In this paper we establish an existence and uniqueness result for a class of non-local elliptic differential equations with the Dirichlet boundary conditions, which, in general, do not accept a maximum principle. We introduce one monotone sequence of lower-upper pairs of solutions and prove uniform convergence of that sequence to the actual solution of the problem, which is the unique solution for some range of γ(the parameter of the problem). The convergence of the iterative sequence is tested through examples with an order of convergence greater than
In the current work, we study a stochastic parabolic problem. The presented problem is motivated by the study of an idealised electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference, a parameter that actually controls the operation of MEMS device. We first present the construction of the mathematical model, and then, we deduce some local existence results. Next for some particular versions of the model, relevant to various boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, which also allows the further investigation the problem with respect to its quenching behaviour.
We consider an initial boundary value problem for the non-local equation, ut = uxx + λf(u)/(∫-11 f(u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e-u, we give an asymptotic estimate: t* ∼ tu(λ - λ*)-1/2 for 0 < (λ - λ*) ≪ 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.
We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass. © 2017 Cambridge University Press
We prove an upper bound for the convergence rate of the homogenization limit epsilon -> 0 for a linear transmission problem for a advection-diffusion(-reaction) system posed in areas with low and high diffusivity, where epsilon is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493-516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.
We wish to understand the macroscopic plastic behaviour of metals by upscaling the micromechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of Gamma-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.