284
(A2) and (A3) and is given by (D2II)(/1, 19)( 7])( s) == v(l + s, .j /1, 19,7]) for 7] E C([-T, 0] X
M) and s E [-T, 0], where v denotes the solution of
C(x )8t v(t, x) - div (k(.) grad v(t, ·))(x) ==
[-/1Q(t, x)82 0' (x, u, J~T f3(s )u(t + s, Xj /1, 'I3)ds) + g'( u)] v(t, x)
(23)
-/1Q(t, x )[hO' (x, u, J~T f3( S )u(t + s, Xj /1, 19)ds) J~T f3( s )v( t + s, x )ds
for t > 0, x E M
V i[-T,olxM== 7].
Again, compared with the cases considered before, there is some extra effort required,
when establishing that the spectral radius r(/1, 19) of (D2II)(/1, 19) is an eigenvalue with a
positive eigenfunction. In fact, (D2II)(/1, ' 13) is not strongly monotone in view of T > 1,
but sufficiently high iterates of this operator are, and so a version of the classical KreinRutman theorem can be employed. Next, we observe that each eigenvalue ( E C of the
complexification of (D2II)(/1, 19) corresponds via ( == e- A to an eigenvalue A E C of
(24) {
c(x)8n(t,x) - div( k(·)grad)'(t, ·))(x) + a(t,xh(t,x) ==
== b(t, x) J~Tf3(s) e-AS)'(t + s,x) ds + AC(X) )'(t,x) (t,x) E [0,1] x M
),(0,·) == )'(1, .).
For convenience we have set
a(t,x) :==
/1Q(t, x) 82 0' (x, u(t, Xj /1, '13), LOT f3( s )u( t + s, Xj /1, 'I3)ds) + g' (u(t, Xj /1, '13)),
The eigenfunctions are related via )'(t, x) == eAtv(t, Xj /1, ' 13, ¢), where), is an eigenfunction
of (24) to the eigenvalue A, iff ¢ is an eigenfunction of (D2II)(/1, ' 13) to the eigenvalue
e- A •
As it is typical for functional differential equations, we find a nonlinear dependence
on the eigenparameter in (24). Nevertheless, the previous observations guarantee that
(24) has a principal eigenvalue AO(/1, ' 13) in the parabolic sense meaning that AO(/1, ' 13) is
simple and has a positive eigenfunction, moreover that ~A ;::: Ao(/1, 19) for all eigenvalues
A of (24) with equality occuring, iff there exists an I E Z with A == >'0(/1,19) + 2i7rl.
Therefore we can formulate:
(A2) and (A3) and is given by (D2II)(/1, 19)( 7])( s) == v(l + s, .j /1, 19,7]) for 7] E C([-T, 0] X
M) and s E [-T, 0], where v denotes the solution of
C(x )8t v(t, x) - div (k(.) grad v(t, ·))(x) ==
[-/1Q(t, x)82 0' (x, u, J~T f3(s )u(t + s, Xj /1, 'I3)ds) + g'( u)] v(t, x)
(23)
-/1Q(t, x )[hO' (x, u, J~T f3( S )u(t + s, Xj /1, 19)ds) J~T f3( s )v( t + s, x )ds
for t > 0, x E M
V i[-T,olxM== 7].
Again, compared with the cases considered before, there is some extra effort required,
when establishing that the spectral radius r(/1, 19) of (D2II)(/1, 19) is an eigenvalue with a
positive eigenfunction. In fact, (D2II)(/1, ' 13) is not strongly monotone in view of T > 1,
but sufficiently high iterates of this operator are, and so a version of the classical KreinRutman theorem can be employed. Next, we observe that each eigenvalue ( E C of the
complexification of (D2II)(/1, 19) corresponds via ( == e- A to an eigenvalue A E C of
(24) {
c(x)8n(t,x) - div( k(·)grad)'(t, ·))(x) + a(t,xh(t,x) ==
== b(t, x) J~Tf3(s) e-AS)'(t + s,x) ds + AC(X) )'(t,x) (t,x) E [0,1] x M
),(0,·) == )'(1, .).
For convenience we have set
a(t,x) :==
/1Q(t, x) 82 0' (x, u(t, Xj /1, '13), LOT f3( s )u( t + s, Xj /1, 'I3)ds) + g' (u(t, Xj /1, '13)),
The eigenfunctions are related via )'(t, x) == eAtv(t, Xj /1, ' 13, ¢), where), is an eigenfunction
of (24) to the eigenvalue A, iff ¢ is an eigenfunction of (D2II)(/1, ' 13) to the eigenvalue
e- A •
As it is typical for functional differential equations, we find a nonlinear dependence
on the eigenparameter in (24). Nevertheless, the previous observations guarantee that
(24) has a principal eigenvalue AO(/1, ' 13) in the parabolic sense meaning that AO(/1, ' 13) is
simple and has a positive eigenfunction, moreover that ~A ;::: Ao(/1, 19) for all eigenvalues
A of (24) with equality occuring, iff there exists an I E Z with A == >'0(/1,19) + 2i7rl.
Therefore we can formulate: