The Stefan problem for the Fisher-KPP equation with unbounded initial range

We consider the nonlinear Stefan problem \left \{ \begin{array} {ll} -d \Delta u=a u-b u^2 \;\; & \mbox{for } x \in \Omega (t), \; t>0, \\ u=0 \mbox{ and } u_t=\mu|\nabla_x u |^2 \;\;&\mbox{for } x \in \partial\Omega (t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in \Omega_0, \end{array}\right. where $\Omega(0)=\Omega_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $\Omega_0$ and $u_0$ vanishes on $\partial\Omega_0$. When $\Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by \cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $\Omega_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $\Omega_0$

Bistable pulsating fronts in slowly oscillating environments *

We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments

Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat

International audienceThis paper is concerned with the existence and qualitative properties of pulsating fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. We focus especially on the influence of the spatial period and, under various assumptions on the reaction terms and by using different types of arguments, we show several existence results when the spatial period is small or large. We also establish some properties of the set of periods for which there exist non-stationary fronts. Furthermore, we prove the existence of stationary fronts or non-stationary partial fronts at any period which is on the boundary of this set. Lastly, we characterize the sign of the front speeds and we show the global exponential stability of the non-stationary fronts for various classes of initial conditions

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

Spreading dynamics and sedimentary process of the Southwest Sub-basin, South China Sea: Constraints from multi-channel seismic data and IODP Expedition 349

© 2015 The Authors. Neotectonic and sedimentary processes in the South China Sea abyssal basin are still debated because of the lack of drilling evidence to test competing models. In this study, we interpreted four multi-channel seismic profiles across the Southwest Sub-basin (SWSB) and achieved stratigraphic correlation with new drilling data from Integrated Ocean Discovery Program (IODP) Expedition 349. Neogene sediments are divided into four stratigraphic units, each with distinctive seismic character. Sedimentation rate and lithology variations suggest climate-controlled sedimentation. In the late Miocene winter monsoon strength and increased aridity in the limited accumulation rates in the SWSB. Since the Pliocene summer monsoons and a variable glacial-interglacial climate since have enhanced accumulation rates. Terrigeneous sediments in the SWSB are most likely derived from the southwest. Three basement domains are classified with different sedimentary architectures and basement structures, including hyper-stretched crust, exhumed subcontinental mantle, and steady state oceanic crust. The SWSB has an asymmetric geometry and experienced detachment faulting in the final stage of continental rifting and exhumation of continental mantle lithosphere. Mantle lithospheric breakup post-dates crustal separation, delaying the establishment of oceanic spreading and steady state crust production