We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen-Cahn and the Cahn-Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across an interface. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses and increases the bistable region. We find existence and non-existence results of single interfaces and striped patterns.
In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is related to "cellular instabilities" occuring in detonation and MHD. Our results extend to multiple dimensions the results of [AS12] on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results of [TZ08a] on transverse Hopf bifurcation, showing that the bifurcating time-periodic solution is in fact a spatially periodic traveling wave. Our technique consists of a Lyapunov-Schmidt type of reduction, which prepares the equations for the application of other bifurcation techniques. For the reduction in transverse modes, a general Fredholm Alternative-type result is derived, allowing us to overcome the unboundedness of the domain and the lack of compact embeddings; this result apply to general closed operators.
We study interfaces in an Allen‐Cahn equation, separating two metastable states. Our focus is on a directional quenching scenario, where a parameter renders the system bistable in a half plane and monostable in its complement, with the region of bistability expanding at a fixed speed. We show that the growth mechanism selects a contact angle between the boundary of the region of bistability and the interface separating the two metastable states. Technically, we focus on a perturbative setting in a vicinity of a symmetric situation with perpendicular contact. The main difficulty stems from the lack of Fredholm properties for the linearization in translation invariant norms. We overcome those difficulties establishing Fredholm properties in weighted spaces and farfield‐core decompositions to compensate for negative Fredholm indices.
In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in our case the interface moves with speed c in such a way that the bistable region grows. According to results from [MS17a, MS17b], several patterns exist when c 0, and here we investigate their persistence for finite c > 0, clarifying the pattern selection mechanism related to the speed c of the quenching front.
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