Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations

Matthieu Alfaro (1, 2) , Arnaud Ducrot (3) , Grégory Faye (4)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 BioSP - Biostatistique et Processus Spatiaux
3 LMAH - Laboratoire de Mathématiques Appliquées du Havre
4 IMT - Institut de Mathématiques de Toulouse UMR5219

We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions).

Mots clés: extinction; propagation; threshold phenomena; sharp threshold phenomena

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Arnaud Ducrot, Grégory Faye. Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations. SIAM Journal on Applied Dynamical Systems, 2020, 19 (2), pp.1291-1311. ⟨10.1137/19M1292187⟩. ⟨hal-02308673⟩