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We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which \begin{document}$ 0 $\end{document} is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczynski (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.

In this paper, we study the generation of interface for the solution of the mass conserved Allen--Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initi...

This paper is concerned with the Cauchy problem $u_{t}=u_{xx}+f(t,u),x\in \mathbb{R} ,t>0$ with initial function $u(0,\cdot)=u_0 (\cdot)\in L^{\infty} (\mathbb{R})$, where $f$ is a rather general n...

We consider the Cauchy problem ut=uxx+f(u), x∈R,t>0,u(x,0)=u0(x), x∈R, where f is a C1 function on R with f(0)=0, and u0 is a nonnegative continuous function on R whose limits at ±∞ are equal to 0....

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R n with n ≥ 2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V γ = κ γ + c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.

In living cells, we can observe a variety of complex network systems such as metabolic network. Studying their sensitivity is one of the main approaches for understanding the dynamics of these biological systems. The study of the sensitivity is done by increasing/decreasing, or knocking out separately, each enzyme mediating a reaction in the system and then observing the responses in the concentrations of chemicals or their fluxes. However, because of the complexity of the systems, it has been unclear how the network structures influence/determine the responses of the systems. In this study, we focus on monomolecular networks at steady state and establish a simple criterion for determining regions of influence when any one of the reaction rates is perturbed through sensitivity experiments of enzyme knock-out type. Specifically, we study the network response to perturbations of a reaction rate j∗ and describe which other reaction rates j′ respond by non-zero reaction flux, at steady state. Non-zero responses of j′ to j∗ are called flux-influence of j∗ on j′. The main and most important aspect of this analysis lies in the reaction graph approach, in which the chemical reaction networks are modelled by a directed graph. Our whole analysis is function-free, ie, in particular, our approach allows a graph theoretical description of sensitivity of chemical reaction networks. We emphasize that the analysis does not require numerical input but is based on the graph structure only. Our specific goal here is to address a topological characterization of the flux-influence relation in the network. In fact we characterize and describe the whole set of reactions influenced by a perturbation of any specific reaction.

We consider semilinear parabolic equations of the form equation(A) ut=uxx+f(u),x∈R,t∈I, where I=(0,∞)I=(0,∞) or I=(−∞,∞)I=(−∞,∞). Solutions defined for all (x,t)∈R2(x,t)∈R2 are referred to as entire solutions. Assuming that f∈C1(R)f∈C1(R) is of a bistable type with stable constant steady states 0 and γ>0γ>0, we show the existence of an entire solution U(x,t)U(x,t) of the following form. For t≈−∞t≈−∞, U(⋅,t)U(⋅,t) has two humps, or, pulses, one near ∞, the other near −∞. As t increases, the humps move toward the origin x=0x=0, eventually “colliding” and forming a one-hump final shape of the solution. With respect to the locally uniform convergence, the solution U(⋅,t)U(⋅,t) is a heteroclinic orbit connecting the (stable) steady state 0 to the (unstable) ground state of the equation uxx+f(u)=0uxx+f(u)=0. We find the solution U as the limit of a sequence of threshold solutions of the Cauchy problem for equation (A).

This work focuses on dynamics arising from reaction-diffusion equations , where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment, under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable non-linearities., Communications in Contemporary Mathematics, World Scientific Publishing, In press

We consider the Cauchy problem where f is a locally Lipschitz function on ℝ with f(0) = 0, and u0 is a nonnegative function in C0(ℝ), the space of continuous functions with limits at ± ∞ equal to 0...

We investigate spreading properties of solutions of a large class of two-component reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known-at least theoretically-about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution-one for the prey and the other for the predator-in some situations., Comment: arXiv admin note: text overlap with arXiv:1812.04440 by other authors

We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shrinking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. We also prove results on the concavity properties of solutions. One of the main tools of this paper is the intersection number principle, however in order to deal with solutions with free boundaries, we introduce what we call “the extended intersection number principle”, which turns out to be exceedingly useful in handling curves with moving endpoints.

