Structural stability and stabilization of solutions of the reversible three‐component Gray‐Scott system

Abstract

This paper is concerned with the structural stability and stabilization of solutions to the three‐component reversible Gray‐Scott system under the Dirichlet or Neumann boundary conditions defined in a bounded domain of Rn for 1 ≤ n ≤ 3. We prove that each solution depends on changes in a coefficient of the ratio of the reverse and forward reaction rates for the autocatalytic reaction as well as proving the continuous dependence on the initial data. We also prove that under Dirichlet's boundary conditions, the system is stabilized to the stationary solution by finitely many Fourier modes.

Entropy Production in Reaction–Diffusion Systems Confined in Narrow Channels

“…Its phase diagram presents a stationary-stable equilibrium point, saddle-node, and Hopf bifurcations. Heterogeneous structures typically arise near these bifurcation curves [ 15 , 16 ].…”

Stability for the Kelvin–Voigt variable order equations backward in time

“…The topic of continuous dependence of a solution to a boundary–initial value problem is very important. This subject, including continuous dependence upon the models themselves is analysed by previous works 23–39 . An increasingly important class of continuous dependence analyses are those pertaining to improperly posed problems, or non‐well‐posed problems; see previous literature 12,13,24,40–53 .…”

“…This subject, including continuous dependence upon the models themselves is analysed by previous works. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] An increasingly important class of continuous dependence analyses are those pertaining to improperly posed problems, or non-well-posed problems; see previous literature. 12,13,24,[40][41][42][43][44][45][46][47][48][49][50][51][52][53] Improperly posed problems have been amenable to analyse especially with the aid of the famous paper of John.…”

Continuous dependence on boundary and Soret coefficients in double diffusive bidispersive convection

“…In many ways, continuous dependence of the solution in changes in the differential equations or boundary conditions is as important as continuous dependence upon the initial data or stability. In fact, continuous dependence on the model occupies much recent research, see for example, Celik and Hoang, 38 Ciarletta et al, 39 Franchi et al, 18 Gentile and Straughan, 21 Harfash, 40 Hoang and Thinh, 41 Kalantarova and Ugurlu, 42 Li et al, 43 Liu, 44 Liu and Xiao, 45 Liu et al, 46 Scott, 47 Varsakelis and Papalexandris, 48 and Wang and Su 49 …”