220
A list of structure assumptions is the following:
p(x) = k(1 - x 2 ) with k > 0,
Ra(x, t, u) = Q(x, t)j3( u) where Q E C([ -1,1] x JR+) satisfies
° < Q(x, t) and j3 is a nondecreasing function such that
1j3(u)1 :::; M Vu E JR, for some M > 0,
}
Re(x,t,u) is a continuous function on x, Lipschitz on t and Re(x,t,') }
is non decreasing as function on u, for any fixed (x, t) E 7 x JR+.
(8)
(9)
(10)
The rest of the work is organized in the following way: The notion of weak solutions
of problem (P) is introduced in Section 2. It is proven that if Uo E Loo(I) there exists at
least one bounded weak solution of (P). This is obtained by two different methods: via
a compactness abstract method and via a regularization argument. Due to the presence
of the degenerate coefficient p(x) the natural energy space is given by V = {w E L2(I) :
Wx E U(I : pn, where U(I : p) is the weighted-Lebesgue space associated to p.
The question of the uniqueness of bounded weak solutions is studied in Section 3. The
answer is positive for the Sellers model (it is enough to require j3 be a locally Lipschitz
continuous function). As in the case of the homogeneous model (see Diaz[1992]) the
Budyko model may have more than one solution. This is explicitly shown in the Subsection
3.1 by means of the construction of a counterexample. Nevertheless, in the Subsection
3.2, it is shown that there is at most one solution of the Budyko model in the class of
solutions satisfying a "nondegeneracy property". The free boundary generated in the case
of Budyko type models is considered in Section 4. Finally, Section 5 is devoted to the
study of the approximate controllability of the problem.
2. On the existence of solutions.
It is well known (see, e.g. Diaz-Herrero [1981] for the special case of p = 1 and
Ra == 0) that if p > 2 the degeneracy of the diffusion operator makes impossible expect
the existence of a classical solution of (P) even for a regular initial datum uo. In order
to make precise the notion of solution we shall study, we start by indicating that the
eventual discontinuous character of the function Ra will be treated by assuming that
Ra(x,t,u) = Q(x,t)j3(u), with Q as in (4) and j3 a }
maximal monotone graph of JR2 such that Izl :::; M
for any z E j3(u), for any u E JR and some M > °
(ll)
(i.e. for example, j3 is given by a nondecreasing real function bas j3(r) = {b(rn if b is
continuous in r or j3(r) = [b(r-), b(r+)] if b has a jump at the point r: see Brezis[1973]).
A usual way to verify the differential equation (at least weakly) is to multiply by a test
function followed by an integration by parts. In doing so we obtain
1 u(x, T)v(x, T)dx - loT 1 u(x, t)Vt(X, t)dxdt
A list of structure assumptions is the following:
p(x) = k(1 - x 2 ) with k > 0,
Ra(x, t, u) = Q(x, t)j3( u) where Q E C([ -1,1] x JR+) satisfies
° < Q(x, t) and j3 is a nondecreasing function such that
1j3(u)1 :::; M Vu E JR, for some M > 0,
}
Re(x,t,u) is a continuous function on x, Lipschitz on t and Re(x,t,') }
is non decreasing as function on u, for any fixed (x, t) E 7 x JR+.
(8)
(9)
(10)
The rest of the work is organized in the following way: The notion of weak solutions
of problem (P) is introduced in Section 2. It is proven that if Uo E Loo(I) there exists at
least one bounded weak solution of (P). This is obtained by two different methods: via
a compactness abstract method and via a regularization argument. Due to the presence
of the degenerate coefficient p(x) the natural energy space is given by V = {w E L2(I) :
Wx E U(I : pn, where U(I : p) is the weighted-Lebesgue space associated to p.
The question of the uniqueness of bounded weak solutions is studied in Section 3. The
answer is positive for the Sellers model (it is enough to require j3 be a locally Lipschitz
continuous function). As in the case of the homogeneous model (see Diaz[1992]) the
Budyko model may have more than one solution. This is explicitly shown in the Subsection
3.1 by means of the construction of a counterexample. Nevertheless, in the Subsection
3.2, it is shown that there is at most one solution of the Budyko model in the class of
solutions satisfying a "nondegeneracy property". The free boundary generated in the case
of Budyko type models is considered in Section 4. Finally, Section 5 is devoted to the
study of the approximate controllability of the problem.
2. On the existence of solutions.
It is well known (see, e.g. Diaz-Herrero [1981] for the special case of p = 1 and
Ra == 0) that if p > 2 the degeneracy of the diffusion operator makes impossible expect
the existence of a classical solution of (P) even for a regular initial datum uo. In order
to make precise the notion of solution we shall study, we start by indicating that the
eventual discontinuous character of the function Ra will be treated by assuming that
Ra(x,t,u) = Q(x,t)j3(u), with Q as in (4) and j3 a }
maximal monotone graph of JR2 such that Izl :::; M
for any z E j3(u), for any u E JR and some M > °
(ll)
(i.e. for example, j3 is given by a nondecreasing real function bas j3(r) = {b(rn if b is
continuous in r or j3(r) = [b(r-), b(r+)] if b has a jump at the point r: see Brezis[1973]).
A usual way to verify the differential equation (at least weakly) is to multiply by a test
function followed by an integration by parts. In doing so we obtain
1 u(x, T)v(x, T)dx - loT 1 u(x, t)Vt(X, t)dxdt