Abstract. We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H:where the map a : R + → R + is nonincreasing and the potential G : H → R is of class C 1 . If the coefficient a(t) is constant and positive, we recover the so-called "Heavy Ball with Friction" system. On the other hand, when a(t) = 1/(t + 1) we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function G is convex, the condition ∞ 0 a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∞ 0 e − t 0 a(s) ds dt < ∞ is necessary to obtain the convergence of the trajectories of (S) toward some minimum point of G. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function G. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.

A new mathematical model for the dynamics of prion proliferation involving an ordinary differential equation coupled with a partial integro-differential equation is analyzed, continuing the work in [J. Prüss, L. Pujo-Menjouet, G.F. Webb, R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. 6 (2006) 225-235]. We show the well-posedness of this problem in its natural phase space Z + := R + × L + 1 ((x 0 , ∞); x dx), i.e., there is a unique global semiflow on Z + associated to the problem. A theorem of threshold type is derived for this model which is typical for mathematical epidemics. If a certain combination of kinetic parameters is below or at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable in Z + ; above the threshold it is unstable, and there is another unique steady state, the disease equilibrium, which inherits that property.

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We give a complete characterization of the behavior of the Anderson acceleration (with arbitrary nonzero mixing parameters) on linear problems. Let ν be the grade of the residual at the starting point with respect to the matrix defining the linear problem. We show that if Anderson acceleration does not stagnate (that is, produces different iterates) up to ν, then the sequence of its iterates converges to the exact solution of the linear problem. Otherwise, the Anderson acceleration converges to a point that is not a solution. Anderson acceleration and GMRES are essentially equivalent up to the index where the iterates of Anderson acceleration begin to stagnate. This result holds also for an optimized version of Anderson acceleration, where at each step the mixing parameter is chosen so that it minimizes the residual of the current iterate.

We obtain comparison results for viscosity solutions of weakly coupled systems of Hamilton‐Jacobi equations. Perron's method is extended to give existence proofs for systems. A stronger definition of sub‐ and supersolutions is introduced, leading to existence results for continuous solutions. The needed sub‐ and supersolutions are constructed for continuous boundary data.