Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria
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Wave equations associated with Liouville‑type problems: global existence in time and blow‑up criteria Weiwei Ao1 · Aleks Jevnikar2 · Wen Yang3,4 Received: 15 January 2020 / Accepted: 28 August 2020 © The Author(s) 2020
Abstract We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality. Keywords Wave equation · Global existence · Blow-up criteria · Liouville-type equation · Mean field equation · Toda system · Sinh-Gordon equation · Moser–Trudinger inequality Mathematics Subject Classification 35L05 · 35J61 · 35R01 · 35A01
* Aleks Jevnikar [email protected] Weiwei Ao [email protected] Wen Yang [email protected] 1
Department of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
2
Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy
3
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, People’s Republic of China
4
Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
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W. Ao et al.
1 Introduction In this paper, we are concerned with wave equations associated with some Liouville-type problems on compact surfaces arising in mathematical physics: sinh-Gordon equation (1.1) and some general Toda systems (1.7). The first wave equation we consider is ( ( ) ) 1 1 e−u eu 2 − − − 𝜌2 on M, 𝜕t u − Δg u = 𝜌1 (1.1) ∫M eu |M| ∫M e−u |M| with u ∶ ℝ+ × M → ℝ , where (M, g) is a compact Riemann surface with total area |M| and metric g, Δg is the Laplace–Beltrami operator, and 𝜌1 , 𝜌2 are two real parameters. Nonlinear evolution equations have been extensively studied in the literature due to their many applications in physics, biology, chemistry, geometry and so on. In particular, the sinh-Gordon model (1.1) has been applied to a wide class of mathematical physics problems such as quantum field theories, non-commutative field theories, fluid dynamics, kink dynamics, solid-state physics, nonlinear optics and we refer to [1, 8, 12, 14, 36, 37, 47, 51] and the references therein. The stationary equation related to (1.1) is the following sinh-Gordon equation: ( ( ) ) 1 1 e−u eu − − − 𝜌2 −Δg u = 𝜌1 . (1.2) ∫M eu |M| ∫M e−u |M| In mathematical physics, the latter equation describ
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