Vol. 314, No. 1, 2021

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On the global weak solution problem of semilinear generalized Tricomi equations, II

Daoyin He, Ingo Witt and Huicheng Yin

Vol. 314 (2021), No. 1, 29–80

In Part I (Calc. Var. Partial Differential Equations 56:2, (2017), 1–24), for the semilinear generalized Tricomi equation t2u tmΔu = |u|p with the initial data (u(0,x),tu(0,x)) =(u0(x),u1(x)), t 0, x n (n 3), p > 1 and m , we have shown that there exists a critical exponent pcrit(m,n) > 1 such that the weak solution u generally blows up in finite time when 1 < p < pcrit(m,n); and meanwhile there exists a conformal exponent pconf(m,n) ( > pcrit(m,n)) such that the weak solution u exists globally when p pconf(m,n) provided that (u0(x),u1(x)) are small. In the present paper, we shall prove that the small data weak solution u of t2u tmΔu = |u|p exists globally when pcrit(m,n) < p < pconf(m,n). 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 t2u tmΔu = |u|p for n 3. Here we point out that the study on the equation t2u tmΔu = |u|p is closely related to those of the semilinear wave equation t2u Δu + μ 1+ttu = |u|p for 0 < μ < 1 or other related physical problems.

generalized Tricomi equation, Fourier integral operator, global existence, weighted Strichartz estimate, weak solution
Mathematical Subject Classification
Primary: 35L65, 35L70
Received: 19 November 2018
Revised: 11 September 2020
Accepted: 22 May 2021
Published: 15 October 2021
Daoyin He
Southeast University
Ingo Witt
University of Göttingen
Huicheng Yin
Nanjing Normal University