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IIT Jodhpur

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm. © 2020 American Institute of Mathematical Sciences. All rights reserved.

Department of Mathematics

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Communications on Pure and Applied Analysis

Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions

We consider reaction diffusion systems where some components react and diffuse on the surface, and others diffuse inside the domain and react with surface components through dynamic boundary conditions. Under reasonable hypotheses, we establish the existence of componentwise non-negative global solutions. © 2020 Khayyam Publishing. All Rights Reserved.

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Differential and Integral Equations

Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions

We consider a question of global existence for two component volume-surface reaction-diffusion systems. The first of the components diffuses in a region, and then reacts on the boundary with the second component, which diffuses on the boundary. We show that if the first component is bounded a priori on any time interval, and the kinetic terms satisfy a generalized balancing condition, then both solutions exist globally. We also pose an open question in the opposite direction, and give some a priori estimates for associated m component systems. © Springer International Publishing AG, part of Springer Nature 2019.

Computational Methods in Applied Sciences

Martin’s problem for volume-surface reaction-diffusion systems

We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d≤3 space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model. © 2016 Elsevier Ltd

Computers and Mathematics with Applications

Finite element approximation of a spatially structured metapopulation PDE model

We consider reaction-diffusion systems where some compo- i nents react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those 011 the boundary through mass transport. We establish criteria guaranteeing that solutions are uniformly bounded in time.

Uniform bounds for solutions to volume-surface reaction diffusion systems

We consider a reaction-diffusion system where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish local well-posedness and global existence of solutions for these systems using classical potential theory and linear estimates for initial boundary value problems. © 2016 Society for Industrial and Applied Mathematics.