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 consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

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 positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

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.

In this article we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for a unique, global-in-time solution of the corresponding rate-independent initial–boundary-value problem. The method of choice is the Yosida approximation and a passage to the limit.

Existence and uniqueness of large, boundary blow-up solutions are obtained for the quasilinear elliptic problem −Δpu = λf(u) in Ω, u = ∞ on ∂Ω via good boundary layer estimates for large λ, where Δp is the p-Laplacian (1 < p < ∞) and Ω ⊂ ℝ N (N ≥ 2) is a bounded smooth domain. The nonlinear term f satisfies f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, with z2 a zero of f of order k. It is shown that, if k ≥ p −1, the unique large solution ūλ is a boundary-layer solution which satisfies ūλ > z2 in Ω; if 0 < k < p −1, the unique large solution ūλ is a boundary-layer solution, but a flat core of ūλ occurs. Furthermore, for sufficiently large λ a small positive boundary blow-up solution uλ is obtained and its asymptotic behaviour as λ → ∞ is discussed.

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.

This paper deals with eigenvalue problems of the form where 0 < σ < τ and V(x) is such that the spectrum of −u″ consists of eigenvalues λ1, λ2,…situated below the continuous spectrum [δ,+∞[.

We analyse the existence of (multiple) solutions for λ < λ1 as well as for λ > λ1 when λ is in a spectral lacuna.

The existence of solutions depends on the weight of μ > 0. Moreover, when λ increases (while μ is kept fixed), some solutions are lost when crossing eigenvalues.

The above results are derived with the help of an abstract approach based on variational techniques for multiple solutions. This approach can even be applied to a wider class of problems, the one presented herein being only a model problem.

A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters ai for which the polynomial xn + a1xn−1 + … + an is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a1 = 0, a2 ≥ −1}.

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.

In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

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.