Comparison Results for Nonlinear Parabolic Equations with Monotone Principal Part

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1 CADERNOS DE MATEMÁTICA 1, April (2) ARTIGO NÚMERO SMA#82 Comparison Results for Nonlinear Parabolic Equations with Monotone Principal Part Alexanre N. Carvalho * Departamento e Matemática, Instituto e Ciências Matemáticas e e Computação, Universiae e São Paulo - Campus e São Carlos, Caixa Postal 668, São Carlos SP, Brazil ancarva@icmsc.sc.usp.br Clauia B. Gentile Departamento e Matemática, Universiae Feeral e Viçosa, Viçosa, MG Brazil gentil .ufv.br In this work we prove an abstract comparison principle for abstract quasilinear parabolic problems with monotone principal part. This result is applie to parabolic problems having the p Laplacian as principal part an with non- Lipschitz perturbations. April, 2 ICMC-USP 1. INTRODUCTION In this work we prove abstract comparison results for solutions of nonlinear parabolic problems with monotone principal part an apply these results to a parabolic problem having the p Laplacian as principal part. To be more precise, let A be a maximal monotone operator in a Hilbert space H, {S(t); t } be the semigroup generate by A an, for i =, 1, consier the associate initial value problem u = Au + f i (t, u) u() = u i. (1) Denote by u(t, u i, f i ) a solution of (1), i =, 1. Assume that H is enowe with an orer (e.g, u in H = L 2 (Ω) if u almost everywhere). A first comparison result in the orere Hilbert space H states that u u 1 implies S(t)u S(t)u 1 for all t. In this case we say that the semigroup S(t) is increasing. We seek for conitions on the maximal monotone operator A that ensures this first comparison result. Once we have obtaine * Research Partially Supporte by CNPq grant 3.889/92-5 an FAPESP grant 97/11323-, Brazil Research Partially Supporte by CNPq an CAPES, Brazil 167 Sob a supervisão CPq/ICMC

2 168 A. N. CARVALHO AND C. B. GENTILE conitions on A ensuring that S(t) is increasing we exten this comparison result to the solutions of the problem (1). The statement of the comparison results that we will present in this paper are similar to those alreay establishe elsewhere for semilinear parabolic equations. In particular the starting point of this theory is a result that can be state in the following way: If the resolvent of A is increasing (in a sense to be explaine later) then, S(t) is an increasing semigroup. This result reas exactly the same as the corresponing result for the case when the maximal monotone operator A is linear. They iffer, however, in the techniques use in the proof. This conition is actually a necessary an sufficient conition an its proof is base on a generalization of a result ue to H. Brèzis that we state in the next section. Concerning the comparison results for (1), the theorem that we woul like to prove shoul rea: If S(t) is increasing, u u 1 an f (t, u) f 1 (t, u) for all t, u then u(t, u, f ) u(t, u 1, f 1 ) as long as both solutions exist. As we will see in the coming sections there are some technical ifficultness that will impair us from obtaining such comparison result. These technical ifficultness will lea to some aitional assumptions on the vector fiels f i. The the comparison results that we will be able to prove for (1) will have statements similar to the statements of the comparison results presente in [1] for semilinear parabolic problems. This paper is organize as follows. Section 2 is evote to the proof of a preliminary result that will be the key to prove the abstract comparison results. Section 3 is evote to the statement an proof of the abstract comparison results for the case of globally Lipschitz perturbations. Section 4 eals with the case of non-lipschitz perturbations of subifferentials. Section 5 is evote to the application of these results to the stuy of a parabolic equation with the p-laplacian perturbe by non-monotone non-lipschitz operators. Finally, in Section 6 we exhibit some more examples of maximal monotone operators with increasing resolvent. 2. A PRELIMINARY RESULT Let H be a Hilbert space with inner prouct, an norm. Let A be a maximal monotone operator in H, an S(t) be the semigroup generate by A. We will employ the following notation. J λ := (I + λa) 1, is the resolvent A; A λ := I J λ, is the Yosia approximation of A; λ A x, is the least norm element of Ax. Let ϕ be a convex, proper an lower semicontinuous (hereafter l.s.c) map efine in H. We efine Sob a supervisão a CPq/ICMC ϕ µ (u) = 1 2µ u (I µ ϕ) 1 u 2 + ϕ((i µ ϕ) 1 u).

