All meromorphic solutions of an ordinary differential equation and its applications

2016 ◽  
Vol 39 (8) ◽  
pp. 2083-2092 ◽  
Author(s):  
Wenjun Yuan ◽  
Fanning Meng ◽  
Jianming Lin ◽  
Yonghong Wu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Meromorphic Solutions

All meromorphic solutions of an auxiliary ordinary differential equation and its applications

2015 ◽  
Vol 35 (5) ◽  
pp. 1241-1250 ◽  
Author(s):  
Wenjun YUAN ◽  
Weiling XIONG ◽  
Jianming LIN ◽  
Yonghong WU
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Meromorphic Solutions

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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