FOURIER PROBLEM FOR WEAKLY NONLINEAR EVOLUTION INCLUSIONS WITH FUNCTIONALS

Mykola M Bokalo,Iryna V Skira

doi:10.15421/141901

Abstract

The Fourier problem or, in other words, the problem without initial conditions for evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained.

Highlights

In this paper we consider problem without initial conditions, or, in other words, the Fourier problem for evolution variational inequalities with functionals
Let L2loc(Q) be the space of functions defined on Q such that their restrictions on any bounded measurable set Q ⊂ Q belong to L2(Q )
We can assume that the initial time is −∞, while 0 is the final time, and initial conditions can be replaced with the behaviour of the solution as time variable turns to −∞

Summary

Introduction

In this paper we consider problem without initial conditions, or, in other words, the Fourier problem for evolution variational inequalities (inclusions) with functionals. 83]) it follows that the problem of finding a solution of variational inequality (1.3) can be written as such subdifferential inclusion: to find a function u ∈ L2loc(S; V ) such that u ∈ L2loc(S; H), condition (1.2) holds and, for a.e. t ∈ S, u(t) ∈ D(∂Φ) and u (t) + ∂Φ(u(t)) + B(t, u(t)) f (t) in H. Tikhonov [23] in the case of heat equation As it was shown by M.M. Bokalo [3], problem without initial conditions for some nonlinear parabolic equations has a unique solution in the class of functions without behavior restriction as time variable terns to −∞. Let us note that problems without initial conditions for variational inequalities or inclusions with functionals have not been considered in the literature, and this serves as one of the motivations for the study of such problems.

Preliminaries

Statement of the problem and main result

Proof of the main result