Exponentially slow motion of interface layers for the one-dimensional Allen–Cahn equation with nonlinear phase-dependent diffusivity

Singular limit of an Allen-Cahn equation with nonlinear diffusion

“…In [11], Fife and Lacey generalized the Allen-Cahn equation, which leads them to a parameter dependent diffusion Allen-Cahn equation. Recently, [12] considered an Allen-Cahn equation with density dependent diffusion in 1 space dimension and showed a slow motion property. However, no rigorous proof on the motion of the interface in the nonlinear diffusion context has been given for larger space dimensions N ≥ 2.…”

Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity

“…Consequently, many extended Cahn-Hilliard or Allen-Cahn models that take these nonlinearities into account have been proposed in the literature (see, e.g., [6,7,9,31]) and which, in turn, have motivated their associated mathematical analyses (cf. [14,15,[18][19][20][21][22]). An interesting mathematical feature appears when the nonlinearity in the diffusion coefficient is degenerate, meaning that diffusion approaches zero when the density u approaches one or two of the pure phases.…”

“…[4,11,12,25]), in this work we apply the energy approach of Bronsard and Kohn [4] to rigorously prove the existence of metastable states for the initial boundary-value problem (IBVP) (1.1), (1.4) and (1.5). In a recent contribution [22], we studied the phenomenon of metastability of solutions to a related equation with density-dependent coefficients. The main differences between the model studied in [22] and (1.1) are (i) that equation (1.1) has a different structure, being the L 2 -gradient flow of a generalized Ginzburg-Landau energy functional (see (2.1) below) whereas the model in [22] is in conservation form; and, (ii) that in the present case the diffusivity is allowed to be degenerate, vanishing at one or both of the pure phases.…”

“…In a recent contribution [22], we studied the phenomenon of metastability of solutions to a related equation with density-dependent coefficients. The main differences between the model studied in [22] and (1.1) are (i) that equation (1.1) has a different structure, being the L 2 -gradient flow of a generalized Ginzburg-Landau energy functional (see (2.1) below) whereas the model in [22] is in conservation form; and, (ii) that in the present case the diffusivity is allowed to be degenerate, vanishing at one or both of the pure phases. Moreover, this work emphasizes the interplay between the diffusion function, D = D(•), and the potential, F = F (•), exemplified by the relation between their (real) exponents m > 1 and n ≥ 2.…”

Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms

“…This phenomenon is known in the literature as metastability, and it was first studied in the context of the classical Allen-Cahn equation in the pioneering works of Carr and Pego [6,7], Fusco and Hale [22] and Bronsard and Kohn [4], which appeared approximately at the same time and which applied different methodologies (see also [10]). Since then, the metastability of transition layer structures has been studied in (and extended to) many other models such as hyperbolic equations [16,17,19], parabolic systems [34], gradient flows [31], viscous conservation laws [20,25,30,33] or reaction-diffusion equations with phase-dependent diffusivities [18], just to mention a few.…”