Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems

Nonautonomous parabolic equations of the form ut − Δu = f(u, t) on a symmetric domain are considered. Using the moving-hyperplane method, it is proved that any bounded nonnegative solution symmetrises as t → ∞. This is then used to show that for nonlinearities periodic in t, any non-negative bounded solution approaches a periodic solutio

Convergence to a steady state for asymptotically autonomous semilinear heat equations on RN

AbstractWe consider parabolic equations of the formut=Δu+f(u)+h(x,t),(x,t)∈RN×(0,∞), where f is a C1 function with f(0)=0, f′(0)<0, and h is a suitable function on RN×[0,∞) which decays to zero as t→∞ (hence the equation is asymptotically autonomous). We show that, as t→∞, each bounded localized solution u⩾0 approaches a set of steady states of the limit autonomous equation ut=Δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations

Loops and Branches of Coexistence States in a Lotka-Volterra Competition Model

Abstract. A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate µ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of µ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches 1 of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types. Key words: Reaction-diffusion, competing species, spatial heterogeneity, bifurcation

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or towards the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence. For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show that the set of continuous initial data corresponding to solutions that converge to a spatially homogeneous equilibrium is prevalent. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.Comment: 18 page

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Stability and convergence in discrete convex monotone dynamical systems

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on $n$ letters, where $n$ is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.Comment: 36 pages, 1 fugur

Complex-valued Burgers and KdV-Burgers equations

Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions

Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m=2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".Comment: 34 page

A new critical curve for the Lane-Emden system

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.Comment: 13 pages, 1 figur