The continuous wavelet transform is studied on certain Gel'fand–Shilov spaces of type S. It is shown that, for wavelets belonging to the one type of S-space defined on R, the wavelet transform is a continuous linear map of the other type of the S-space into a space of the same type (latter type) defined on R × R+. The wavelet transforms of certain ultradifferentiable functions are also investigated.

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

In this paper we investigate the critical Fujita exponent for the initial-value problem of the degenerate and singular nonlinear parabolic equation with a non-negative initial value, where p > m ≥ 1 and 0 ≤ λ1 ≤ λ2 < p(λ1 + 1) − 1. We prove that, for m < p ≤ pc = m + (2 + λ2)/(n + λ1), every non-trivial solution blows up in finite time, while, for p > pc, there exist both global and non-global solutions to the pro

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.

Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

A three spectral inverse Sturm–Liouville problem is considered arising in the theory of vibrating strings. It is shown that in some special cases the solution of this problem is unique.

We study the heat flow projected on a manifold M ⊂ L2(Ω). This manifold is defined by the condition that the integrals ∫Ωuk (t,x) dx, k = 1,…,N, are constants of motion. We show that solutions to this problem converge to a steady state as time tends to +∞. We use in an essential way a variant of the Łojasiewicz inequality.

The Lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume-filling effect. For some typical case it is proved that the ω-limit set of any trajectory consists of regular stationary solutions. Some lower and upper bounds on the stationary solutions are found. For a given range of parameters there are stationary solutions which are inhomogeneous in space.

In this paper we consider the Navier–Stokes equations in Rn, n ≥ 3. We prove the asymptotic stability for weak solutions in the marginal class u ∈ L2(0, ∞; BMO), where ‘BMO’ denotes the bounded mean oscillation function, with arbitrary initial and external perturbations.

Lp–Lq decay estimates of solutions to Cauchy problems of linear thermoelastic systems with second sound in three spatial variables are studied. By carefully analysis of the different effects of higher, middle and lower frequencies in phase space, the asymptotic behaviour of characteristic roots of coefficient matrices is obtained. Then, with the help of the information on characteristic roots and the interpolation theorem, a decay estimate of parabolic type for the coupled system of the potential part of displacement, temperature and heat flux is obtained. Finally, Lp–Lq decay estimates of hyperbolic type for the original thermoelastic systems with second sound are obtained.

We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

We discuss the spectral properties of a class of sequences of what we call ‘spectral’ type. We introduce an effective method to calculate the zeta invariants for this type of sequence. Such invariants are given in terms of some new and old special functions, and we consider a number of examples in which we study the properties of these special functions.