共查询到20条相似文献,搜索用时 15 毫秒
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Mohammad Ibrahim El Smaily 《Annali di Matematica Pura ed Applicata》2010,189(1):47-66
This paper is concerned with some nonlinear propagation phenomena for reaction–advection–diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $u_t =\nabla\cdot(A(z)\nabla u)\;+q(z)\cdot\nabla u+\,f(z,u),\qquad t\in\mathbb{R},\quad z\in\Omega,$ propagating with a speed c. In the case of a “combustion” nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min–max and a max–min formula for this speed c. On the other hand, in the case of a “ZFK” or a “KPP” nonlinearity, there exists a minimal speed of propagation c*. In this situation, we give a min–max formula for c*. Finally, we apply this min–max formula to prove a variational formula involving eigenvalue problems for the minimal speed c* in the “KPP” case. 相似文献
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We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one. 相似文献
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P. V. Gordon C. B. Muratov M. Novaga 《Calculus of Variations and Partial Differential Equations》2013,47(3-4):683-709
We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions. 相似文献
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We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c? such that for each wave speed c≤c?, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c?. 相似文献
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Andrej Zlatoš 《Journal de Mathématiques Pures et Appliquées》2012,98(1):89-102
We use a new method in the study of Fisher–KPP reaction–diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction–diffusion equations in several spatial dimensions. Our method is based on the construction of sub- and super-solutions to the non-linear PDE from solutions of its linearization at zero. 相似文献
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J. A. Ferreira 《Applicable analysis》2013,92(12):1231-1246
In this article the qualitative properties of numerical traveling wave solutions for integro- differential equations, which generalize the well known Fisher equation are studied. The integro-differential equation is replaced by an equivalent hyperbolic equation which allows us to characterize the numerical velocity of traveling wave solutions. Numerical results are presented. 相似文献
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In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient
is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the
discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers
present in the solution. Using a Petrov–Galerkin finite element formulation, a fitted finite difference operator is shown
to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally
convergent in the pointwise maximum norm. 相似文献
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Xavier Cabré Anne-Charline Coulon Jean-Michel Roquejoffre 《Comptes Rendus Mathematique》2012,350(19-20):885-890
We are interested in the time asymptotic location of the level sets of solutions to Fisher–KPP reaction–diffusion equations with fractional diffusion in periodic media. We show that the speed of propagation is exponential in time, with a precise exponent depending on a periodic principal eigenvalue, and that it does not depend on the space direction. This is in contrast with the Freidlin–Gärtner formula for the standard Laplacian. 相似文献
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《Comptes Rendus Mathematique》2008,346(17-18):951-956
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L. Monsaingeon A. Novikov J.-M. Roquejoffre 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c∈]c?,+∞[, where c?>0 is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where u=0 and a region where u>0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u>0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed. 相似文献
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P. Broadbridge B. H. Bradshaw-Hajek 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2016,67(4):93
Reaction–diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms. 相似文献
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We state and discuss a number of fundamental asymptotic properties of solutions to one-dimensional advection–diffusion equations of the form , , , assuming initial values for some . To cite this article: P. Braz e Silva, P.R. Zingano, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
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We study a singularly perturbed periodic problem for the parabolic reaction–advection–diffusion equation with small advection. We consider the case in which there exists an internal transition layer under the conditions of balanced nonlinearity. An asymptotic expansion of the solution is constructed. To substantiate this asymptotics, we use the asymptotic method of differential inequalities. The Lyapunov asymptotic stability of the periodic solution is analyzed. 相似文献