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Abstract

In Part I
(Calc. Var. Partial Differential Equations 56:2,
(2017), 1–24), for the semilinear generalized Tricomi equation
${\partial}_{t}^{2}u{t}^{m}\Delta u=u{}^{p}$ with the
initial data
$\left(u\left(0,x\right),{\partial}_{t}u\left(0,x\right)\right)=\left({u}_{0}\left(x\right),{u}_{1}\left(x\right)\right)$,
$t\ge 0$,
$x\in {\mathbb{R}}^{n}$
$\left(n\ge 3\right)$,
$p>1$ and
$m\in \mathbb{N}$,
we have shown that there exists a critical exponent
${p}_{crit}\left(m,n\right)>1$ such that the weak
solution
$u$ generally blows
up in finite time when
$1<p<{p}_{crit}\left(m,n\right)$;
and meanwhile there exists a conformal exponent
${p}_{conf}\left(m,n\right)$
$\left(>{p}_{crit}\left(m,n\right)\right)$ such that the weak
solution
$u$ exists
globally when
$p\ge {p}_{conf}\left(m,n\right)$
provided that
$\left({u}_{0}\left(x\right),{u}_{1}\left(x\right)\right)$
are small. In the present paper, we shall prove that the small data weak solution
$u$ of
${\partial}_{t}^{2}u{t}^{m}\Delta u=u{}^{p}$ exists globally
when
${p}_{crit}\left(m,n\right)<p<{p}_{conf}\left(m,n\right)$.
Hence, collecting the results in this paper and the previous paper, we have given a
basically systematic study on the blowup or global existence of small data weak solution
$u$ to the
equation
${\partial}_{t}^{2}u{t}^{m}\Delta u=u{}^{p}$
for
$n\ge 3$.
Here we point out that the study on the equation
${\partial}_{t}^{2}u{t}^{m}\Delta u=u{}^{p}$
is closely related to those of the semilinear wave equation
${\partial}_{t}^{2}u\Delta u+\frac{\mu}{1+t}{\partial}_{t}u=u{}^{p}$ for
$0<\mu <1$ or
other related physical problems.

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Keywords
generalized Tricomi equation, Fourier integral operator,
global existence, weighted Strichartz estimate, weak
solution

Mathematical Subject Classification
Primary: 35L65, 35L70

Milestones
Received: 19 November 2018
Revised: 11 September 2020
Accepted: 22 May 2021
Published: 15 October 2021

