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- The Redner–Ben-Avraham–Kahng coagulation system with constant coefficients: the finite dimensional casePublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe study the behaviour as t → ∞ of solutions (cj (t)) to the Redner–Ben-Avraham–Kahng coagulation system with positive and compactly supported initial data, rigorously proving and slightly extending results originally established in [4] by means of formal arguments.
- On the convergence to critical scaling profiles in submonolayer deposition modelsPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelIn this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $n\geq 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $x=\tau$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $\xi:= (x − \tau )/ \tau$ , corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $\Phi_{2,n}(\xi)$ when $x, \tau \to +\infty$ with $\xi$ fixed, as well as the rate at which the limit is approached.
- The Redner - Ben-Avraham - Kahng cluster systemPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe consider a coagulation model first introduced by Redner, Ben-Avraham and Kahng in [11], the main feature of which is that the reaction between a j-cluster and a k-cluster results in the creation of a |j − k|-cluster, and not, as in Smoluchowski’s model, of a (j + k)-cluster. In this paper we prove existence and uniqueness of solutions under reasonably general conditions on the coagulation coefficients, and we also establish differenciability properties and continuous dependence of solutions. Some interesting invariance properties are also proved. Finally, we study the long-time behaviour of solutions, and also present a preliminary analysis of their scaling behaviour.
- Scaling behaviour in a coagulation-annihilation model and Lotka-Volterra competition systemsPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, Rafael; Roessel, Henry J. vanIn a recent paper, Laurencot and van Roessel (2010 J. Phys. A: Math. Theor., 43, 455210) studied the scaling behaviour of solutions to a two-species coagulation–annihilation system with total annihilation and equal strength coagulation, and identified cases where self-similar behaviour occurs, and others where it does not. In this paper, we proceed with the study of this kind of system by assuming that the coagulation rates of the two different species need not be equal. By applying Laplace transform techniques, the problem is transformed into a two-dimensional ordinary differential system that can be transformed into a Lotka–Volterra competition model. The long-time behaviour of solutions to this Lotka–Volterra system helps explain the different cases of existence and nonexistence of similarity behaviour, as well as why, in some cases, the behaviour is nonuniversal, in the sense of being dependent on initial conditions.
- Convergence to self-similarity in an addition model with power-like time-dependent input of monomersPublication . Costa, Fernando Pestana da; Sasportes, Rafael; Pinto, João TeixeiraIn this note we extend the results published in Ref. 1 to a coagulation system with Becker-Doring type interactions and time-dependent input of monomers $J_{1}(t)$ of power–like type: $J_{1}(t)/(\alpha t^{\omega }) \rightarrow 1$ as $t \rightarrow \infty$, with $\alpha > 0$ and $\omega > − \frac{1}{2}$. The general framework of the proof follows Ref. 1 but a different strategy is needed at a number of points.
- Modelling silicosis: dynamics of a model with piecewise constant rate coefficientsPublication . Antunes, Pedro R. S.; Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe study the dynamics about equilibria of an infinite dimensional system of ordinary differential equations of coagulation–fragmentation–death type that was introduced recently by da Costa et al. (Eur J Appl Math 31(6):950–967, 2020) as a model for the silicosis disease mechanism. For a class of piecewise constant rate coefficients an appropriate change of variables allows for the appearance of a closed finite dimensional subsystem of the infinite-dimensional system and the analysis of the eigenvalues of the linearizations of this finite dimensional subsystem about the equilibria is then used to obtain the results on the stability of the equilibria in the original infinite dimensional model.
- Steady state solutions in a model of a cholesteric liquid crystal samplePublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Grinfeld, Michael; Mottram, Nigel; Xayxanadasy, KedtysackMotivated by recent mathematical studies of Fréedericksz transitions in twist cells and helix unwinding in cholesteric liquid crystal cells [(da Costa et al. in Eur J Appl Math 20:269–287, 2009), (da Costa et al. in Eur J Appl Math 28:243–260, 2017), (McKay in J Eng Math 87:19–28, 2014), (Millar and McKay in Mol Cryst Liq Cryst 435:277/[937]–286/[946], 2005)], we consider a model for the director configuration obtained within the framework of the Frank-Oseen theory and consisting of a nonlinear ordinary differential equation in a bounded interval with non-homogeneous mixed boundary conditions (Dirichlet at one end of the interval, Neumann at the other). We study the structure of the solution set using the depth of the sample as a bifurcation parameter. Employing phase space analysis techniques, time maps, and asymptotic methods to estimate integrals, together with appropriate numerical evidence, we obtain the corresponding novel bifurcation diagram and discuss its implications for liquid crystal display technology. Numerical simulations of the corresponding dynamic problem also provide suggestive evidence about stability of some solution branches, pointing to a promising avenue of further analytical, numerical, and experimental studies.
- Kickback in nematic liquid crystalsPublication . Costa, Fernando Pestana da; Grinfeld, Michael; Langer, Mathias; Mottram, Nigel J.; Pinto, João TeixeiraWe describe a nonlocal linear partial differential equation arising in the analysis of dynamics of a nematic liquid crystal. We confirm that it accounts for the kickback phenomenon by decoupling the director dynamics from the flow. We also analyse some of the mathematical properties of the decoupled director equation.
- Modelling silicosis: existence, uniqueness and basic properties of solutionsPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.
- Bifurcations analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions: the asymmetric casePublication . Costa, Fernando Pestana da; Méndez, Maria Isabel; Pinto, João TeixeiraIn the paper, Bifurcation analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions (2009 Eur. J. Appl. Math. 20, 269–287) by da Costa et al. the twist-Fréedericksz transition in a nematic liquid-crystal one-dimensional cell of lenght L was studied, imposing an antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper, we extend that study to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions. We use phase-plane analysis tools and appropriately defined time maps to obtain the bifurcation diagrams of the model when L is the bifurcation parameter, and related these diagrams with the one in the antisymmetric situation. The stability of the bifurcating solutions is investigated by applying the method of Maginu (1978 J. Math. Anal. Appl. 63, 224–243).