6 Well-posedness in local uniform topology and the theory of semiflow 24 6.1 Well-posedness in local uniform topology . . . . . . . . . . . . 24 6.2 Steady-states of the equation . . . . . . . . . . . . . . . . . . 27 6.3 Semiflow of the mild solutions and existence of minimal wave speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.4 Linearized equation and semiflow . . . . . . . . . . . . . . . . 32

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where \({\Omega(t) \subset \mathbb{R}^{n}}\) (\({n \geqq 2}\)) is bounded by the free boundary \({\Gamma(t)}\) , with \({\Omega(0) = \Omega_0}\) , μ and d are given positive constants. The initial function u0 is positive in \({\Omega_0}\) and vanishes on \({\partial \Omega_0}\) . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary \({\Gamma(t)}\) is smooth outside the closed convex hull of \({\Omega_0}\) , and as \({t \to \infty}\) , either \({\Omega(t)}\) expands to the entire \({\mathbb{R}^n}\) , or it stays bounded. Moreover, in the former case, \({\Gamma(t)}\) converges to the unit sphere when normalized, and in the latter case, \({u \to 0}\) uniformly. When \({g(u) = au - bu^2}\) , we further prove that in the case \({\Omega(t)}\) expands to \({{\mathbb R}^n}\) , \({u \to a/b}\) as \({t \to \infty}\) , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists \({\mu^* \geqq 0}\) such that \({\Omega(t)}\) expands to \({{\mathbb{R}}^n}\) exactly when \({\mu > \mu^*}\) .

We study a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with spatially undulating boundary. The law of motion is given by V = κ + A , where V is the normal velocity of the curve, κ is the curvature, and A is a positive constant. The boundary undulation is assumed to be almost periodic, or, more generally, recurrent in a certain sense. We first introduce the definition of recurrent traveling waves and establish a necessary and sufficient condition for the existence of such traveling waves. We then show that the traveling wave is asymptotically stable if it exists. Next we show that a regular traveling wave has a well-defined average speed if the boundary shape is strictly ergodic. Finally we study what we call “virtual pinning”, which means that the traveling wave propagates over the entire cylinder with zero average speed. Such a peculiar situation can occur only in non-periodic environments and never occurs if the boundary undulation is periodic.

We construct new examples of non-uniqueness of positive solutions of the Cauchy problem for the Fujita equation. The solutions we find are not self-similar and some of them blow up in finite time. Heteroclinic connections and ancient solutions of a rescaled equation play the key role in our construction.

International audience; Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

We consider degenerate reaction diffusion equations of the form ut = Δum + f(x, u), where f(x, u) ~ a(x)up with 1 ≤ p 0 at least in some part of the spatial domain, so that \({u \equiv 0}\) is an unstable stationary solution. We prove that the unstable manifold of the solution \({u \equiv 0 }\) has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as \({t\to -\infty}\) while its support shrinks to an arbitrarily chosen point x* in the region where a(x) > 0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.

We investigate the singular limit, as \({\varepsilon \to 0}\), of the Allen-Cahn equation \({u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}\), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of \({{\varepsilon}}\). It is known that this equation converges to the generalized motion by mean curvature — in the sense of viscosity solutions—defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions \({u^{\varepsilon}}\) are sandwiched between two sharp “interfaces” moving by mean curvature, provided that these “interfaces” sandwich at t = 0 an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.

We consider the Allen–Cahn equation in R n (with n ⩾ 2 ) and study how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. We first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t = + ∞ . Using this observation, we then show that the planar front is asymptotically stable in L ∞ ( R n ) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. As a by-product of our analysis, we present a result of a rather general nature, which states that, for a large class of evolution equations, the unique ergodicity of the initial data is inherited by the solution at any later time.

We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of "quasipositivity principle" and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.

We consider the dead-core problem for the semilinear heat equation with strong absorption and with positive boundary values in a ball. We investigate the dead-core rate, i.e. the rate at which the solution reaches its first zero. We first show, as in the one-dimensional case, that the dead-core rate is always faster than the self-similar rate. By using some special solutions and the braid group theory, we then derive the exact dead-core rates for a large class of initial data.

We study the Cauchy problem utDuxxCf.u/ .t > 0; x2 R 1 /; u.0;x/Du0.x/ .x2 R 1 /; wheref.u/ is a locally Lipschitz continuous function satisfyingf.0/D 0. We show that any non- negative bounded solution with compactly supported initial data converges to a stationary solution as t! 1. Moreover, the limit is either a constant or a symmetrically decreasing stationary solu- tion. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solutionu , we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even iff has a jumping discontinuity atuD 1.

We consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity. The goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space ℝN. As a first step, we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle. This causes disturbance on the shape of the front, but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction, leaving the obstacle behind. Whether the recovery is uniform in space is shown to depend on the shape of the obstacle. © 2008 Wiley Periodicals, Inc.

