222
Problem (P) can be considered as a perturbed problem associated to
1
Ut-(p(x)IUxIP-2ux)x=0, xE(-1,1),t>0,
(P*) p(x)luxI P - 2 ux = 0,
X = ±1,t > 0,
u(x,O)=Uo(X),
xE(-1,1).
The abstract Cauchy problem associated to (P') is given by
(CP.)J ~~(t)+Au(t)=O, inL2(1), fort>O,
1 u(O) = Uo
where we are identifying u(t) E U(I) with u(·, t). The operator A : D(A) -+ L2(1),
with D(A) C L2(1), is described in the following result giving also the existence and
uniqueness of the solution of (CP*).
Proposition 1 . (a) Consider the functional cp : L2( I) -+ 1R U {+oo} given by
() {
~ ( p(x)luxIPdx ifu E V
cpu= ph
= +00
otherwise.
(14)
Then cp =1= +00, cp is convex and lower semicontinuous.
(b) Let A(u) = or.p(u). Then D(A) C V, D(A) is dense in L2(1) and
Au = -(p(x)lux lp-2ux )x for any u E D(A).
(15)
(c) For any Uo E L2(1) there exists a unique function u E C([O, T] : L2(1)), for T > °
arbitrary, such that u(t) E D(A) for a.e. t > 0, d*t E L 2 (0,T : L2(1)) and satisfies
(CP·). Moreover if Uo E Lq(l) with 1 ~ q ~ +00 then u(t) E Lq(l). Finally, the
application S(t)uo = u(t) is a semigroup of contractions on L2(1).
Proof. (a) To prove that cp =1= +00 and that r.p is convex is obvious. The lower semicontinuity of r.p can be shown, for instance, using the reflexivity of the space P(I : p), and
that the norm is l.s.c. for the weak convergence.
(b) It is clear that V = D(cp)(=. {w E U(I) : cp(w) < oo}) is a dense subspace of L2(1)
(notice that C8"(I) c V). Then as D(ocp) C D(cp) and D(ocp) = D(cp) (see Brezis [1973])
we have that D( ocp) = L2(1). On the other hand it is a routin matter to see that r.p is
Gateux differentiable in V and that
'( )
. cp( u + )"h) - cp( u) f I I 2
< r.p u ,h >v'v= lIm
)..
=
p(x) Ux P- uxhxdx.
A~O
I
As or.p(u) is a maximal monotone operator we obtain (15).
(c) The existence of u with the indicated regularity is now a consequence of the abstract
Hille-Yosida theorem given in Brezis [1973]. If Uo E Lq(1) we multiply the equation by
Problem (P) can be considered as a perturbed problem associated to
1
Ut-(p(x)IUxIP-2ux)x=0, xE(-1,1),t>0,
(P*) p(x)luxI P - 2 ux = 0,
X = ±1,t > 0,
u(x,O)=Uo(X),
xE(-1,1).
The abstract Cauchy problem associated to (P') is given by
(CP.)J ~~(t)+Au(t)=O, inL2(1), fort>O,
1 u(O) = Uo
where we are identifying u(t) E U(I) with u(·, t). The operator A : D(A) -+ L2(1),
with D(A) C L2(1), is described in the following result giving also the existence and
uniqueness of the solution of (CP*).
Proposition 1 . (a) Consider the functional cp : L2( I) -+ 1R U {+oo} given by
() {
~ ( p(x)luxIPdx ifu E V
cpu= ph
= +00
otherwise.
(14)
Then cp =1= +00, cp is convex and lower semicontinuous.
(b) Let A(u) = or.p(u). Then D(A) C V, D(A) is dense in L2(1) and
Au = -(p(x)lux lp-2ux )x for any u E D(A).
(15)
(c) For any Uo E L2(1) there exists a unique function u E C([O, T] : L2(1)), for T > °
arbitrary, such that u(t) E D(A) for a.e. t > 0, d*t E L 2 (0,T : L2(1)) and satisfies
(CP·). Moreover if Uo E Lq(l) with 1 ~ q ~ +00 then u(t) E Lq(l). Finally, the
application S(t)uo = u(t) is a semigroup of contractions on L2(1).
Proof. (a) To prove that cp =1= +00 and that r.p is convex is obvious. The lower semicontinuity of r.p can be shown, for instance, using the reflexivity of the space P(I : p), and
that the norm is l.s.c. for the weak convergence.
(b) It is clear that V = D(cp)(=. {w E U(I) : cp(w) < oo}) is a dense subspace of L2(1)
(notice that C8"(I) c V). Then as D(ocp) C D(cp) and D(ocp) = D(cp) (see Brezis [1973])
we have that D( ocp) = L2(1). On the other hand it is a routin matter to see that r.p is
Gateux differentiable in V and that
'( )
. cp( u + )"h) - cp( u) f I I 2
< r.p u ,h >v'v= lIm
)..
=
p(x) Ux P- uxhxdx.
A~O
I
As or.p(u) is a maximal monotone operator we obtain (15).
(c) The existence of u with the indicated regularity is now a consequence of the abstract
Hille-Yosida theorem given in Brezis [1973]. If Uo E Lq(1) we multiply the equation by