On semilinear Tricomi equations in one space dimension

Abstract

For 1-D semilinear Tricomi equation ${\partial }_t^{2}u-t{\partial }_x^{2}u=|u{|}^p$ 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 $p_{\mathrm{crit}}=5$ such that the small data weak solution $u$ exists globally when $p>p_{\mathrm{crit}}$; on the other hand, the weak solution $u$, in general, blows up in finite time when $1<p<p_{\mathrm{crit}}$. We specially point out that for 1-D semilinear wave equation ${\partial }_t^{2}v-{\partial }_x^{2}v=|v{|}^p$, 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 $\alpha$.

22 pages. arXiv admin note: text overlap with arXiv:1611.07606