Global attractors for two-phase stefan problems in one-dimensional space

T Aiki

doi:10.1155/s1085337597000262

Abstract

In this paper we consider one-dimensional two-phase Stefan problems for a class of parabolic equations with nonlinear heat source terms and with nonlinear flux conditions on the fixed boundary. Here, both time-dependent and time-independent source terms and boundary conditions are treated. We investigate the large time behavior of solutions to our problems by using the theory for dynamical systems. First, we show the existence of a global attractor𝒜of autonomous Stefan problem. The main purpose in the present paper is to prove that the set𝒜attracts all solutions of non-autonomous Stefan problems as time tends to infinity under the assumption that time-dependent data converge to time-independent ones as time goes to infinity.

Highlights

Let us consider a two-phase Stefan problem SP = SP (ρ; a; bt0, bt1; β, g, f0, f1, u0, 0) described as follows: Find a function u = u(t, x) on Q(T )= (0, T ) × (0, 1), 0 < T < ∞, and a curve x = (t), 0 < < 1, on [0, T ] satisfying (1.1) ρ(u)t − a(ux)x + ξ + g(u) =f0 f1 in Q(0)(T ), in Q(1)(T ), Q(0)(T ) = {(t, x); 0 < t < T, 0 < x < (t)}, Q(1)(T ) = {(t, x); 0 < t < T, (t) < x < 1}, 1991 Mathematics Subject Classification
(0) = 0, where ρ : R → R and a : R → R are continuous increasing functions; β is a maximal monotone graph in R × R; g : R → R is a Lipschitz continuous function; fi(i = 0, 1) is a given function on (0, ∞) × (0, 1); bti(i = 0, 1) is a proper l.s.c. convex function on R for each t ≥ 0 and ∂bti denotes its subdifferential in R; u0 is a given initial function and 0 is a number with 0 < 0 < 1
SP (ρ; a; b0, b1; β, g, f0∗, f1∗, u0, 0) where fi∗ ∈ L2(0, 1) and bi is a proper l.s.c. convex function on R for i = 0, 1

Summary

Introduction

In order to consider the large time behavior of solutions we discuss a global attractor for the problem SP. There are many interesting results dealing with a global attractor of autonomous nonlinear partial differential equations N f (t), t > 0, in H, was considered, where H is a Hilbert space, φt is a proper l.s.c. convex function on H for t > 0, ∂φt is its subdifferential, N : H → H is Lipschitz continuous and f is a given function. They gave a more general answer for that question. We refer to Brezis [3] for definitions and basic properties concerned with convex analysis

Main results

Preliminaries and known results

Uniform estimates

Global existence and global attractor