3 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 169 ϕ µ thus efine is convex, Frechét-ifferentiable an ϕ µ = ( ϕ) µ. Furthermore, ϕ µ (x) ϕ(x) as µ, x H, (see Proposition 2.11 in [3], page 39). The following theorem can be foun in [3], Theorem 4.4, page 13. Theorem 2.1. for any x H: Let ϕ be a convex proper an l.s.c. map efine in H an assume that (H-1) ϕ(proj D(A) x) ϕ(x). Then, the following are equivalent: (i) ϕ((i + λa) 1 x) ϕ(x) x H, λ > ; (ii) A λ x, z z ϕ(x), λ > ; (iii) A λ x, ϕ µ (x) x H, λ, µ > ; (iv) A x, ϕ µ (x) x H, µ > ; (v) ϕ µ (S(t)x) ϕ µ (x) t, x D(A), µ > ; (vi) ϕ(s(t)x) ϕ(x) t, x D(A). Remark Note that, in the statement of Theorem 2.1, the operator A in general is not the subifferential ϕ of ϕ. The following theorem is a variation of Proposition 4.7, page 134, [3], an follows as a consequence of Theorem 2.1. The equivalent statements of the next theorem are the base to all comparison results that will be prove in this paper. Theorem 2.2. Let ϕ be a convex proper an l.s.c. function in H an let A, B be maximal monotone operators in H satisfying (H-2) ϕ(proj D(A) x Proj D(B) y) ϕ(x y), x, y H. Sob a supervisão CPq/ICMC

4 17 A. N. CARVALHO AND C. B. GENTILE Then, the following are equivalent: (i) ϕ((i + λa) 1 x (I + λb) 1 y) ϕ(x y) x, y H, λ > ; (ii) A λ x B λ y, z z ϕ(x y), λ > ; (iii) A λ x B λ y, ϕ µ (x y) x, y H, λ, µ > ; (iv) A x B y, ϕ µ (x y) x, y H, µ > ; (v) ϕ µ (S A (t)x S B (t)y) ϕ µ (x y) (vi) ϕ(s A (t)x S B (t)y) ϕ(x y) t, µ >, x, y D(A); t >, x D(A), y D(B). Proof: Let H = H H, be the Hilbert space with the inner prouct ([u, ū], [v, v]) = u, ū + v, v, an let C : H H, be the operator efine by C[u, ū] = [Au, Bū]. The operator C is monotone, since (C[u, ū] C[v, v], [u, ū] [v, v]) = ([Au, Bū] [Av, B v], [u v, ū v]) = ([Au Av, Bū B v], [u v, ū v]) = Au Av, u v + Bū v, ū v. Let us verify that R(I + C) = H. In fact, for all [u, v] H there exists [x, y] H such that then (I + A)x = u e (I + B)y = v, (I + C)[x, y] = [x, y] + C[x, y] = [x, y] + [Ax, By] = [x + Ax, y + By] = [(I + A)x, (I + B)y] = [u, v]. Hence, C is maximal monotone in H. Besies that we have: 1. J λ [u, ū] = [J A λ u, J B λ ū], where J λ is the resolvent of C; 2. C λ [u, ū] = [A λ u, B λ ū], where C λ is the Yosia approximation of C; 3. S(t)[u, ū ] = [S A (t)u, S B (t)ū ], where S(t) is the semigroup generate by C, S A (t) is the semigroup generate by A an S B (t) is the semigroup generate by B. Let ϕ be a convex, proper an l.s.c. function efine in H, an let ψ be the function given by ψ[x, y] = ϕ(x y). Then ψ is convex proper an l.c.s. an: Sob a supervisão a CPq/ICMC

