Local existence inC b 0, 1 and blow-up of the solutions of the Cauchy Problem for a quasilinear hyperbolic system with a singular source term

2001 ◽  
Vol 32 (3) ◽  
pp. 343-357
Author(s):  
Jo�o-Paulo Dias ◽  
M�rio Figueira
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic System ◽  
Source Term ◽  
Blow Up ◽  
Local Existence ◽  
Quasilinear Hyperbolic System ◽  
The Cauchy Problem ◽  
Singular Source

scholarly journals Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zenggui Wang
Keyword(s):  
Cauchy Problem ◽  
Life Span ◽  
Hyperbolic System ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Classical Solutions ◽  
Quasilinear Hyperbolic System ◽  
Inverse Mean Curvature Flow ◽  
The Cauchy Problem

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.

Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

2019 ◽  
Vol 18 (02) ◽  
pp. 333-358
Author(s):  
Ben Duan ◽  
Zhen Luo ◽  
Yan Zhou
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Force Term ◽  
The Cauchy Problem ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

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