Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity
Date
2024
Authors
Miller, T.
Tam, A.K.Y.
Marangell, R.
Wechselberger, M.
Bradshaw Hajek, B.H.
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Studies in Applied Mathematics, 2024; 153(1, article no. e12685):1-23
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Abstract
Reaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field (Formula presented.) according to diffusion and net local changes. Usually, the diffusivity is positive for all values of (Formula presented.), which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity (Formula presented.) that is negative for (Formula presented.). We use a nonclassical symmetry to construct analytic receding time-dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, 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 (Formula presented.) and (Formula presented.) constant solutions, and prove for certain (Formula presented.) and (Formula presented.) that receding traveling waves are spectrally stable. In addition, 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 nonsymmetric diffusivity it results in a different shock position.
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Copyright 2024 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. (http://creativecommons.org/licenses/by/4.0/)