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.

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 is obtained and its asymptotic behaviour as λ → ∞ is discussed.

We are interested in Lp-estimates and scattering rates for the dissipative wave equation with time-dependent coefficients in an exterior domain outside a star-shaped obstacle. We want to notice the case that the support of dissipation expands strictly less than the wave speed. We develop a new cut-off method, which is time dependent. For this, we shall obtain the local energy decay over the time-dependent subdomain

Let us consider the set SA(Rn) of rapidly decreasing functions G: Rn → A, where A is a separable C*-algebra. We prove a version of the Calderón–Vaillancourt theorem for pseudodifferential operators acting on SA(Rn) whose symbol is A-valued. Given a skew-symmetric matrix, J, we prove that a pseudodifferential operator that commutes with G(x + JD), G ∈ SA(Rn), is of the form F(x − JD), for F a C∞-function with bounded derivatives of all orders.

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.

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 [Λ,+∞[.

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.

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

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.

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ = 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.

In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.

Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every f ∈ LΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.

Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

We prove a sharp Hölder estimate for solutions of linear, two-dimensional, divergence-form, elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has unit determinant. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.

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}.

This paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is where Ω ⊂ Rn (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪ Γ1, m > 1, 2 ≤ p < r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n = 1, 2 and u0 ∈ H1(Ω), u0 = 0 on Γ0. We prove local existence of the solutions in H1(Ω) when m > r/(r + 1−p) or n = 1, 2 and global existence when p ≤ m or the initial datum is inside the potential well associated to the stationary problem.

We give a complete characterization of the analytic and Darbouxian first integrals of the Rikitake system, which serves as a model for the reversal of polarity of the Earth's electromagnetic field. Our approach uses the Darboux theory of integrability.

We prove uniform Morrey–Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted L2-estimates for the solution. We also prove a uniform L2-estimate without weight for the trace of the solution on the interface.

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.

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 ≤ ∞.

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

In this paper we study the Morse functions on Grassmann manifolds. It is shown that many well-known calibrations can be used to construct degenerate Morse functions on Grassmann manifolds. In the study we use Clifford algebras and Lie groups.