5 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS ψ[x, y] = {[z, z]; z ϕ(x y)}; 2. ψ λ [x, y] = [( ϕ) 2λ (x y), ( ϕ) 2λ (x y)]; 3. ψ λ [x, y] = (ϕ) 2λ (x y). With the consierations above the result now follows applying Theorem 2.1 to the operator C efine in H, an the convex, proper an l.s.c. function ψ. Remark The hipothesis (H-2) is automatically satisfie if D(A) = H an D(B) = H. 3. COMPARISON RESULTS In this section we first prove abstract monotonicity results for semigroups of nonlinear contractions generate by maximal monotone operators using the results of previous section. Then, we use these monotonicity results to obtain comparison results for abstract nonlinear parabolic problems with monotone principal part an non-monotone perturbations of various types (time-epenent, globally Lipschitz an non-globally Lipschitz). We start introucing a concept of orer in a Banach space which is suitable for the esire comparison results: Definition 3.1. An orere Banach space is a pair (X, ), where X is a Banach space an is a orer relation in X satisfying: 1. x y implies x + z y + z, x, y, z X; 2. x y implies λx λy, x, y X an λ R; 3. The positive cone C = {x X : x } is close in X. Definition 3.2. Let (X, ) an (Y, ) be orere Banach spaces. We say that a map T : X Y is increasing if an only if x y implies T (x) T (y). We say that T is positive if an only if x implies T (x). Let (H, ) be an orere Hilbert space an let A be a maximal monotone operator in H. Let C be the positive cone efine in H, an let I C be the inicator map: if x C, I C = + if x C. I C is a convex, proper an l.s.c. map, an therefore I C is a maximal monotone operator in H. In what follows we characterize (I λ I C ) 1, ( I C ) λ an (I C ) λ, for etails see [3, 2]. Sob a supervisão CPq/ICMC

6 172 A. N. CARVALHO AND C. B. GENTILE the resolvent operator (I λ I C ) 1 of I C is the projection in C. In other wors, y = (I λ I C ) 1 x y = Proj C x; the Yosia approximation of I C is given by ( I C ) λ = 1 λ (x Proj C x); (I C ) λ (x) = 1 2λ x Proj C x 2. A semigroup of nonlinear contractions {S(t) : t } is increasing if, for each t, S(t) is an increasing map. In what follows we give e necessary an sufficient conition so that the semigroup {S(t); t } generate by a maximal monotone operator A be increasing: Theorem 3.1. Let A be a maximal monotone operator in H satisfying (H-3) I C (Proj D(A) x Proj D(A) y) I C (x y), x, y H. Then, the following are equivalent: (i) x y (I + λa) 1 x (I + λa) 1 y x, y H, λ > ; (ii) x y S(t)x S(t)y t, x, y D(A). Furthermore, if S(t) =, t we have that (iii) x S(t)x t, x D(A). an in this case we say that the semigroup is positive. Proof: It is enough to consier A = B an ϕ = I C in Theorem 2.2, an the result will follow as a consequence of the equivalence (i) (vi). More generally we have: Theorem 3.2. Uner the hypothesis (H-3). Denote by u, ū, the respective solutions of the following initial value problems: t u + Au y u() = u, e tū + Aū ȳ ū() = ū. If y, ȳ, u, ū H, y ȳ an u ū, an if (I + λa) 1 is increasing, then u(t) ū(t) t >. Proof: In fact, it is enough to note that, if y H, the operator A y : H H given by A y u = Au y is maximal monotone. Then the result follows from Theorem 2.2 with ϕ = I C, A = A y e B = Aȳ, observing that w = (I + λa y ) 1 u w = (I + λa) 1 (u + λy), Sob a supervisão a CPq/ICMC

7 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 173 w = (I + λaȳ) 1 ū w = (I + λa) 1 (ū + λȳ), in such a way that u ū e y ȳ implies (I + λa y ) 1 u (I + λaȳ) 1 ū, < λ R, that is, = I C ((I + λa y ) 1 u (I + λaȳ) 1 ū) I C (u ū) =, u, ū H, λ >. To exten the above results to the solutions of the non-homogeneous equations, we shall employ the same limiting proceure use to obtain existence of solutions for these equations. Theorem 3.3. Let T > an let f(t), f(t) L 1 (, T ; H), u D(A). Suppose that u, ū are, respectively, the weak solutions of the following initial value problems: u + Au f(t) t u() = u, an + Aū f(t) tū ū() = ū. If A is a maximal monotone operator in H with increasing resolvent an satisfying the hypothesis (H-3), f(t) f(t) a.e. in [, T ], u an ū H, then u ū implies u(t) ū(t) t [, T ]. Proof: There are sequences {f n } an { f n } of step functions in [, T ] such that f n f an f n f in L 1 (, T ; H). We may assume, without loss of generality that for each n N, given a partition of the interval [, T ], = t < t 1 <... < t kn = T, f n a ni an f n ā ni in [t i 1, t i ] with a ni, ā ni R, a ni ā ni. Let u n e ū n be strong solutions of the equations t u n + Au n f n an tūn + Aū n f n respectively, with u n () = u, an ū n () = ū. It follows from Theorem 3.2 that u n (t) ū n (t) t [, T ], n N. Since u n (t) u m (t) f n (τ) f m (τ) τ t [, T ], an ū n (t) ū m (t) f n (τ) f m (τ) τ t [, T ], then u n u an ū n ū uniformly in C(, T ; H), an since the positive cone is close, u(t) ū(t). Once we have the above result we start to compare solutions of non-linearly perturbe problems. As in the preceing proof the comparison is obtaine through the same proceure use to obtain existence of solutions. Sob a supervisão CPq/ICMC

