We study the asymptotic behavior of solutions to the delayed monostable equationOur basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on R, every solution of equation ( * ) with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sharp form of the theory of spreading speeds for equation ( * ).AMS classification scheme numbers: 34K12, 35K57, 92D25
We propose a new approach for proving uniqueness of semi-wavefronts in generally nonmonotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.
We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation ( * ) u t (t, x) = u xx (t, x) − u(t, x) + g(u(t − h, x)), x ∈ R, t > 0, considered with Lipschitz continuous reaction term g : R + → R + . We are also assuming that g is C 1,α -smooth in some neighbourhood of the equilibria 0 and κ > 0 to ( * ). In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of g so that equation ( * ) can possess pushed traveling fronts. Our first main result says that the non-critical wavefronts of ( * ) with monotone g are globally nonlinearly stable. In the special and easier case when the Lipschitz constant for g coincides with g ′ (0), we present a series of results concerning the exponential [asymptotic] stability of non-critical [respectively, critical] fronts for monostable model ( * ). As an application, we present a criterion of the absolute global stability of non-critical wavefronts to the diffusive Nicholson's blowflies equation.
We give an iterative method to estimate the disturbance of semi-wavefronts of the equation:As a consequence, we show the exponential stability, with an unbounded weight, of semiwavefronts with speed c > 2 √ 2 and h > 0. Under the same restriction of c and h, the uniqueness of semi-wavefronts is obtained.If φ c (+∞) = 1 semi-wavefronts are called wavefronts.If h = 0, the questions on existence, uniqueness, geometry and stability of wavefronts have been satisfactory responded (see, e.g.[11] and [15] and references therein). In this case the general conclusions are: (i) semi-wavefronts are indeed monotone wavefronts existing for all c ≥ 2, (ii) two wavefronts with same speed are unique up to translations and (iii) wavefronts are stable under suitable perturbations.However, for h > 0 it has been only recently established the existence of semiwavefronts on the domain {(h, c) ∈ R 2 : h ≥ 0 and c ≥ 2} (see [7] and [2]). The
We study the large time asymptotic behavior of the solutions of the linear parabolic equation with delay ( * ):As an application we get estimates on the measure of level sets of non local KPP type equations with delay. For this type of nonlinear equations we prove that, in contrast with the classical case, the solution to the initial value problem with data of compact support may not be persistent.
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