Luisa Toscano

Long-Time Behaviour of Solutions for Autonomous Evolution Hemivariational Inequality with Multidimensional “Reaction-Displacement” Law

We consider autonomous evolution inclusions and hemivariational inequalities with nonsmooth dependence between determinative parameters of a problem. The dynamics of all weak solutions defined on the positive semiaxis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. New properties of complete trajectories are justified. We study classes of mathematical models for geophysical processes and fields containing the multidimensional “reaction-displacement” law as one of possible application. The pointwise behavior of such problem solutions on attractor is described.

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

Abstract In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.

Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations

In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.

On the solvability of a class of general systems of variational equations with nonmonotone operators

Abstract For a wide class of systems of variational equations depending on parameters with nonmonotone operators in a space real reflexive Banach space, we study the solvability, the existence of solutions with every components different to zero, the existence of multiple solutions and, in the omogeneous case, the existence of solutions with every positive components when Wl is a vector lattice according to the fibering method. We obtain results which have many different applications. For example, they can be used in order to study Dirichlet and Neumann problems and to check ODE periodic solutions.

Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components

We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.

On the Solvability of Some Nonlinear Systems of Elliptic Equations

Abstract We study the solvability and the existence of multiple solutions of nonlinear systems of elliptic equations.

On the solvability of some nonlinear Neumann problems

Abstract In this paper we study the existence of solutions of some nonlinear Neumann problems and the main tool used is the fibering method.

Blow-up Results for Parabolic Inequalities with Chipot–Weissler Nonlinearity in the Gradient Term

Abstract We obtain blow-up results for a wide class of nonlinear parabolic problems with nonlinearity of the Chipot-Weissler type in the gradient term. Some of these answer an open question concerning the nonexistence of positive solutions to the problem where λ > 0 is small, u0 ∈ Lloc(ℝN) with u0 ≥ 0, when . For this purpose we use the method of “test function”.

A New Result Concerning the Solvability of a Class of General Systems of Variational Equations with Nonmonotone Operators: Applications to Dirichlet and Neumann Nonlinear Problems

A new result of solvability for a wide class of systems of variational equations depending on parameters and governed by nonmonotone operators is found in a Banach real and reflexive space with applications to Dirichlet and Neumann problems related to nonlinear elliptic systems.

Nonstationary problems of oscillations for elastic system with unilateral external interaction

We study dynamics of elastic systems with external unilateral interaction (fric-tion, recovering force), which is given as a function defined by inequalities relative to unknown variables. The analytical-numerical method for investigation oscillations of the mentioned systems was constructed. We consider transient regimes of the system motion. It was shown that the unilateral interaction promotes conditions for preventing development of resonance and can be considered as a mean for active suppression of oscillations.