8 174 A. N. CARVALHO AND C. B. GENTILE Theorem 3.4. Let A be a maximal monotone operator in an orere Hilbert space H, an let be its orer relation. Let B, B be globally Lipschitz maps in H, with Lipschitz constants ω an ω respectively. Let u, ū D(A), an let u, ū be weak solutions of the problems u(t) + Au(t) Bu(t) t u() = u, an + Aū(t) Bū(t) tū(t) ū() = ū. respectively. Suppose that A has increasing resolvent, satisfies the hypothesis (H-3), an there exists G : D(A) H, increasing, such that Bu Gu Bu, u D(A). Then u ū implies u(t) ū(t) t. The proof of the above theorem is similar to the proof of the next theorem an therefore will be omitte. Theorem 3.5. Let ϕ be a convex, proper an l.s.c. in the orere Hilbert space H, let be its orer. If B, B are maps from [, T ] D(ϕ) in H, each of them satisfying the conitions (1) an (2) of the Proposition 3.13 in [3], page 17, with Lipschitz constants ω e ω respectively. Let u an ū be solutions of the problems u(t) + ϕ(u(t)) B(t, u(t)) t u() = u, an tū(t) + ϕū(t) B(t, ū(t)) ū() = ū. respectively. Suppose that ϕ has increasing resolvent, satisfies the hypothesis (H-3), an there exist G : [, T ] D(ϕ) H, increasing in the variable u for each t [, T ], an such that for all t [, T ], B(t, u) G(t, u) B(t, u), u D(ϕ). Then u ū implies u(t) ū(t) t. Proof: We consier the sequences u n e ū n efine by u (t) u an ū (t) ū, u n e ū n are solutions of the equations t u n(t) + ϕ(u n (t)) B(t, u n 1 (t)) u n () = u, an tūn(t) + ϕ(ū n (t)) B(t, ū n 1 (t)) ū n () = ū. Sob a supervisão a CPq/ICMC

9 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 175 respectively. By Theorem 3.3 u 1 (t) ū 1 (t), t >. Assume by inuction that u n 1 (t) ū n 1 (t), t >. Then by Theorem 3.3, u n (t) ū n (t) n N, t >, an since u n+1 (t) u n (t) B(τ, u n (τ)) B(τ, u n 1 (τ)) τ ω u n (τ) u n 1 (τ) τ ū n+1 (t) ū n (t) B(τ, ū n (τ)) B(τ, ū n 1 (τ)) τ then ω u n+1 (t) u n (t) ū n (τ) ū n 1 (τ) τ (ωt )n u 1 u L (,T ;H) n! ( ωt )n ū n+1 (t) ū n (t) ū 1 ū L n! (,T ;H) where u n u an ū n ū uniformly in [, T ], in such a way that u(t) ū(t), t >. 4. MORE GENERAL PERTURBATIONS OF SUBDIFFERENTIALS This section is evote to the proof of comparison results for problems with subifferential principal part an for perturbations which are not globally Lipschitz. To accomplish that we will use the existence results prove in [9] which we briefly escribe in Subsection 4.1. The proof of the comparison results are given in Subsection The Existence In this subsection we state a particular case of a result ue to Mitsuharu Ôtani, [9]. That result assures the existence of at least one strong solution for the equation u(t) + Au(t) Bu(t), < t < T, (2) t where some of the restrictions are mae on the operator A so that the hypotheses in the perturbation B may be relaxe. Since the proof of this result is use to prove our comparison results we shall inicate its main steps. We will assume that A = ϕ, with ϕ satisfying: (O-1) ϕ is a convex, proper an l.s.c. from H into [, + ], such that for each L (, + ), the set {u H; ϕ(u) + u 2 L} is compact in H. Sob a supervisão CPq/ICMC

