A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.     

Semilinear subdiffusion with memory in multidimensional domains

For , we investigate the nonlinear integro‐differential equation on a multidimensional space domain where is the Caputo fractional derivative and and are uniform elliptic operators with smooth coefficients depending on time. Under suitable conditions on the nonlinearity, the global existence and uniqueness of the classical solution to the related initial and boundary value problems are established.     

A convergence–divergence test for series of nonnegative terms

We present a convergence–divergence test for series of nonnegative terms. Our proof is elementary, and yet we show examples of application to some apparently difficult cases.     

Uniform estimates of Gronwall type

We discuss a Gronwall-type lemma depending on a small parameter ε>0, based on an integral inequality that predicts blow-up in finite time of the involved unknown function for any fixed ε. The result permits to establish uniform estimates even if the function itself depends on ε.     

Functions of regular variation and freely stable laws

Freely stable laws and their domains of attraction are characterized using the theory of functions of regular variation. The results show a complete analogy with classical probability. In fact they can be used to provide an alternative proof of the corresponding classical arguments. The first author was supported in part by a grant from the National Science Foundation.     

On the exponential stability of linear thermoelasticity

Received March 28, 2000     

On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation

We consider the one-dimensional Cahn-Hilliard equation with an inertial term ?utt, for ??0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S?(t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors {M?}, whose common basins of attraction are the whole phase-space.     

Stability properties of an abstract system with applications to linear thermoelastic plates

This paper deals with an abstract version of the evolution system $$\left\{\begin{array}{ll}u_{tt}-\omega \Delta u_{tt}+\Delta (\Delta u + \theta)=0 \\\theta_t - \Delta \theta -\Delta u_t=0\end{array}\right.$$ with ω ≥  0, ruling the dynamics of linear thermoelastic plates. In particular, we will consider a more general coupling term depending on a real parameter σ. The (exponential and polynomial) decay properties of the related solution semigroup are investigated in dependence of the values of σ.     

Asymptotic behavior of a semilinear problem in heat conduction with memory

This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved. Received March 23, 1997 - Revised version received November 12, 1997     

Robust exponential attractors for the strongly damped wave equation with memory. I

We consider the singular limit of the semilinear strongly damped wave equation with memory