This paper is concerned with propagation phenomena in a nonlocal delayed reaction-diffusion equation in time almost periodic media. More specifically, we investigate the existence and nonexistence of transition waves and the upper and lower bound estimates of spreading speeds. By studying properties of exponential decaying solutions of the corresponding linearized problem, constructing appropriate upper and lower solutions, and using comparison arguments, we first prove that no matter the birth rate function is monotone or not, there is a critical wave speed such that a transition wave exists as soon as the mean value of the wave speed is greater than this critical speed. Moreover, in the monotone case we confirm the asymptotic behavior of the transition wave. Under some further assumptions, we establish the upper and lower bound estimates of spreading speeds for solutions of the Cauchy problem with slow and fast exponential decay initial values and also obtain the nonexistence of transition waves. Finally, a brief discussion is given to demonstrate that the critical wave speed obtained in the present paper coincides with the minimum speed and spreading speeds observed by others for some special cases.
We study “change of weights” maps between loci of the stack of (φ, Γ)-modules over the Robba ring with integral Hodge–Tate–Sen weights. We show that in the GL2(ℚp) case these maps can realize translations of (φ, Γ)-modules geometrically. The motivation is to investigate translations of locally analytic representations under the categorical p-adic Langlands correspondence.
For general hyperbolic metric spaces, we introduce a new notion of a dual system (extending the influential notion from the context of normed linear spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Using this abstract notion of duality, we lift various notions from convex analysis into this nonlinear setting, including Fréchet differentiability and Bregman distances. Further, we introduce a notion of a monotone operator relative to a given dual system and, using the new Fréchet derivatives, we study corresponding resolvents relative to a given gradient, generalizing the seminal notion of Eckstein from the linear setting. These resolvents are then related to corresponding notions of Bregman nonexpansive mappings which are introduced relative to this generalization of the classical Bregman distance and we prove a convergence result of an analogue of the proximal point algorithm. For that, using methods from proof mining, we even provide quantitative results on its convergence in very general settings.
We fully characterize well-posedness in vector-valued Lebesgue–Bochner spaces of a non-local in time Moore–Gibson–Thompson equation in terms of the R-boundedness of the associated operator-valued symbol by using operator valued Fourier multipliers techniques. From our results, we are able to ensure Lp well-posedness when dealing with selfadjoint operators on Hilbert spaces or sectorial ones. This allows to include several classes of operators other than the Laplacian.
We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length sclG and the stable mixed commutator length sclG,N are not bi-Lipschitzly equivalent for the surface group G = π_1(Σ_ℓ) of genus at least 2 and its commutator subgroup N = [π_1(Σ_ℓ), π_1(Σ_ℓ)] . We also show the non-equivalence for a pair (G, N) such that G is the fundamental group of a 3-dimensional closed hyperbolic mapping torus. These pairs serve as the first family of examples of such (G, N) in which G is finitely generated.
