On the extinction of the solution for an elliptic equation in a cylindrical domain

We obtain some conditions under which there is extinction in finite time of the solution of an elliptic equation in a cylindrical domain. We also show how the solution of our elliptic equation approaches its stationary solution.


Introduction
Let 03A9 be a bounded domain in IRn with smooth boundary ~03A9. We consider the following boundary value problem for t): 2u ~t2 + 0394u -03BBf(u) = 0 in 03A9 (4, oo), The proof of the proposition is based on the following lemma Comparison lemma 1.2 (maximum principle). Let u, v E C(03A9 x [0, oo)) satisfying the following inequalities: Then we have u(x, t) > v(x, t) in S2 x (0, oo). Proof. Assume that w = u-v takes negative values and let m = inf w(x, t) 0. We have L(t, C) m(t) in ~0, oo) for C large and the infimum of all constants C with this property, call it Co, has the property that Co > 0, L(t, Co) m(t), there is to > 0 with L(to, Co) = m(to).
Let xo be such that m(to) = w(xo, to). Since w(xo, to) 0, i.e u(xo, to) v(xo, to), it follows from the second inequality of the lemma that xo is in H. Now we have f (u(xa, ta)) , w(x, to) > w(xo, to) in 03A9 ~ (Lw)(xo, to) > 0, L(t, Co) with equality for t = to ==~ 0 in contradiction to the assumed first inequality of the lemma. 0 Proof of Proposition 1.1. Since Problem (1.1)-(1.3) has a comparison lemma, to prove our proposition, it suffices to show that ( 1.1 ) -(1.3) has both a supersolution and a subsolution (see [10]). Let us notice that 0 is a subsolution and 1 is a supersolution, which leads us to the result. D By a stationary solution of (1.1) -(1.3), we mean a function s(x) satisfying a f (s(x)) = 0 in fI, (1.5) role in its asymptotic behavior as t ~ oo. This phenomenon is well known in the parabolic case (see, for instance [3], [4]). Our paper is organized as follows.
In the next section, we give some conditions of extinction and in the last section, we study the asymptotic behavior of the solution u as t ~ oo.

Conditions of extinction
In this section, we show that under some conditions, there is extinction in finite 3). We say that there exists a dead core for 03C6(x) if there exists a set no c 03A9 such that 03C6(x) = 0 for x E 03A9o. The following lemma which may be found in [4] will be used later. It gives us information on the following steady-state problem: h(x) on ~03A9. The maximum principle implies that w(x, t) > u(x,t) in 03A9 x (0, oo), and the result follows from (2.5) and (2.9). The proof of Theorem 2.5 (b) is as the proof of Theorem 2.2 (b). 0 3. Asymptotic behavior.
The maximum principle implies that z(x,t) > u(~,t) in S2 x (O,oc), which gives the second result. 0