Stability properties of solutions to convection-reaction equations with nonlinear diffusion

Alessandro Alla; Alessandra De Luca; Raffaele Folino; Marta Strani; Alessandro Alla; Alessandra De Luca; Raffaele Folino; Marta Strani

doi:10.3934/mine.2025023

Mathematics in Engineering

2025, Volume 7, Issue 5: 553-585. doi: 10.3934/mine.2025023

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Research article

Stability properties of solutions to convection-reaction equations with nonlinear diffusion

1.
Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 5, Roma 00185, Italy
2.
Dipartimento di Matematica "Giuseppe Peano", Università di Torino, Via Verdi 8, Torino 10124, Italy
3.
Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, C.P. 04510 Cd. de México, México
4.
Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia Mestre, Campus Scientifico, Via Torino 155, Venezia Mestre 30170, Italy

Received: 02 June 2025 Revised: 02 September 2025 Accepted: 02 September 2025 Published: 08 September 2025

In this paper we study a convection-reaction-diffusion equation of the form $ u_t = \varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \; \; t > 0, $ with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $ \varepsilon > 0 $ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $ \varepsilon $) time in a transition non-stable phase, before converging to a stable configuration.
- nonlinear diffusion,
- steady-states,
- asymptotic behavior,
- metastability,
- numerical approximation
Citation: Alessandro Alla, Alessandra De Luca, Raffaele Folino, Marta Strani. Stability properties of solutions to convection-reaction equations with nonlinear diffusion[J]. Mathematics in Engineering, 2025, 7(5): 553-585. doi: 10.3934/mine.2025023

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Abstract

In this paper we study a convection-reaction-diffusion equation of the form $ u_t = \varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \; \; t > 0, $ with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $ \varepsilon > 0 $ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $ \varepsilon $) time in a transition non-stable phase, before converging to a stable configuration.

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