Introduction

In the following we consider simplified versions of the above model by setting

Jp = Jn and w = wh(t) := a(uh(t)) on all inflow boundary sections so that

(1.4)—(1.5) become

u = uh(t) on <9f2T> w = wh(t) on d£l~, (1-6)

and we rewrite (1.3) as

v = —sign(w)F(|Vii|)Vii, (1.7)

where we assume F is a given Lipschitz continuous function such that

0 < F(r) < 1 r > 0; and 0 < f(r) — f(s) < r — s V r > s , (1*8)

where /(r) = rF(|r|). We also assume that uh(t) and wh(t) £ C 1(0,T).