32,006 research outputs found
Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations
This paper investigates the existence of generalized transition fronts for
Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that
the linearized elliptic operator near the unstable steady state admits an
almost periodic eigenfunction, we show that such fronts exist if and only if
their average speed is above an explicit threshold. This hypothesis is
satisfied in particular when the reaction term does not depend on x or (in some
cases) is small enough. Moreover, except for the threshold case, the fronts we
construct and their speeds are almost periodic, in a sense. When our hypothesis
is no longer satisfied, such generalized transition fronts still exist for an
interval of average speeds, with explicit bounds. Our proof relies on the
construction of sub and super solutions based on an accurate analysis of the
properties of the generalized principal eigenvalues
Transition Fronts in Time Heterogeneous and Random Media of Ignition Type
The current paper is devoted to the investigation of wave propagation
phenomenon in reaction-diffusion equations with ignition type nonlinearity in
time heterogeneous and random media. It is proven that such equations in time
heterogeneous media admit transition fronts or generalized traveling wave
solutions with time dependent profiles and that such equations in time random
media admit generalized traveling wave solutions with random profiles.
Important properties of generalized traveling wave solutions, including the
boundedness of propagation speeds and the uniform decaying estimates of the
propagation fronts, are also obtained
Transition fronts for periodic bistable reaction-diffusion equations
International audienceThis paper is concerned with the existence and qualitative properties of transition fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. The notion of transition fronts connecting two stable steady states generalizes the standard notion of pulsating fronts. In this paper, we prove that the time-global solutions in the class of transition fronts share some common features. In particular, we establish a uniform estimate for the mean speed of transition fronts, independently of the spatial scale. Under the a priori existence of a pulsating front with nonzero speed or under a more general condition guaranteeing the existence of such a pulsating front, we show that transition fronts are reduced to pulsating fronts, and thus are unique up to shift in time. On the other hand, when the spatial period is large, we also obtain the existence of a new type of transition fronts which are not pulsating fronts. This example, which is the first one in periodic media, shows that even in periodic media, the notion of generalized transition fronts is needed to describe the set of solutions connecting two stable steady states
Propagation phenomena for time heterogeneous KPP reaction-diffusion equations
We investigate in this paper propagation phenomena for the heterogeneous
reaction-diffusion equation , ,
, where f=f(t,u) is a KPP monostable nonlinearity which depends in a
general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t)
u(1-u), with m bounded and having positive infimum. We first prove the
existence of generalized transition waves (recently defined by Berestycki and
Hamel, Shen) for a given class of speeds. As an application of this result, we
obtain the existence of random transition waves when f is a random stationary
ergodic function with respect to t. Lastly, we prove some spreading properties
for the solution of the Cauchy problem
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