Increasing Evolutionary Inequalities

We derive absolute observation stability and instability results for controlled evolutionary inequalities which are based on frequency-domain characteristics of the linear part of the inequalities. The uncertainty parts of the inequalities (nonlinearities which represent external forces and constitutive laws) are described by certain local and integral quadratic constraints. Other terms in the considered evolutionary inequalities represent contact-type properties of a mechanical system with dry friction.

The mathematical model of the injection molding process is characterized by evolutionary inequalities with an elliptic principal part and a Volterra term. Existence and uniqueness of the solution of these problems are derived by means of an argument based on the Banach fixed point theorem. Regularity results are obtained by using the penalty and Rothe method. The results can be considered as a starting point for the numerical treatment and for the application to injection molding.

The present part of the paper continues the study of the abstract evolution inequality from the first part. Theorem 1 states the existence and uniqueness of a weak solution to the evolution inequality under consideration. The proof is based on the method of approximation of the weak solution by a sequence of strong solutions. Theorem 2 yields two regularity results for the strong solution.

On Some Evolution Inequalities in a Hilbert Space

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.

We show some existence et uniqueness theorems for the bounded and almost periodic solution of some non linear equations and inequalities of evolution.

Nonexistence for higher order evolution inequalities with Hardy potential in [formula omitted

We consider an evolution unilateral problem with time-dependent obstacles. The problem studies the temperature-field in an open bounded set filled with a continuous medium. We obtain a convenient regularity theorem and using the finite affine triangular element method, we discretize the corresponding evolution inequality. After all we obtain, via the regularity theorem, an estimate of the discretization error.

We study a class of non-clamped dynamical problems for visco-elastic materials, the contact condition is modeled by a normal compliance, with friction, damage and heat exchange. The weak formulation leads to a general system defined by a second-order quasi-variational evolution inequality on the displacement field coupled with a nonlinear evolutional inequality on temperature field and a parabolic variational inequality on the damage field. We present and establish an existence and uniqueness result of different fields, by using general results on evolution variational inequalities, with monotone operators and fixed point methods. Then, we present a fully discrete numerical scheme of approximation and derive an error estimate. Finally, various numerical computations are developed.

Chapter 4: Evolution Inequalities

Higher order evolution inequalities with convection terms in an exterior domain of [formula omitted

Blow-up results for a class of first-order nonlinear evolution inequalities

A mathematical model that describes a dynamic frictionless contact of a viscoelastic body with a rigid‐plastic foundation is considered. The constitutive law is assumed to be a nonlinear constitutive law with long memory. We derive a variational formulation of the model, and the global existence and uniqueness of a solution are proved. The proof is based on arguments of evolutionary variational inequalities, a classical existence and uniqueness result of nonlinear first‐order evolution inequalities, and Banach fixed point theorem. Finally, we apply the multiplier method to establish the exponential energy decay of the solution.

The purpose of this work is to establish a Liouville-type comparison principle for solutions of higher order differential evolution inequalities of the form in 𝕊 := {(t, x): t > 0, x ∈ ℝ n }, without any initial conditions on the solutions on the hyperplane t = 0 being involved, where L is a linear lth-order differential operator of the form , a α(t, x) are measurable bounded functions in 𝕊, |α| = α1 + ··· + α n , l ≥ 1 and n ≥ 1 are natural numbers, and q ≥ 1 is a real number. The principal examples of the operator L are the Laplacian Δ and the mth-order polyharmonic operator Δ m := Δ1(Δ m−1), where Δ1 := Δ and m ≥ 2 is a natural number.

In this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.