Non-Local Partial Differential Equations for Engineering and Biology

Kavallaris, Nikos I.; Suzuki, Takashi

doi:10.1007/978-3-319-67944-0

Cited by 41 publications

(36 citation statements)

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“…In particular, it follows from Proposition 1.2.2 page 12 in Kavallaris and Suzuki [11] together with Lemma 2.1 that equation…”

Section: Existence Of a Solution Trapped In S * (T )mentioning

confidence: 91%

“…We kindly ask the reader to self-check. [11]), we can derive the local existence and uniqueness of classical solutions of equations (2.5) and (2.8). Since nonnegativity is preserved for these equation, we will assume thatū and U are nonegative.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…According to the above mentioned reference in [11], the local Cauchy problem of (1.2) is solved. Then, either our solution is global in time or there exists T > 0 such that lim inf We can see that if the above condition occurs, the right-hand side of (1.2) may become singular.…”

Section: Introductionmentioning

confidence: 99%

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Profile of a touch-down solution to a nonlocal MEMS model

Duong

Zaag

2019

Math. Models Methods Appl. Sci.

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In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on (0, T ) × Ω :where Ω is a bounded domain in R n and λ, γ > 0. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point a ∈ Ω. In particular, we give a description of the quenching behavior according to the following final profile1 3 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

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“…In particular, it follows from Proposition 1.2.2 page 12 in Kavallaris and Suzuki [11] together with Lemma 2.1 that equation…”

Section: Existence Of a Solution Trapped In S * (T )mentioning

confidence: 91%

Section: Formulation Of the Problemmentioning

confidence: 99%

See 1 more Smart Citation

Profile of a touch-down solution to a nonlocal MEMS model

Duong

Zaag

2019

Math. Models Methods Appl. Sci.

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“…In a biological context such terms can account for both local and nonlocal interactions between cells and their surroundings. We refer to [64,113] for a rich variety of nonlocal reaction models in engineering and biology. Several model classes have also been developed featuring integral terms that describe nonlocality with respect to one or several other variables, including age, phenotype, internal cell state, velocity, etc.…”

Section: (34)mentioning

confidence: 99%

Mathematical models for cell migration: a non-local perspective

Chen

Painter

Surulescu

et al. 2020

Phil. Trans. R. Soc. B

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We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell’s motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell–cell and cell–tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

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“…However, the integral terms allow us to consider PDEs with non-local behavior. In fact, such PDEs model various processes in engineering and biology like microelectro-mechanical-systems, Ohmic heating phenomena, chemotaxis, and cell dynamics [23].…”

Section: Problem Statementmentioning

confidence: 99%

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

Gahlawat

Valmórbida

2020

IEEE Trans. Automat. Contr.

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We present a method for the stability analysis of a large class of linear Partial Differential Equations (PDEs) in one spatial dimension. We rely on Lyapunov analysis to establish the exponential stability of the systems under consideration. The proposed test for the verification of the underlying Lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations. We illustrate the application of this method using a PDE actuated by a backstepping computed feedback law. Furthermore, for the case of PDEs defined by polynomial data, we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically. We show the effectiveness of the proposed numerical methodology using examples of different types of PDEs.

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Non-Local Partial Differential Equations for Engineering and Biology

Cited by 41 publications

References 0 publications

Profile of a touch-down solution to a nonlocal MEMS model

Profile of a touch-down solution to a nonlocal MEMS model

Mathematical models for cell migration: a non-local perspective

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

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