scholarly journals The behavior of the free boundary for reaction–diffusion equations with convection in an exterior domain with Neumann or Dirichlet boundary condition

2016 ◽  
Vol 260 (6) ◽  
pp. 5075-5102
Author(s):  
Ross G. Pinsky
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Dirichlet Boundary Condition ◽  
Exterior Domain ◽  
Dirichlet Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

A diffusive Holling–Tanner prey–predator model with free boundary

2018 ◽  
Vol 11 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Chenglin Li
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Global Existence ◽  
Comparison Principle ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Threshold Value ◽  
Spreading Coefficient ◽  
Initial Size ◽  
Predator Model

This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out.

Positive solutions for diffusive Logistic equation with refuge

2019 ◽  
Vol 9 (1) ◽  
pp. 1092-1101 ◽  
Author(s):  
Jian-Wen Sun
Keyword(s):  
Boundary Condition ◽  
Weight Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Value ◽  
Stationary Solutions ◽  
Logistic Equation ◽  
Diffusion Equations ◽  
Diffusion Operator ◽  
Diffusive Logistic Equation

Abstract In this paper, we study the stationary solutions of the Logistic equation $$\begin{array}{} \displaystyle u_t=\mathcal {D}[u]+\lambda u-[b(x)+\varepsilon]u^p \text{ in }{\it\Omega} \end{array}$$ with Dirichlet boundary condition, here 𝓓 is a diffusion operator and ε > 0, p > 1. The weight function b(x) is nonnegative and vanishes in a smooth subdomain Ω0 of Ω. We investigate the asymptotic profiles of positive stationary solutions with the critical value λ when ε is sufficiently small. We find that the profiles are different between nonlocal and classical diffusion equations.

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

2012 ◽  
Vol 67 (8-9) ◽  
pp. 479-482 ◽  
Author(s):  
Junping Zhao
Keyword(s):  
Boundary Condition ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Existence Theorems ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Gradient Term

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)▽u)+ f (x;u; |▽u|2; t) under nonlinear boundary condition ¶u=¶n+g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f .

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