scholarly journals Nonlinear delay reaction-diffusion systems with nonlocal initial conditions having affine growth

2016 ◽  
pp. 1
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Initial Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Nonlinear Delay

scholarly journals A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

2015 ◽  
Vol 61 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Fixed Point ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Compactness Arguments ◽  
Fixed Point Techniques

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

scholarly journals Reaction-diffusion coupled inclusions with variable exponents and large diffusion

2021 ◽  
Vol 41 (4) ◽  
pp. 539-570
Author(s):  
Jacson Simsen ◽  
Mariza Stefanello Simsen ◽  
Petra Wittbold
Keyword(s):  
Initial Conditions ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Global Attractors ◽  
Coupled Systems ◽  
Variable Exponents ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Large Diffusion

This work concerns the study of asymptotic behavior of coupled systems of \(p(x)\)-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity.

scholarly journals Isolating Patterns in Open Reaction–Diffusion Systems

2021 ◽  
Vol 83 (7) ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Philip K. Maini ◽  
Denis Headon ◽  
Eamonn A. Gaffney
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
Mixed Boundary ◽  
Cell Signalling ◽  
Reaction Diffusion ◽  
Mixed Boundary Conditions ◽  
Three Dimensions ◽  
Spatiotemporal Pattern ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

AbstractRealistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

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