226
3.1. The Sellers model.
The following result shows the uniqueness and others properties of solutions for the
Sellers model.
Theorem 2 Let p > 1, and assume that
Ra satisfies (11) with f3 a locally Lipschitz function of u.
(23)
Then given Uo E Loo(1) there exists at most one bounded weak solution of (P).
Idea of the proof. First of all we point out that Ut E LV' (0, T : V'). This can be obtained
from the definition of bounded weak solutions and the characterization of the dual space
V' (see e.g. Ivanov [1981], Lemma V.2.1). Moreover, if we define w = e-Ctu, w satisfies
(in a weak sense) the equation
Wt - e-C(p-2)t(p(x)lwxIP-2wx)x = e-CtQ(x, t)f3(we Ct ) - e- Ct Re(x, t, we Ct ) - Cwo
Since f3 is assumed locally Lipschitz we can choose C large enough such that the function
F(x,t,w) = e- Ct Q(x,t)f3(we Ct ) - Cw
is a strictly decreasing function of v for fixed (x, t). Now assume that we have another
solution u* of (P) corresponding to the same datum Uo. We take w - w* (w* = e-Ctu*)
as test function in the difference of the identities satisfied by wand w* (see the definition
of bounded weak solution). We have that
< Wt(t) - w;(t), w(t) - w*(t) >v/v= ! i Iw(t) - w*(t)1 2 dx
(see e.g. Temam [1988]). Moreover, there exists K > 0 such that if p 2: 2
ip(x)(lwxIP-2wx -lw;IP-2W ;)(wx - w;)dx 2: K ip(x)lwx - w;IPdx
(24)
For 1 < p < 2 the right-hand side term must be replaced by
K ip(x)lwx - w;1 2 (lwxI 2 - P + Iw;1 2 -P)dx
(see, e.g., Diaz[1985] Lemma 4.10). Using the monotonicity of R.(·,·, u) and F(·,·, w) we
obtain that
! i Iw(t) - w*(tWdx ~ 0
and so necessarily u = U·.
•
Corollary 2. Assume (23). Let Uo, Uo E Loo(1) and let u, u be weak solutions of
(P) corresponding to the energy emmision functions Ra(x,t,u) = ,(u) + f(x,t) and
Ra(x,t,u) = ,(u) + j(x,t) satisfying the condition (11). Then there exists a constant
K = K(T) ;:: 0 such that
II [u(t) - u(t)]+ 11£2(1)~
~ e Kt (II [uo - uo]+ 11£2(1) + lot e- Kt II [f( s) - j( s )]+ IIL2(1) dS) .
(25)
In particular Uo ~ uo, f ~ j imply u ~ u.
3.1. The Sellers model.
The following result shows the uniqueness and others properties of solutions for the
Sellers model.
Theorem 2 Let p > 1, and assume that
Ra satisfies (11) with f3 a locally Lipschitz function of u.
(23)
Then given Uo E Loo(1) there exists at most one bounded weak solution of (P).
Idea of the proof. First of all we point out that Ut E LV' (0, T : V'). This can be obtained
from the definition of bounded weak solutions and the characterization of the dual space
V' (see e.g. Ivanov [1981], Lemma V.2.1). Moreover, if we define w = e-Ctu, w satisfies
(in a weak sense) the equation
Wt - e-C(p-2)t(p(x)lwxIP-2wx)x = e-CtQ(x, t)f3(we Ct ) - e- Ct Re(x, t, we Ct ) - Cwo
Since f3 is assumed locally Lipschitz we can choose C large enough such that the function
F(x,t,w) = e- Ct Q(x,t)f3(we Ct ) - Cw
is a strictly decreasing function of v for fixed (x, t). Now assume that we have another
solution u* of (P) corresponding to the same datum Uo. We take w - w* (w* = e-Ctu*)
as test function in the difference of the identities satisfied by wand w* (see the definition
of bounded weak solution). We have that
< Wt(t) - w;(t), w(t) - w*(t) >v/v= ! i Iw(t) - w*(t)1 2 dx
(see e.g. Temam [1988]). Moreover, there exists K > 0 such that if p 2: 2
ip(x)(lwxIP-2wx -lw;IP-2W ;)(wx - w;)dx 2: K ip(x)lwx - w;IPdx
(24)
For 1 < p < 2 the right-hand side term must be replaced by
K ip(x)lwx - w;1 2 (lwxI 2 - P + Iw;1 2 -P)dx
(see, e.g., Diaz[1985] Lemma 4.10). Using the monotonicity of R.(·,·, u) and F(·,·, w) we
obtain that
! i Iw(t) - w*(tWdx ~ 0
and so necessarily u = U·.
•
Corollary 2. Assume (23). Let Uo, Uo E Loo(1) and let u, u be weak solutions of
(P) corresponding to the energy emmision functions Ra(x,t,u) = ,(u) + f(x,t) and
Ra(x,t,u) = ,(u) + j(x,t) satisfying the condition (11). Then there exists a constant
K = K(T) ;:: 0 such that
II [u(t) - u(t)]+ 11£2(1)~
~ e Kt (II [uo - uo]+ 11£2(1) + lot e- Kt II [f( s) - j( s )]+ IIL2(1) dS) .
(25)
In particular Uo ~ uo, f ~ j imply u ~ u.