10 176 A. N. CARVALHO AND C. B. GENTILE (O-2) For all u D( ϕ), Bu is a convex subset of H; (O-3) B is semi-close in the following sense: for all interval [a, b] [, T ] the following hols: if u n u in C(a, b; H), ϕ(u n ( )) g n ( ) g( ) ϕ(u( )) in L 2 (a, b; H) an if Bu n ( ) b n ( ) b( ) in L 2 (a, b; H), then b( ) Bu( ); that is, b(t) Bu(t) for almost all [a, b]; (O-4) There exists a real, increasing an positive L ( ), a constant γ (, 1) an a constant c such that: Bu 2 H γ ϕ (u) 2 + c L (ϕ(u) + u ), u D( ϕ), where ϕ (u) is the smallest norm element in ϕ(u) e Bu H = sup{ b, b Bu}. In the proof of the local existence result the following version of Schauer-Tychonoff Theorem is use: Theorem 4.1 (Brower). Let K be a compact convex subset of a locally convex topological vector space X. Let T be a multi-value l.s.c. map from K into X such that for each x K, T (x) is a close convex subset X such that T (x) K Ø. Then T has a fixe point in K; that is, there is an element x K such that x T (x ). Definition 4.1. For each u D( ϕ), h(t) L 2 (, T ; H), an S (, T ], we enote by E u,s(h) the unique strong solution of the equation u(t) + ϕ(u(t)) h(t) (3) t in [, S] satisfying u() = u. Let R be a fixe positive real number an let K R,S be the set K R,S := {u L 2 (, S; H); u L 2 (,S;H) R}, enowe with the weak topology of L 2 (, S; H). Let B u,s,r : K R,S K R,S efine in the following manner: given h K R,S, if there exist b K R,S such that b(t) B(E u,s(h)(t)) almost everywhere (, S), then b B u,s,r(h), an D(B u,s,r) = {h K R,S ; B u,s,r(h) }. Lemma 4.1. Suppose that the hypotheses (O-1), (O-2) an (O-3) are satisfie. Then, the graph G(B u,s,r), of B u,s,r, is close in K R,S K R,S. Besies that, if h D(B u,s,r) then B u,s,r(h) is a close convex subset of K R,S. Sob a supervisão a CPq/ICMC

11 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 177 Theorem 4.2. Suppose that the hypotheses (O-1), (O-2), (O-3) an (O-4) are satisfie. If u D(ϕ) the problem (2) has a strong solution in [, T ] for some T (, T ]. Sketch of the Proof: The iea of the proof is to show that the operator B u,t,r has at least one fixe point in K R,T by making appropriate choice for the values of R an T. In that way, there exist h K R,T such that B u,t,r(h) = h an, from the efinition of B u,t,r it follows that E u,t (h) is a solution of the problem (2) in [, T ) Comparison In what follows we are going to compare solutions of problems with perturbations more general then Lipschitz. As before, we are going to appeal to the techniques use to obtain the existence an Theorem 3.3. We assume throughout this section that (H 3) is satisfie. Theorem 4.3. Let ϕ satisfying the hypothesis (O-1), u, ū D(ϕ), u ū. Let B, B be two functions efine in D(ϕ) taking values in H an satisfying the hypotheses (O-2), (O-3), an (O-4). Suppose that ϕ has increasing resolvent, satisfies the hypothesis (H- 3), an that there exist G : D(ϕ) H, increasing, such that Bu Gu Bu, u D(ϕ). Then given a solution u(t) of the problem u(t) + ϕ(u(t)) Bu(t), < t < T, t u() = u D(ϕ), (4) there exists a solution ū(t) of the problem tū(t) + ϕ(ū(t)) Bū(t), < t < T, ū() = ū D(ϕ), (5) ū(t) efine in [, T ) for some T >, an u(t) ū(t) t [, T ). Proof: In fact, let T 1 > such that u(t) is solution of (4) in [, T 1 ), let T T 1 be a positive real number to be etermine, an let K R,T, B u,t,r an Bū,T,R as in the Definition 4.1 with R 2 = 2 γ ϕ(ū ) + ε, (1 γ ) where γ is as in the hypothesis (O-4); that is Bū 2 H γ ϕ (ū) 2 + c L (ϕ(ū) + ū ), ū D( ϕ), an ε is an arbitrary positive number. By efinition of B u,t,r an u, h := Bu is a fixe point B u,t,r in K R,T. We consier the restriction B K of Bū,T,R to the following subset Sob a supervisão CPq/ICMC

