The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.

We study the phenomenon of the infinite-time stabilization of classical global solutions of nonlinear reaction–diffusion equations to an unbounded (singular) stationary state and we present a new case of such an asymptotic singularity pattern formation. We concentrate on the most famous parabolic model, namely the semi-linear Frank-Kamenetskii equation from combustion theory,where B is a ball in RN. Our goal is to show that a new asymptotic problem arises precisely in dimension 10, not being available in other dimensions (which were studied earlier). For N = 10, we fix the ball B = {|x| < 4} and take bounded initial data u0 below the singular stationary solution Us(x) = ln(16/|x|2), which is unbounded at the origin x = 0.

We establish a sharp estimate on the rate of convergence u(x, t) → Us(x) as t → ∞ on compact subsets bounded away from x = 0 and also at the singularity point. We show that u(0, t) = α0t + O(ln t) → ∞, where the positive constant α0 is given by the first eigenvalue of the associated linear differential operator. We present formal asymptotic results showing that a detailed asymptotic analysis depends on a quite involved balance between various linear and nonlinear terms. Moreover, similar critical asymptotic behaviour is shown to exist in various related nonlinear second- and higher-order parabolic equation.

We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

Sufficient conditions are given for an autonomous differential systemto have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, RN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in RN.