This work deals with the existence and non-existence of travelling wave solutions for a two-dimensional reaction-diffusion equation coupled with ordinary differential equations. The system models the spatial spread of a fungal disease over a field of crops whose spatial configuration exhibits a periodic stripe pattern, such as a vineyard. The standard comparison principle does not hold for this system. We establish a sharp criterion for the existence of (directional) travelling waves in terms of what we call the epidemic threshold \({\mathscr {R}}_0\), which is independent of the direction of the travelling wave. We then study the minimal speed \(c^*_\theta \) of travelling waves for each direction \(\theta \in [0,2\pi )\) and prove its monotone dependence on \(\theta \). Our analysis is based on the fixed point argument, a variational characterization of principal eigenvalues and Harnack type inequalities for elliptic and parabolic problems.

We consider reaction diffusion equations of the prototype form u t = u xx + λ u + |u| p-1 u on the interval 0 1 and λ > m 2. We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u 1, . . . ,u m . Since u k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x 1, . . . ,x m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation).

We discuss the generation and the motion of internal layers for a Lotka-Volterra competition-diffusion system with spatially inhomogeneous coefficients. We assume that the corresponding ODE system has two stable equilibria (ū, 0) and (0, ) with equal strength of attraction in the sense to be specified later. The equation involves a small parameter ϵ, which reflects the fact that the diffusion is very small compared with the reaction terms. When the parameter ϵ is very small, the solution develops a clear transition layer between the region where the u species is dominant and the one where the v species is dominant. As ϵ tends to zero, the transition layer becomes a sharp interface, whose motion is subject to a certain law of motion, which is called the “interface equation”. A formal asymptotic analysis suggests that the interface equation is the motion by mean curvature coupled with a drift term. We will establish a rigorous mathematical theory both for the formation of internal layers at the initial stag...

Recent papers have studied the existence of phase transition solutions for model equations of Allen-Cahn type equations. These solutions are either single or multi-transition spatially heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition of the construction of the multi-transition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.

In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = 1 u + |u| p−1 u either on R N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, p > ps := N + 2 N − 2 . We prove that if ps p ∗ , the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. W e then establish useful estimates for the so-called incomplete blowup, which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. c 2004 Wiley Periodicals, Inc.

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.

We give a necessary and sufficient condition for the existence of L1-connections between equilibria of a semilinear parabolic equation. By an L1-connection from an equilibrium φ− to an equilibrium φ+ we mean a function u(⋅, t) which is a classical solution on the interval (−∞, T) for some T ∈ ∝ and blows up at t = T but continues to exist in the space L1 for t ∈ [T, ∞) and satisfies u(⋅, t) → φ± (in a suitable sense) as t → ±∞. The main tool in our analysis is the zero number.

We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of $L^1$ connections among equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium $\phi^+$ we mean a function $u$($.,t$) which is a classical solution on the interval $(-\infty,T)$ for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a certain weak sense for $t\in [T,\infty)$ and satisfies $u$($.,t$)$\to \phi^\pm$ as $t\to\pm\infty$ in a suitable sense. The main tool in our analysis is the zero number argument; namely to count the number of intersections between the graph of a given solution and that of various specific solutions.

This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.

We study reaction-diffusion equations with a spatially inhomogeneous reaction term. If the coefficient of these reaction term is much larger than the diffusion coefficient, a sharp interface appears between two different phases. We show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations. In particular, we show that the same rich spatial patterns observed for a chemotaxis-growth model can be realized by a system without a drift term.

We study the global attractor of semilinear parabolic equations of the form $u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$ Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.

In this paper we investigate the large time behavior of solutions of the semilinear heat equation. Where u{sub 0} is the initial data, N {ge} 1 and p > 1. It can be easily checked that if u(t,x) satisfied (1.1), then for {gamma} > 0 the rescaled functions u{sub {gamma}}(t,x) satisfies (1.1), then for {gamma}>0 the rescaled functions define a one parameter family of solutions to (1.1). A solution u {equivalent_to} 0 is said to be self-similar, when u{sub {gamma}} {equivalent_to} u for all {gamma} > 0. For instance, for any fixed p > 1, w{sup *}(t,x):=((p-1)t){sup {minus}1/(p-1)} is such a solution. Actually, it has been proved by H.Brezis, L.A. Peletier & D. Terman that for 1 {infinity}. 15 refs.

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.