12 178 A. N. CARVALHO AND C. B. GENTILE of K R,T : K = {g K R,T ; g h}. K is convex an compact. Let us verify that B K : K K. Well, if g K, then g h an by Theorem 3.3 u g (t) u h (t), t >. Thus B(u g ) G(u g ) G(u h ) B(u h ), an consequently B(g) B(h) = h, g K. It remains to conclue that given g(t) K, if b(t) B(g(t)) almost everywhere in [, T ], then b(t) L 2 (,T ;H) R. For that we first observe that if k K, an v = Eū,T (k), then by Lemma 3.3 in [3], page 73, for almost all t [, T ], we have: therefore t ϕ(v(t)) = t v(t) + k(t), t v(t) = t v(t) + k(t) 2 + t v(t) + k(t), k(t), t v(t) + k(t) 2 + t ϕ(v(t)) t v(t) + k(t) k(t) an therefore In one sie t v(t) + k(t) k(t) 2, t v(s) + k(s) 2 s + ϕ(v(t)) ϕ(ū ) t v(s) + k(s) 2 s 2ϕ(ū ) + k(s) 2 s. k(s) 2 s 2ϕ(ū ) + R 2, (6) an in another That is ϕ(v(t)) ϕ(ū ) k(s) 2 s ϕ(ū ) R2. max{ϕ(v(t)); t [, T ]} C, where C = ϕ(ū ) R2. (7) Besies that, from the efinition of subifferential, ϕ(v(t)) ϕ(ū ) t v(t) + k(t), v(t) ū ϕ(v(t)) ϕ(ū ) 1 2 t v(t) ū 2 + k(t), v(t) ū Sob a supervisão a CPq/ICMC

13 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS t v(t) ū 2 +ϕ(v(t)) ϕ(ū ) + k(t) v(t) ū 1 2 v(t) ū 2 From which we conclue that ϕ(ū )s + k(s) v(s) ū s. v(t) u() 2tϕ(ū ) + ( ) 1/2 t k(s) 2 s v(t) 2tϕ(ū ) + tr + ū, an therefore max{ v(t) ; t T } C 1, where C 1 = ( 2ϕ(ū ) + R)T 1/2 + ū. (8) Now, if b(t) Bv(t), then by the hypothesis (O-4), by (8) an by (7), Hence an by 6 From the choice of R, Choosing T small enough so that we obtain b(t) 2 γ ϕ (v(t)) 2 + K, where K = c L (C 1 + C ). b(t) 2 L 2 (,T ;H) γ ϕ (v(t)) 2 L 2 (,T ;H) + T K, b(t) 2 L 2 (,T ;H) γ [2ϕ(ū ) + R 2 ] + T K. b(t) 2 L 2 (,T ;H) (1 γ)r2 ε + γr 2 + T K. T K ε, b(t) L 2 (,T ;H) R. That is, if k K, v = Ēū,T (k), an b(t) Bv(t) almost everywhere in [, T ], then b K. In this way Bū,T,R( K) K. Besies that B K satisfy all the hypotheses of the Theorem 4.1 since B satisfy, from which one conclues by Lemma 4.1 an Theorem 4.1, that h K satisfying h = Bū,T,R( h). Hence, there exists a solution ū for the problem (5) in the interval [, T ]. Since Bū(t) Bu(t) follows by Theorem 3.3 that ū(t) u(t) almost everywhere in [, T ). Sob a supervisão CPq/ICMC

