260
and 'lj1(0,·) = {}, then each local maximum of 'lj11(0, 00) x M is ~ g-l(IIQll oo [1 - inf oj)
and u(·,·j 1', t9) :S ' lj1. Moreover, we get ' lj1 ~ and'lj1 exfst on ~ x M therefore u(','j 1', t9).
A more ca.{eful reasoning actually establishes the following
Absorbing Property. Let (HO) - (H3) be satisfied and {} E C+(M). Denote by
u = u(t, Xj 1', t9) the maximal solution of (1), u(O, x) = {}(x). Then u is a nonnegative,
bounded function on ~ x M, and each w-limit function w ofu satisfies g-l(1' inf Q [1lIalloo]):S infw:S supw:S g-l(1' IIQlloo [I-info]).
Recall that w(u) = {w E C(M): 3(tn )withtn -+ ooandu(tn ,·) -+ w}. As in
the case of ordinary differential equations we can use a Liapunov functional in order
to derive that w( u) consists of stationary solutions of (1), only. To this end, we fix
I' E [0,00) and set J(I',x,y) = g [I'Q(x)[l- a(x,7])]- g(7])]d7] and
rl'(w) =! r k(x) Igradw(x)1 2 dx - r J(I',X,w(x))dx.
21M
1M
for w E C1(M). r I' is bounded from below and satisfies
r~(w)(¢» = iM k(x) grad w(x)·grad ¢>(x) dx
- i)J1.Q(X)[l- a(x,w(x))]- g(w(x))J¢>(x)dx =
= - iM (div(kgradw)(x) + I'Q(x)[l- a(x,w(x))]- g(w(x)))¢>(x)dx
for w E C 2 (M) and ¢> E C1(M). Therefore we have
!rl' ou(t,·) = (r~ ou(t,·))Otu(t,·) =
iM {-div (k grad u(t, ·))(x) -I' Q(x)[l - a(x, u(t, x))] + g(u(t, x))}Otu(t, x) dx =
- iM c(~) {-div (k grad u(t,· ))(x) - I' Q(x )[1 - a(x, u(t, x))] + g( u(t, x))}2 dx :S 0,
whenever u is a solution of (1), with "= 0", iff u is a stationary solution of (1) (unique
solvability). This says intuitively that the family of solution curves t9 H u("'j 1', t9)
intersects the nondegenerate level hypersurfaces of the potential r I' transversally, hence
the w-limit of every solution is contained in a critical level set of r I'" In mathematical
terms, t9 H u(','j J1., t9) is called the solution semiflow of (1), and one says that such a
and 'lj1(0,·) = {}, then each local maximum of 'lj11(0, 00) x M is ~ g-l(IIQll oo [1 - inf oj)
and u(·,·j 1', t9) :S ' lj1. Moreover, we get ' lj1 ~ and'lj1 exfst on ~ x M therefore u(','j 1', t9).
A more ca.{eful reasoning actually establishes the following
Absorbing Property. Let (HO) - (H3) be satisfied and {} E C+(M). Denote by
u = u(t, Xj 1', t9) the maximal solution of (1), u(O, x) = {}(x). Then u is a nonnegative,
bounded function on ~ x M, and each w-limit function w ofu satisfies g-l(1' inf Q [1lIalloo]):S infw:S supw:S g-l(1' IIQlloo [I-info]).
Recall that w(u) = {w E C(M): 3(tn )withtn -+ ooandu(tn ,·) -+ w}. As in
the case of ordinary differential equations we can use a Liapunov functional in order
to derive that w( u) consists of stationary solutions of (1), only. To this end, we fix
I' E [0,00) and set J(I',x,y) = g [I'Q(x)[l- a(x,7])]- g(7])]d7] and
rl'(w) =! r k(x) Igradw(x)1 2 dx - r J(I',X,w(x))dx.
21M
1M
for w E C1(M). r I' is bounded from below and satisfies
r~(w)(¢» = iM k(x) grad w(x)·grad ¢>(x) dx
- i)J1.Q(X)[l- a(x,w(x))]- g(w(x))J¢>(x)dx =
= - iM (div(kgradw)(x) + I'Q(x)[l- a(x,w(x))]- g(w(x)))¢>(x)dx
for w E C 2 (M) and ¢> E C1(M). Therefore we have
!rl' ou(t,·) = (r~ ou(t,·))Otu(t,·) =
iM {-div (k grad u(t, ·))(x) -I' Q(x)[l - a(x, u(t, x))] + g(u(t, x))}Otu(t, x) dx =
- iM c(~) {-div (k grad u(t,· ))(x) - I' Q(x )[1 - a(x, u(t, x))] + g( u(t, x))}2 dx :S 0,
whenever u is a solution of (1), with "= 0", iff u is a stationary solution of (1) (unique
solvability). This says intuitively that the family of solution curves t9 H u("'j 1', t9)
intersects the nondegenerate level hypersurfaces of the potential r I' transversally, hence
the w-limit of every solution is contained in a critical level set of r I'" In mathematical
terms, t9 H u(','j J1., t9) is called the solution semiflow of (1), and one says that such a