\mathrm{\partial }}_x^{2}v=\mathrm{\mid }v{\mathrm{\mid }}^p$, 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 ${\mathrm{\partial }}_t^{2}w-t{\mathrm{\partial }}_x^{2}w=F(t,x)$ with $(w(0,x),{\mathrm{\partial }}_{t}w(0,x))=(0,0)$, i.e., an inequality with the weight $(\frac{4}{9}{t}^3-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\alpha }$ between the solution $w$ and the function $F$ is derived for some real numbers $\alpha$.