تقرير
Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity
| العنوان: | Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity |
|---|---|
| المؤلفون: | Miller, Thomas, Tam, Alexander K. Y., Marangell, Robert, Wechselberger, Martin, Bradshaw-Hajek, Bronwyn H. |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Dynamical Systems, 35K57 (Primary), 35B35 |
| الوصف: | Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u,$ which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a nonclassical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u = 1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position. Comment: 36 pages, 10 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2309.00204 |
| رقم الأكسشن: | edsarx.2309.00204 |
| قاعدة البيانات: | arXiv |
كن أول من يترك تعليقا!