Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications

In this paper, we consider fully discrete approximations of abstract evolution equations, by means of a quasi non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Hence, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We set the problem in the very general and abstract setting of pseudo-monotone operators, which allows for a unified treatment of several evolution problems. The examples — which fit into our setting and which motivated our research — are problems describing the motion of incompressible fluids, since the quasi non-conforming approximation allows to handle problems with prescribed divergence. Our abstract results for pseudo-monotone operators allow to show convergence just by verifying a few natural assumptions on the operator time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be easily performed. The results of some numerical experiments are reported in the final section.

Set-valued perturbation to second-order evolution problems with time-dependent subdifferential operators

The main purpose of this work is to study the existence of solutions for a perturbed second-order evolution inclusion involving time-dependent subdifferential operators. Under suitable conditions on the set-valued perturbation, the main result of the paper is proved in the context of a separable Hilbert space. A second-order evolution quasi-variational inequality is also investigated.

STUDY OF FRACTIONAL ORDER DELAY CAUCHY NON-AUTONOMOUS EVOLUTION PROBLEMS VIA DEGREE THEORY

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

Regularity and time behavior of the solutions to weak monotone parabolic equations

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.