14 18 A. N. CARVALHO AND C. B. GENTILE Remark If in the above theorem we can ensure that the problem (5) has a unique solution ū(t), then all solution u(t) of the problem (4) satisfy u(t) ū(t), t [, T ]. 5. ONE APPLICATION Let Ω R n be a boune smooth omain with bounary Ω, an let H = L 2 (Ω) be enowe with the following orer relation: if u, v L 2 (Ω) then u v u(x) v(x) a.e in Ω. Consier the following secon orer nonlinear partial ifferential equation: t u pu = Bu, p > 1, (9) where p u = iv( u p 2 u) is the p-laplacian. In this section we aim to apply the abstract results evelope in the previous sections to obtain comparison results for (9). To that en we assume that the operator B is single value an can be ecompose into the sum of two other operators, that is, B = B 1 + B 2, with B 1 an B 2 satisfying the following conitions: (A-1) p u B 1 (u) is the subifferential of a convex, proper an lower semicontinuous function ϕ : H R such that ϕ(u) u W 1,p (Ω), D(ϕ) = W 1,p (Ω); (A-2) B 1 is the Nemitskiĭ operator associate to a function b 1 : R R which is continuous an non-increasing an satisfies b 1 () = ; (A-3) B 2 is the Nemitskiĭ operator associate to a function b 2 : R R Uner these conitions we rewrite the equation (9) as: t u(t) + ϕ(u(t)) = B 2(u(t)), t >. (1) Assume that B 2 satisfies all the hypothesis require in Theorem 4.2. Then, given u W 1,p (Ω) there exist T > an at least one function u C(, T ; L 2 (Ω)) satisfying the equation (9) a.e. in (, T ) an such that u() = u. Note that H = D(ϕ) = D( ϕ); that is, ϕ is ensely efine in H. Sob a supervisão a CPq/ICMC

15 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 181 Let us show that the operator Au = p u B 1 u has increasing resolvent. In fact, we have Au Av, ( I C ) µ (u v) = (Au Av)(u v) x Ω = ( u p 2 u v p 2 v)( u v)x Ω + Ω ( B 1 u ( B 1 v))(u v)x, where Ω = {x Ω; u(x) v(x) < }. From hypothesis (A-2) an Lemma 4.4, [5], we can conclue that Au Av, ( I C ) µ (u v). Then it follows from Theorem 2.2, (iv) (i), that A = p B 1 has increasing resolvent. Next consier the following auxiliary equation t w(t) + ϕ(w(t)) = B+ 2 (w(t)), t >. (11) where for B + 2 we assume that (A-4) B 2 + is the Nemitskiĭ operator associate to a function b+ 2 : R R which is continuous, non-ecreasing, satisfies b 2 (s) b + 2 (s) for all s R an is such that all conitions of Theorem 4.2 are satisfie for the problem 11 As a consequence of the results in the previous sections we obtain the following result Theorem 5.1 (Comparison Theorem). Assume (A 1), (A 2),(A 3) an (A 4) are satisfie an that u, w W 1,p (Ω), u w. If a solution u(t, u ) of (1) is efine for t T then there exists a solution w(t, w ) of (11) efine on [, T ], < T T, such that u(t, u ) w(t, w ), t T. Remark This result is use to compare the solutions of (1) to equations with simpler right han sie B 2 +. In [4] this result is use to obtain global attractors for problems of the form (1) by comparing it to equations for which B 2 + is given by b+ 2 (s) = c 1s + c 2, where c 1 an c 2 are constants. A similar result will also hol for comparison from bellow. In what follows we make assumptions on the real value functions b 1, b 2 an b + 2 ensure the conitions necessary to apply Theorem 5.1: that 1. p > n, or 2. p = n an b 1, b 2 an b + 2 grow polynomially, or Sob a supervisão CPq/ICMC

