Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn't summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion pro\-blems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).Comment: arXiv admin note: text overlap with arXiv:1510.0420

2012 IMSAloquium, Student Investigation Showcase

Through SIR and its partnerships, IMSA students engage in rich opportunities to pursue compelling questions of interest, conduct investigations, engage with extraordinary advisors, communicate findings, and ultimately impact society.https://digitalcommons.imsa.edu/archives_sir/1004/thumbnail.jp

Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models

Bayesian inference for linear stochastic differential equations with application to biological processes

Ph. D. Thesis.Stochastic differential equations (SDEs) provide a natural framework for describing the stochasticity inherent in physical processes that evolve continuously over time. In this thesis, we consider the problem of Bayesian inference for a specific class of SDE – one in which the drift and diffusion coefficients are linear functions of the state. Although a linear SDE admits an analytical solution, the inference problem remains challenging, due to the absence of a closed form expression for the posterior density of the parameter of interest and any unobserved components. This necessitates the use of sampling-based approaches such as Markov chain Monte Carlo (MCMC) and, in cases where observed data likelihood is intractable, particle MCMC (pMCMC). When data are available on multiple experimental units, a stochastic differential equation mixed effects model (SDEMEM) can be used to further account for between-unit variation. Integrating over this additional uncertainty is computationally demanding. Motivated by two challenging biological applications arising from physiology studies of mice, the aim of this thesis is the development of efficient sampling-based inference schemes for linear SDEs. A key contribution is the development of a novel Bayesian inference scheme for SDEMEMs