From transient elastic linkages to friction: a complete study of a penalized fourth order equation with delay
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910721395130368 |
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| author | Milisic, Vuk Souplet, Philippe |
| author_facet | Milisic, Vuk Souplet, Philippe |
| contents | In this paper we consider a fourth order nonlinear parabolic delayed problem modelling a quasi-instantaneous turn-over of linkages in the context of cell-motility. The model depends on a small parameter $ε$ which represents a typical time scale of the memory effect. We first prove global existence and uniqueness of solutions for $ε$ fixed. This is achieved by combining suitable fixed-point and energy arguments and by uncovering a nonlocal in time, integral conserved quantity. After giving a complete classification of steady states in terms of elliptic functions, we next show that every solution converges to a steady state as $t \to \infty$. When $ε\to 0$, we then establish convergence results on finite time intervals, showing that the solution tends in a suitable sense towards the solution of a parabolic problem without delay. Moreover, we establish the convergence of energies as $ε\to 0$, which enables us to show that, for $ε$ small enough, the $ε$-dependent problem inherits part of the large time asymptotics of the limiting parabolic problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From transient elastic linkages to friction: a complete study of a penalized fourth order equation with delay Milisic, Vuk Souplet, Philippe Analysis of PDEs 35B40, 35B25, 35B45, 45K05, 35K55, 35K35, 35Q80 In this paper we consider a fourth order nonlinear parabolic delayed problem modelling a quasi-instantaneous turn-over of linkages in the context of cell-motility. The model depends on a small parameter $ε$ which represents a typical time scale of the memory effect. We first prove global existence and uniqueness of solutions for $ε$ fixed. This is achieved by combining suitable fixed-point and energy arguments and by uncovering a nonlocal in time, integral conserved quantity. After giving a complete classification of steady states in terms of elliptic functions, we next show that every solution converges to a steady state as $t \to \infty$. When $ε\to 0$, we then establish convergence results on finite time intervals, showing that the solution tends in a suitable sense towards the solution of a parabolic problem without delay. Moreover, we establish the convergence of energies as $ε\to 0$, which enables us to show that, for $ε$ small enough, the $ε$-dependent problem inherits part of the large time asymptotics of the limiting parabolic problem. |
| title | From transient elastic linkages to friction: a complete study of a penalized fourth order equation with delay |
| topic | Analysis of PDEs 35B40, 35B25, 35B45, 45K05, 35K55, 35K35, 35Q80 |
| url | https://arxiv.org/abs/2401.01139 |