scholarly journals Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations

2019 ◽  
Vol 39 (5) ◽  
pp. 2877-2891
Author(s):  
H. A. Erbay ◽  
◽  
S. Erbay ◽  
A. Erkip ◽  
Keyword(s):  
Wave Equations ◽  
Existence Of Solutions ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence ◽  
Nonlocal Nonlinear

scholarly journals The lifespan for 3-dimensional quasilinear wave equations in exterior domains

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
John Helms ◽  
Jason Metcalfe
Keyword(s):  
Wave Equations ◽  
Energy Estimates ◽  
Exterior Domains ◽  
Small Initial Data ◽  
3 Dimensional ◽  
Long Time Existence ◽  
Homogeneous Wave ◽  
Long Time ◽  
Quasilinear Wave Equations ◽  
Time Existence

AbstractThis article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

scholarly journals Long time existence of solutions to an elastic flow of networks

2020 ◽  
Vol 45 (10) ◽  
pp. 1253-1305 ◽  
Author(s):  
Harald Garcke ◽  
Julia Menzel ◽  
Alessandra Pluda
Keyword(s):  
Existence Of Solutions ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Long-Time Existence of Solutions to Boussinesq Systems

2012 ◽  
Vol 44 (6) ◽  
pp. 4078-4100 ◽  
Author(s):  
Mei Ming ◽  
Jean Claude Saut ◽  
Ping Zhang
Keyword(s):  
Existence Of Solutions ◽  
Long Time Existence ◽  
Long Time ◽  
Boussinesq Systems ◽  
Time Existence

scholarly journals The parabolic Monge–Ampère equation on compact almost Hermitian manifolds

2020 ◽  
Vol 2020 (761) ◽  
pp. 1-24 ◽  
Author(s):  
Jianchun Chu
Keyword(s):  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Limit Function ◽  
Uniqueness Of Solutions ◽  
Long Time Existence ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Long Time ◽  
Monge Ampere ◽  
Time Existence

AbstractWe prove the long time existence and uniqueness of solutions to the parabolic Monge–Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in {C^{\infty}} topology as {t\rightarrow\infty}. Up to scaling, the limit function is a solution of the Monge–Ampère equation. This gives a parabolic proof of existence of solutions to the Monge–Ampère equation on almost Hermitian manifolds.

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