Stability, Uniqueness and Recurrence of Generalized Traveling Waves in Time Heterogeneous Media of Ignition Type

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave

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

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

AMS(MOS) subject classifications: 34C27, 34D05, 35B15, 35B40, 35K57, 54H20.The current series of papers, which consists of three parts, are devoted to the study of almost automorphic dynamics in differential equations. By making use of techniques from abstract topological dynamics, we show that almost automorphy, a notion which was introduced by S. Bochner in 1955, is essential and fundamental in the qualitative study of almost periodic differential equations. Fundamental notions from topological dynamics are introduced in the first part. Harmonic properties of almost automorphic functions such as Fourier series and frequency module are studied. A module containment result is provided. In the second part, we study lifting dynamics of w-limit sets and minimal sets of a skew-product semiflow from an almost periodic minimal base flow. Skewproduct semiflows with (strongly) order preserving or monotone natures on fibers are given a particular attention. It is proved that a linearly stable minimal set must be almost automorphic and become almost periodic if it is also uniformly stable. Other issues such as flow extensions and the existence of almost periodic global attractors, etc. are also studied. The third part of the series deals with dynamics of almost periodic differential equations. In this part, we apply the general theory developed in the previous two parts to study almost automorphic and almost periodic dynamics which are lifted from certain coefficient structures (e.g., almost automorphic or almost periodic) of differential equations. It is shown that (harmonic or subharmonic) almost automorphic solutions exist for a large class of almost periodic ordinary, parabolic and delay differential equations.Partially supported by NSF grants DMS-9207069, DMS-9402945 and DMS-9501412