We investigate the nonhomogeneous initial-boundary value problem for the fourth-order nonlinear Schro & uml;dinger equation posed on the finite spatial interval [0, L], incorporating both nonlinear effects and nonhomogeneous boundary conditions. Specifically, we derive an explicit solution formula for the linear inhomogeneous problem via Laplace transform, which, together with the contraction mapping principle, is used to establish the local well-posedness of the initial-boundary value problem in C (0, T; Hs(0, L))boolean AND L2 (0, T; Hs+2(0, L)).
Understanding how forest ecosystems respond to megafires arising from climate change is a major challenge, as forests are well known to be an important reserve of carbon and biodiversity. In this paper, we present a new hybrid model that combines a continuous and deterministic mechanism governed by reaction-diffusion equations modeling the biological evolution of the forest, coupled with a discrete and probabilistic process reproducing the impact of forest fires. This original model allows us to study the competition and the ecological transitions between the temperate forest and the boreal forest. We prove that the hybrid model admits relevant equilibrium states of extinction, coexistence and invasion, and generates global solutions. We perform a complete stability analysis of the extinction equilibrium, establishing a non trivial parameter regime where the deterministic component of the model is unstable, with a stabilization under the action of the probabilistic process. We show that well chosen trajectories of the model can reach an invasion equilibrium where the temperate forest colonizes the boreal forest. We also prove that the model is able to reproduce spatial heterogeneity with the formation of patterns corresponding to enclaves of temperate forest within the boreal forest.
Given a language generated by an automaton over an alphabet of real (or complex) numbers, and a real or complex number beta in the open unit disk, we consider the beta-direct valuation and reversed valuation sets of this language. For the k-Pell, k-Fibonacci and k-metallic languages, we compute and describe the topological properties of these valuation sets when beta is the smaller root of the polynomial associated with these languages. Furthermore, we introduce the connectivity locus for the valuations of these languages, establish bounds for these sets, and exhibit some points in their boundary
