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Register a new userPropagation phenomena in periodic patchy landscapes with interface conditions
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Authors
Franc{c}ois Hamel
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This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, c * , exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for c * by using the principal eigenvalues of certain linear periodic eigenvalue problems.
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We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $mathbb{R}^Nsetminus K$, where $Ksubsetmathbb{R}^N$ is a compact obstacle. The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as $tto-infty$ and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.
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We give a detailed study of attractors for measure driven quintic damped wave equations with periodic boundary conditions. This includes uniform energy-to-Strichartz estimates, the existence of uniform attractors in a weak or strong topology in the energy phase space, the possibility to present them as a union of all complete trajectories, further regularity, etc.
We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the recently suggested method of spatio-temporal averaging.
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