Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations
نویسنده
چکیده
This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It first introduces the concept of generalized traveling wave solutions of time recurrent and space periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It then proves that in the direction of any unit vector ξ, there is c∗(ξ) such that for any c > c∗(ξ), a generalized traveling wave solution in the direction of ξ with averaged propagation speed c exists. It also proves that if the time recurrent and space periodic monostable equation is indeed time periodic, then c∗(ξ) is the minimal wave speed in the direction of ξ and the generalized traveling wave solution in the direction of ξ with averaged speed c > c∗(ξ) is a periodic traveling wave solution with speed c, which recovers the existing results on the existence of periodic traveling wave solutions in the direction of ξ with speed greater than the minimal speed in that direction.
منابع مشابه
Uniqueness of monostable pulsating wave fronts for time periodic reaction-diffusion equations
Keywords: Reaction–diffusion equations Pulsating wave fronts KPP and monostable nonlinearities Uniqueness a b s t r a c t We establish the uniqueness of pulsating wave fronts for reaction–diffusion equations in time periodic media with monostable nonlinearities. For the Kolmogorov–Petrovsky– Piskunov (KPP) type nonlinearity, this result provides a complete classification of all types of KPP pul...
متن کاملGeneralized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic reactions the fronts are proved to propagat...
متن کاملExistence of traveling waves for Lipschitz discrete dynamics. Monostable case as a limit of bistable cases
We study discrete monostable dynamics with general Lipschitz non-linearities. This includes also degenerate non-linearities. In the positive monostable case, we show the existence of a branch of traveling waves solutions for velocities c ≥ c+, with non existence of solutions for c < c+. We also give certain sufficient conditions to insure that c+ ≥ 0 and we give an example when c+ < 0. We as we...
متن کاملTraveling Waves of Some Symmetric Planar Flows of Non-Newtonian Fluids
We present some variants of Burgers-type equations for incompressible and isothermal planar flow of viscous non-Newtonian fluids based on the Cross, the Carreau and the power-law rheology models, and on a symmetry assumption on the flow. We numerically solve the associated traveling wave equations by using industrial data and in order to validate the models we prove existence and uniqueness of ...
متن کاملPlanar Traveling Waves for Nonlocal Dispersion Equation with Monostable Nonlinearity
In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in n-dimensional space ut − J ∗ u+ u+ d(u(t, x)) = ∫ Rn fβ(y)b(u(t− τ, x− y))dy, u(s, x) = u0(s, x), s ∈ [−τ, 0], x ∈ Rn, where the nonlinear functions d(u) and b(u) possess the monostable characters like Fisher-KPP type, fβ(x) is the heat kernel, and the kernel J(x) satisfies Ĵ(ξ) = 1 − K|ξ|α + o(|ξ|...
متن کامل