scholarly journals Some free boundary problems involving non-local diffusion and aggregation

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140275 ◽  
Author(s):  
José Antonio Carrillo ◽  
Juan Luis Vázquez
Keyword(s):  
Parabolic Equations ◽  
Free Boundary Problems ◽  
Diffusion Theory ◽  
Degenerate Parabolic Equations ◽  
Free Boundaries ◽  
Boundary Problems ◽  
Recent Trends ◽  
Non Local ◽  
Degenerate Parabolic

We report on recent progress in the study of evolution processes involving degenerate parabolic equations which may exhibit free boundaries. The equations we have selected follow two recent trends in diffusion theory: considering anomalous diffusion with long-range effects, which leads to fractional operators or other operators involving kernels with large tails; and the combination of diffusion and aggregation effects, leading to delicate long-term equilibria whose description is still incipient.

Coexistence Solutions for a Periodic Competition Model with Singular–Degenerate Diffusion

2016 ◽  
Vol 60 (4) ◽  
pp. 1065-1075
Author(s):  
Yifu Wang ◽  
Jingxue Yin ◽  
Yuanyuan Ke
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Parabolic Equations ◽  
Competition Model ◽  
Degenerate Parabolic Equations ◽  
Degenerate Diffusion ◽  
Volterra Type ◽  
Non Local ◽  
Degenerate Parabolic

AbstractWe investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

scholarly journals A direct algorithm in some free boundary problems

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Cornel M. Murea ◽  
Dan Tiba
Keyword(s):  
Parabolic Equations ◽  
Obstacle Problem ◽  
Free Boundary Problems ◽  
Stability And Convergence ◽  
Free Boundaries ◽  
Convergence Properties ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Examples ◽  
Coincidence Set

AbstractIn this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one- and two- phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

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