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- W2899408504 abstract "For 1-D semilinear Tricomi equation $partial_t^2 u-tpartial_x^2u=|u|^p$ with initial data $(u(0,x), partial_t u(0,x))$ $=(u_0(x), u_1(x))$, where $tge 0$, $xinmathbb{R}$, $p>1$, and $u_iin C_0^infty(mathbb{R})$ ($i=0,1$), 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 $partial_t^2 u-tDelta u=|u|^p$ 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 $partial_t^2 w-tpartial_x^2 w=F(t,x)$ 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 $alpha$." @default.
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- W2899408504 date "2018-10-27" @default.
- W2899408504 modified "2023-10-18" @default.
- W2899408504 title "On semilinear Tricomi equations in one space dimension" @default.
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- W2899408504 doi "https://doi.org/10.48550/arxiv.1810.12748" @default.
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