Methods and Applications of Analysis

Volume 28 (2021)

Number 3

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part II

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Finite time blowup for the 1-D semilinear generalized Tricomi equation with subcritical or critical exponents

Pages: 313 – 324

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n3.a4

Authors

Daoyin He (School of Mathematics, Southeast University, Nanjing, China)

Ingo Witt (Mathematical Institute, University of Göttingen, Germany)

Huicheng Yin (School of Mathematical Sciences, Nanjing Normal University, Nanjing, China)

Abstract

For the 1-D semilinear generalized Tricomi equation with the subcritical or critical exponents\[\partial^2_t u - t^m \partial^2_x u = {\lvert u \rvert}^p, \qquad (u(0, x), \partial_t u(0, x)) = (u_0(x), u_1(x)),\]where $t \gt 0, x \in \mathbb{R}, 1 \lt p \leq p_m = 1+ \frac{4}{m}$ and $u_i \in C^\infty_0 (\mathbb{R}) (i = 0, 1)$, we shall prove that the weak solution $u$ generally blows up in finite time. Note that for the 1‑D equation $\partial^2_t u - t^m \partial^2_x u = {\lvert u \rvert}^p$ with $p \gt p_m$, the global existence of small value weak solution u has been obtained by us. By this paper and our previous papers, we have given a systematic study on the blowup or global existence of small value solution $u$ to the equation $\partial^2_t u - t^m \Delta u = \lvert u \rvert p$ for all space dimensions and all $p \gt 1$. One main ingredient in this paper is to apply the explicit solution formula of linear generalized Tricomi equation to derive the crucial inequality $G(t) = \int_\mathbb{R} u(t, x)dx \geq Ct \ln t$ for large $t \gt 0$ and $p = p_m$ when $u_0 (x) \geq 0, u_1 (x) \geq 0$ and $(u_0 (x), u_1 (x)) \not \equiv 0$.

Keywords

generalized Tricomi equation, subcritical exponent, critical exponent, hypergeometric function, blowup

2010 Mathematics Subject Classification

35L65, 35L67, 35L70

Received 11 October 2020

Accepted 28 December 2020

Published 10 June 2022