16 182 A. N. CARVALHO AND C. B. GENTILE 3. p < n an there are constants c 1, c 2 such that b 1, b 2 an b + 2 satisfy the growth conitions b 1 (s) + b + 2 (s) c 1( s r ), r 1 np n p ; b 2 (s) b 2 ( s) c 2 s s ( s r2 1 + s r ), s, s R, r 2 min{ np 2(n p), n(p 1)+p n p }. Remark Uner the above assumptions on b 1 an b 2 it is not har to see that all conitions of Theorem 4.2 are satisfie an therefore the initial value problem for (1) has at least one local solution. Concerning the initial value problem for (11), the above conitions on the growth an monotonicity of b 1 an b + 2 ensure that it has a unique solution that epens continuously on the initial conitions. Besies uniqueness an continuity with respect to initial conitions we also have that the solutions of (11) converge to an equilibrium for (11), that is, a solution of ϕ(η) B 1 (η) B 2 + (η) =. All these conclusions follow from the fact that ϕ B 1 B 2 + is the subifferential of a convex, proper an lower semicontinuous function an from basic results on the theory of ifferential equations with maximal monotone principal part (see, [2, 3, 1]). 6. MORE EXAMPLES OF INCREASING RESOLVENT OPERATORS In this section we give some other examples of operators which have increasing resolvent. Equations having one of these operators as main part may have comparison results similar to the ones evelope in the previous sections. One can verify that these operators have increasing resolvent in the same way we have shown the p-laplacian operator has increasing resolvent. Let Ω R n be a boune smooth omain with bounary Ω. Example 6.1. Consier H = H 1 (Ω) enowe with the inner prouct given by the extension to H H of the bilinear form u, v = ( u, ( D ) 1 v ) L 2 (Ω), u, v L2 (Ω), where D enotes the Laplacian operator with homogeneous Dirichlet bounary conition. For p 2, an V = L p (Ω), we have V H V with continuous an ense inclusions. Let A : V V be the operator efine by the form a(, ), given by (see [8], Ex. 3.2, p.191). a(u, v) = 1 u p 2 uvx. p 1 Ω Sob a supervisão a CPq/ICMC

17 COMPARISON RESULTS FOR NONLINEAR PARABOLIC EQUATIONS 183 Example 6.2. Consier the operator A 1 given by ( n u A 1 u = x i x i 1 p 2 u x i ). The operator A 1 is the subiferential φ of a convex, proper, l.s.c application φ efine on H = L 2 (Ω), 1 n p u p φ(u) = 1 Ω x i x, if u W 1,p (Ω) +, otherwise. Example 6.3. Let A 2 be the operator given by A 2 u = iv(a( u p ) u p 2 u), where a C 1 (R + ) such that a (σ), σ R + an δσ p 1 a(σ p )σ p 1 ασ p 1 + β, with δ, α, β. The operator A 2 is the subiferential φ of a convex, proper, l.s.c application φ efine on H = L 2 (Ω), where A(u) = u 1 n p φ(u) = 1 +, a(σ)σ. A( u p )x, if u W 1,p (Ω) Ω otherwise, The operator A satisfies the conitions of Theorem 3.4, an the operators A 1 an A 2 satisfy the conitions of Theorem 4.3 with perturbations satisfying the same growth conitions consiere in the previous section. REFERENCES 1. Arrieta, J.; Carvalho, A.N. Roriguez-Bernal, A. Attractors of Parabolic Problems with Nonlinear Bounary Conition. Uniform Bouns. Communications in Partial Differential Equations, to appear. 2. Barbu, V., Nonlinear Semigroups an Differential Equations in Banach Spaces, Noorhoff International Publishing, Brèzis, H., Operateurs Maximaux Monotones et Semi-groupes e Contractions ans les Espaces e Hilbert, North-Hollan Publishing Company, Amsteram, Sob a supervisão CPq/ICMC

18 184 A. N. CARVALHO AND C. B. GENTILE 4. Carvalho, A. N. an Gentile C. B., Asymptotic Behavior of Parabolic Equations with Subifferential Principal Part. Preprint 5. DiBenneeto, E., Degenerate Parabolic Equations, Springer-Verlag, New York, Layzhenskaya, O., Attractors for Semigroups an Evolution Equations, Cambrige university Press, Cambrige, Layzhenskaya, O., Ural tseva, N., Linear an Quasilinear Elliptic Equations, Acaemic Press, Lonon, Lions, J.L., Quelques Méthoes e Résolution es Problèmes aux Limites non Linéaires, Duno, Paris, Ôtani, M., Nonmonotone Perturbations for Nonlinear Parabolic Equations Associate with Subifferential Operators, Cauchy Problems, Journal of Differential Equations 46, , Pazy, A., Semi-Groups of Nonlinear Contractions an Their Asymptotic Behaviour, Nonlinear Analysis an mechanics: Heriot-watt Symposium, vol III, Research Notes in Mathematics, Pitman Publishing Limite, Lonon, Temam, R., Infinite - Dimensional Dynamical Systems in Mechanics an Physics, Springer-Verlag, New York, Vrabie, I.I., Compactness Methos for Nonlinear Evolutions, Pitman Monographs an Surveys in Pure an Applie Mathematics, Lonon, Sob a supervisão a CPq/ICMC