ABSTRACT
Then it is not difficult to see that Q is positively invariant to the flows of Eq. (1.1). The solution uM(t,a) of Eq. (1.1) satisfying the initial condition uM(0, a) = 1 is a maximum solution in Q. Moreover uM(t, •) G Ü for t > 0 and the monotonicity implies that uM(t, •) is decreasing and is bounded below by zero. Consequently uM(t, •) either converges to zero or converges to a positive equilibrium w+. For the former case, we conclude that all solutions with initial functions given in ÇI will go to zero as t goes to oo because all solutions are dominated by uM. For the later case, let u+ be a positive equilibrium, then
If we let A\ : D (A) -> Ll(0,u) be defined as
then 0 G ap(A\). Notice that A</> > A\<\> for all 0 G D (A) with 0 > 0. It follows that so(A) > SQ(A\) > 0. With a certain condition posed on the functions K\ and K2 such that so(A) is strictly larger than so(A\) when u+ is positive. One then concludes that Eq. (1.1) does not have a positive equilibrium in Q when so(A) < 0, and hence the zero solution is globally stable. On the other hand if so(A) > 0, then A has a positive eigenfunction 0o corresponding to eigenvalue s0{A), and the solution near origin along the direction of eigenfunction 0o is ejective. Hence by using the monotonicity we can prove that for sufficiently small e > 0, the solution ue(t, •) with initial condition e0o is increasing. Thus ue(t, •) <uM(t,-) for all t > 0 implies that u+ = lim^oo uM(t, •) is a positive equilibrium. A simplest case is that the positive equilibrium is unique that requires an additional condition on the functions K\ and K2. In this situation one can partition Q into
2 Preliminaries
(2.1)
(2.2)
Since TN is compact, we know that if À = r(TN), the spectral radius of TN\ is larger than 0 (in fact we can show that A > 0), then À is an eigenvalue of TN and the corresponding eigenfunction <j> is nonnegative. It follows that a — r(T) = \l/N, and
gives a positive eigenfunction of T corresponding to the eigenvalue r(T). We further assume that operator T satisfies the following hypothesis.
