Mathematical modelling of surface vibration with volume constraint and its analysis

A variational scheme for hyperbolic obstacle problems

“…In this section, as a first step towards obstacle problems, we extend to the fractional wave equation a time-disretized variational scheme which traces back to Rothe [25] and since then has been extensively applied to many different hyperbolic type problems, see e.g. [31,21,32,11].…”

“…Localizing everything on A δ , we can prove v t = u tt in A δ so that u tt ∈ L ∞ (0, T ; H −1 (A δ )), and equation (2) follows by passing to the limit as done in the proof of Theorem 3 (cf. [32,11]).…”

The Allen–Cahn equation with dynamic boundary conditions and mass constraints

“…More precisely, we require that the solution u satisfieswhere k , k are given constants fulfilling k Ä k , and w and w are prescribed weight functions on and , respectively. For example, in the case when w Á 1, w Á 0, and k D k , (1.3) represents the conservation of the volume R u.x, t/dx D k , for all t 2 OE0, T, a condition which instead arises naturally from the problem in the framework of a Cahn-Hilliard system (see, e.g., [13, 17] and [18] for a different type of natural mass conservation).The analysis of the abstract theory for this kind of constraint was developed in [19] and motivated from the generalization of concrete problems [20][21][22] (see also [17], where the essential structure of possible constraints has been discussed for Cahn-Hilliard equation). In the abstract approach by [19], the constraint and in particular the barriers k and k in (1.3) are allowed to depend on time.…”

Wave-type threshold dynamics and the hyperbolic mean curvature flow

“…1. Prescribe initial radius r 0 , initial velocity v 0 and time step τ = t e /N , where t e is the extinction time (16) and N ∈ N.…”