259
nonlinear mapping ryt : R+ x C(M) -t C(M), the net radiation forcing, by ryt(ll, w )(x) =
c/x) (IlQ(X )[1 - a(x, w(x ))] - g( w(x ))) for Il E R+, w E C(M) and x E M. It turns out
that ryt is a Lipschitz function on bounded subsets ofR+ xC(M). Employing the method
of variation of constants we can write the initial value problem (1), u(O,·) = () E C(M)
as an abstract Volterra integral equation
u(t,·) = S(t){} + [S(t-S)ryt(Il,U(S"))dS.
Standard techniques using Banach's fixed point theorem yield unique maximal solutions
in C(M), which depend continuously on the initial data. We refer to [Amann (1984),
Daners-Koch/Medina (1993), Hess (1991), Mora (1983)] for much more general results
on semilinear evolution equations. Regularity results show that these so-called mild
solutions are in fact classical solutions of the initial value problem. We now restrict
our attention again to nonnegative initial data and study the global solvability of the
initial value problem and the boundedness of those global solutions. This can be done
by utilizing standard comparison techniques.
Consider e.g. the initial value problem
{
c(x )Ot x EM, t E (0,00),
If inf 0, we have
Ot which shows

where U(t,Xjll,{)) denotes the maximal solution of (1) satisfying u(O,') = (), we obtain
C(x)Otv(t,x)-div(kgradv(t,·))(x) = IlQ(x)[l- a(x,u(t,Xjll,{)))l
-Ilinf Q[l-llallool - g(u(t,x)) + g(ip(t, x)) ::::: g(ip(t,x)) - g(u(t,x))
on dom( v). Therefore inf v < 0 can be excluded as before, since v(O,') == 0 and 9 is
strictly increasing, thus ip :::; u(·,·j Il, ()).
Likewise, let 1f; fulfils
C(x)Ot1f;(t,x) - div(k grad1f;(t, ·))(x) = f-l IIQlloo [1 - info:l - g(1f;(t,x))