Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control

This is the peer reviewed version of the following article: Kavallaris, N. I. (2018). Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control. Mathematical Methods in the Applied Sciences, 41(3), 1074-1082. doi:10.1002/mma.4176, which has been published in final form at DOI: 10.1002/mma.4176. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-ArchivingIn the current paper, we consider a stochastic parabolic equation which actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system (MEMS). We first present the derivation of the mathematical model. Then after establishing the local well-posedeness of the problem we investigate under which circumstances a {\it finite-time quenching} for this SPDE, corresponding to the mechanical phenomenon of {\it touching down}, occurs. For that purpose the Kaplan's eigenfunction method adapted in the context of SPDES is employed

A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding

This is the author's PDF version of an article published in Mathematical Modelling of Natural Phenomena© 2015. The definitive version is available at http://www.mmnp-journal.org/articles/mmnp/abs/2015/06/mmnp2015106p90/mmnp2015106p90.htmlIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm

An analytic approach to the normalized Ricci flow-like equation: Revisited

NOTICE: this is the author’s version of a work that was accepted for publication in Applied Mathematics Letters. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Applied Mathematics Letters, 44, June 2015, pp. 30-33, http://dx/doi.org/10.1016/j.aml.2014.12.009In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence

Finite-time blow-up of a non-local stochastic parabolic problem

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up

A Novel Averaging Principle Provides Insights in the Impact of Intratumoral Heterogeneity on Tumor Progression

Typically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a well-known Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account

Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta_c is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point

Delineating the role of βIV-Tubulins in pancreatic cancer: βIVb-tubulin inhibition sensitizes pancreatic cancer cells to vinca alkaloids

Pancreatic cancer (PC) is a lethal disease which is characterized by chemoresistance. Components of the cell cytoskeleton are therapeutic targets in cancer. βIV-tubulin is one such component that has two isotypes—βIVa and βIVb. βIVa and βIVb isotypes only differ in two amino acids at their C-terminus. Studies have implicated βIVa-tubulin or βIVb-tubulin expression with chemoresistance in prostate, breast, ovarian and lung cancer. However, no studies have examined the role of βIV-tubulin in PC or attempted to identify isotype specific roles in regulating cancer cell growth and chemosensitivity. We aimed to determine the role of βIVa- or βIVb-tubulin on PC growth and chemosensitivity. PC cells (MiaPaCa-2, HPAF-II, AsPC1) were treated with siRNA (control, βIVa-tubulin or βIVb-tubulin). The ability of PC cells to form colonies in the presence or absence of chemotherapy was measured by clonogenic assays. Inhibition of βIVa-tubulin in PC cells had no effect chemosensitivity. In contrast, inhibition of βIVb-tubulin in PC cells sensitized to vinca alkaloids (Vincristine, Vinorelbine and Vinblastine), which was accompanied by increased apoptosis and enhanced cell cycle arrest. We show for the first time that βIVb-tubulin, but not βIVa-tubulin, plays a role in regulating vinca alkaloid chemosensitivity in PC cells. The results from this study suggest βIVb-tubulin may be a novel therapeutic target and predictor of vinca alkaloid sensitivity for PC and warrants further investigation

A stationary free boundary problem modeling electrostatic MEMS

A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero