We study a reaction–diffusion mathematical model for the evolution of atherosclerosis as an inflammation process by combining analytical tools with computer-intensive numerical calculations. The computational work involved the calculation of more than sixty thousand solutions of the full reaction–diffusion system and lead to the complete characterisation of the \(\omega \)-limit for every initial condition. Qualitative properties of the solution are rigorously proved, some of them hinted at by the numerical study.

This work studies the sensitivity of a global climate model with deep ocean effect to the variations of a Solar parameter $Q$. The model incorporates a dynamic and diffusive boundary condition. We study the number of stationary solutions according to the positive parameter $Q$.

We present some results on the mathematical treatment of a global two-dimensional diffusive climate model with land - sea distribution. The model is based on a long time averaged energy balance and leads to a nonlinear parabolic equation for the averaged surface temperature. The spatial domain is a compact two-dimensional Riemannian manifold without boundary simulating the Earth surface with land - sea configuration. In the oceanic areas the model is coupled with a deep ocean model. The coupling is given by a dynamic and diffusive boundary condition. We study the existence of a bounded weak solution and its numerical approximation.

We consider a coupled model surface-deep ocean effect, where an Energy Balance Model (EBM) is used for modelling the surface temperature and a two-dimensional heat equation represents the evolution of the temperature of the deep ocean. Although the model under study is based on that proposed by Watts & Morantine (1990), here we consider a modified model that incorporates other processes, such as the nonlinear diffusion and the action of coalbedo, depending on the temperature. The stationary states of the model under study, taking the solar constant as the parameter, are numerically attained. The results of the simulation are depicted in a $\left\{\right(Q,u\left)\right\}$ plot where u is the temperature in the surface and Q is the solar constant. The numerical solution is achieved by means of a finite volume scheme with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space and third order Runge-Kutta scheme, which verifies the Total Variation Diminishing (TVD) property, for time integration. The equilibrium states are accomplished by evolving in time the numerical solution until the stationary solutions are reached. The main novel results of this work concern the numerical obtention of the stationary solutions of both the EBM and the coupled model EBM-deep ocean and the agreement of these results with the theoretically obtained in previous works, where an interval of values of the solar constant Q was obtained with the existence of at least three stationary solutions. In this work, we have numerically obtained more than three stationary solutions for such interval of Q.

We consider an initial boundary value problem of the complex Ginzburg-Landau equation with some delayed feedback terms proposed for the control of chemical turbulence in reaction diffusion systems. We consider the equation in a bounded domain \(\Omega\subset\mathbb{R}^{N}\) (\(N\leq3\)), $$ \frac{\partial u}{\partial t}-(1+i\epsilon)\Delta u +(1+i\beta) | u| ^2u-(1-i\omega) u=F(u(x,t-\tau)) $$ for t>0, with $$ F(u(x,t-\tau)) =e^{i\chi_0}\big\{ \frac{\mu}{| \Omega| }\int_{\Omega}u(x,t-\tau) dx+\nu u(x,t-\tau) \big\} , $$ where \(\mu\), \(\nu\geq0\), \(\tau>0\) but the rest of real parameters \(\epsilon\), \(\beta\), \(\omega\) and \(\chi_0\) do not have a prescribed sign. We prove the existence and uniqueness of weak solutions of problem for a range of initial data and parameters. When \(\nu=0\) and \(\mu>0\) we prove that only the initial history of the integral on \(\Omega\) of the unknown on \((-\tau,0)\) and a standard initial condition at t=0 are required to determine univocally the existence of a solution. We prove several qualitative properties of solutions, such as the finite extinction time (or the zero exact controllability) and the finite speed of propagation, when the term \(|u| ^2u\) is replaced by \(|u| ^{m-1}u\), for some \(m\in(0,1)\). We extend to the delayed case some previous results in the literature of complex equations without any delay. For more information see https://ejde.math.txstate.edu/Volumes/2020/40/abstr.html