Nonlocal dynamic problems with singular nonlinearities and applications to MEMS (original) (raw)

Nonlocal Singular Problems and Applications to MEMS

—We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechan-icalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory.

PERIODIC SOLUTIONS TO NONLOCAL MEMS EQUATIONS In loving memory of Alfredo

Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices. New insights are also derived for the stationary problem improving previous existence results by removing smallness assumptions on the domain.

Existence and dynamic properties of a parabolic nonlocal MEMS equation

arXiv (Cornell University), 2008

Let Ω ⊂ R n be a C 2 bounded domain and χ > 0 be a constant. We will prove the existence of constants λ N ≥ λ * N ≥ λ * (1 + χ Ω dx 1−w *) 2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1 + χ Ω 1/(1 − v)dx) 2 in Ω, v = 0 on ∂Ω, such that a solution exists for any 0 ≤ λ < λ * N and no solution exists for any λ > λ N where λ * is the pull-in voltage and w * is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ր λ *. Moreover λ N < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the parabolic nonlocal MEMS equation u t = ∆u + λ/(1 − u) 2 (1 + χ Ω 1/(1 − u) dx) 2 in Ω × (0, ∞), u = 0 on ∂Ω × (0, ∞), u(x, 0) = u 0 in Ω. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.

The existence and dynamic properties of a parabolic nonlocal MEMS equation

Nonlinear Analysis: Theory, Methods & Applications, 2011

Let Ω ⊂ R n be a C 2 bounded domain and χ > 0 be a constant. We will prove the existence of constants λ N ≥ λ * N ≥ λ * (1 + χ Ω dx 1−w *) 2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1 + χ Ω 1/(1 − v)dx) 2 in Ω, v = 0 on ∂Ω, such that a solution exists for any 0 ≤ λ < λ * N and no solution exists for any λ > λ N where λ * is the pull-in voltage and w * is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ր λ *. Moreover λ N < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the parabolic nonlocal MEMS equation u t = ∆u + λ/(1 − u) 2 (1 + χ Ω 1/(1 − u) dx) 2 in Ω × (0, ∞), u = 0 on ∂Ω × (0, ∞), u(x, 0) = u 0 in Ω. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.

GLOBAL EXISTENCE FOR NONLOCAL MEMS PROBLEMS

We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of MicroElectroMechanicalSystems. The existence result, obtained within a suitable Implicit Function Theorem framework, is established under rather general boundary conditions and for bounded domains whose diameter is fairly small.

Singular perturbations of integro-differential equations

Applied Mathematics and Computation, 2006

We study the singular perturbation problem (E) 2 u (t) + u (t) = Au (t) + (K * Au)(t) + f (t), t ≥ 0, > 0, for the integrodifferential equation (E) w (t) = Aw(t) + (K * Aw)(t) + f (t), t ≥ 0, in a Banach space, when → 0 +. Under the assumption that A is the generator of a strongly continuous cosine family and under some regularity conditions on the scalar-valued kernel K we show that problem (E) has a unique solution u (t) for each small > 0. Moreover u (t) converges to u(t) as → 0 + , the unique solution of equation (E).

On solvability of the integro differential hyperbolic equation with purely non local conditions

In this study, we prove the existence, uniqueness, and continuous dependence upon the data of solution to integro-differential hyperbolic equation with purely nonlocal (integral) conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution using a numerical technique (Stehfest algorithm) by inverting the Laplace transform. Key words Integro-differential hyperbolic equation; approximate solution; nonlocal purely conditions 2010 MR Subject Classification 44A10; 34A12; 35L10

Some existence results for a nonlinear hyperbolic integrodifferential equation with singular kernel

Journal of Integral Equations and Applications, 1991

We consider the nonlinear Volterra integrodifferential equation u t (t, x) − t 0 a(t − s)σ(u x (s, x)) x ds = f (t, x), t ≥ 0, x ∈ R, with initial function u(0, x) = u 0 (x). We prove existence of global (in time) smooth solutions in the case where the data are small, assuming only a ∈ L 1 (R +) and strong positivity on the kernel. A local existence result for large data is obtained. The proofs use approximating kernels, uniformly of strong positive type and energy estimates.

Existence, uniqueness and a priori estimates for a nonlinear integro-differential equation

Ricerche di Matematica, 2008

The paper deals with the explicit calculus and the properties of the fundamental solution K of a parabolic operator related to a semilinear equation that models reaction diffusion systems with excitable kinetics. The initial value problem in all of the space is analyzed together with continuous dependence and a priori estimates of the solution. These estimates show that the asymptotic behavior is determined by the reaction mechanism. Moreover it's possible a rigorous singular perturbation analysis for discussing travelling waves with their characteristic times.