Mathematical Journal of Okayama University

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1 Mthemtil Journl of Okym University Volume 41, Issue Artile 1 JANUARY 1999 Nonliner Semigroups Anlyti on Setors Gen Nkmur Toshitk Mtsumoto Shinnosuke Ohru Mtsue Ntionl College of Tehnology Hiroshim University Hiroshim University Copyright 1999 by the uthors. Mthemtil Journl of Okym University is produed by The Berkeley Eletroni Press (bepress).

2 Nkmur et l.: Nonliner Semigroups Anlyti on Setors Mth. J. Okym Univ. 41 (1999), NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU 0. Introdution. Of onern in this pper is typil lss of nonliner evolution equtions of the form (EE) (d/dξ)u(ξ) = Au(ξ), ξ Σ C in omplex Bnh spe (X, ). Here the Bnh spe X stnds for n pproprite spe of omplex-vlued funtions nd A represents possibly nonliner differentil opertor in X. Also, Σ denotes n open setor in the omplex plin C Σ = {se iφ + te iψ : s, t > 0}, where π/2 < φ < 0 < ψ < π/2. The losure of Σ is denoted by Σ, nmely, Σ = {se iφ + te iψ : s, t 0}. We here disuss the genertion nd hrteriztion of semigroup of the solution opertors to (EE) whih provide solutions nlyti in Σ. In this pper the totlity of dmissible initil dt is denoted by D nd is lssified in terms of lower semiontinuous funtionl Φ : X [0, + ] in suh wy tht D D(Φ) = {v X : Φ(v) < + } nd D = α>0 D α, where D α = {v D : Φ(v) α}, α > 0. Evolution eqution (EE) is onsidered under the initil ondition (IC) lim u(ξ) = v D ξ Σ, ξ 0 nd the solution u(ξ; v) to the evolution problem (EE)-(IC) is sought s n nlyti funtion in Σ. Now the growth of the nlyti solution u(ξ; v) is restrited by mens of the nonnegtive-vlued funtion Φ(u(ξ; v)) on Σ. Also, the solution opertors W (ξ), ξ Σ, re onstruted on D in suh 1991 Mthemtis Subjet Clssifition. 47H20. Key words nd phrses. nlyti semigroups, boundry semigroups, Morer s theorem. Produed by The Berkeley Eletroni Press,

3 Mthemtil Journl of Okym University, Vol. 41 [1999], Iss. 1, Art GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU wy tht to eh α > 0 nd eh τ > 0 there orresponds onstnt M(α, τ) > 0 suh tht (LL) S(ξ)v S(ξ)w M(α, τ) v w for v, w D α nd ξ Σ with ξ τ. The losed setor Σ is n dditive semigroup in the sense tht 0 is the pex of Σ nd η, ξ Σ imply η + ξ Σ. Therefore the solution opertors {W (ξ) : ξ Σ } forms nonliner opertor semigroup on D suh tht u(ξ; v) = W (ξ)v is solution of (EE)-(IC) nd nlyti in Σ. Nonliner semigroups nlyti on setors s mentioned bove ws first disussed by Hyden nd Mssey III [3]. Our results re onerned with solutions nlyti on setors. These results n be extended to the se of solutions nlyti in neighborhoods of the rel hlf line. Erly work in this diretion ws done in Mssey III [5], Ōuhi [8] nd Promislow [9]. An extension in this diretion nd genertion of rel nlyti semigroups will be treted in the forthoming pper [6]. This pper is orgnized s follows: Setion 1 is devoted to the study of semigroups nlyti in Σ. In this setion lss of nlyti semigroups on Σ is introdued nd the hrteriztion of those nlyti semigroups is disussed in terms of three different types of onditions. Setion 2 is onerned with the proof of the bove hrteriztion theorem. A hrteristi feture of this rgument is to pply Morer s theorem for vetor-vlued nlyti funtions. Applition of Morer s theorem to nonliner nlyti semigroups ws first mde by K. Furuy [1]. It should be noted tht the lss D of initil-dt is subdivided by mens of the lower semiontinuous funtionl Φ, nd tht our nlyti semigroup is Lipshitz ontinuous on eh D α. 1. Anlyti semigroups on setors. In this setion we introdue the notion of nonliner semigroup on subset D of X whih provides nlyti solutions of the evolution problem (EE)-(IC) nd disuss the nlytiity in time of the semigroups. Here A is possibly nonliner opertor in omplex Bnh spe (X, ), D D(A) stnds for given lss of initil dt in X, nd φ, ψ re n rbitrry but fixed pir of ngles stisfying (1) π/2 < φ < 0 < ψ < π/2. The union of Σ nd its boundry lines Γ φ {se iφ : s 0} nd Γ ψ {te iψ : t 0} is its losure Σ. In wht follows, the spe of ontinuous funtions from subset of C into D is denoted by C( ; D). 2

4 Nkmur et l.: Nonliner Semigroups Anlyti on Setors NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 139 In order to introdue lss of opertors whih re Lipshitz ontinuous in lol sense on D, we employ lower semiontinuous funtionl Φ : X [0, + ] suh tht D {v X : Φ(v) < + }. For eh α > 0 we put (2) D α = {v D : Φ(v) α} nd the lss D of initil dt is subdivided in the sense tht D = α>0 D α. Throughout this pper we ssume tht for eh α > 0 there exists β > 0 suh tht D α D(A) D β. Our im in this setion is twofold. First, we introdue lss of opertor semigroups W = {W (ξ)} on D suh tht given n initil-vlue v D, u(ξ; v) W (ξ)v is unique nlyti solution of (EE)-(IC) on Σ. Seondly, we disuss the hrteriztion of suh semigroup on D by mens of the boundry semigroups defined s below. Our lss of nonliner nlyti semigroups is formulted s follows: Definition. A one-prmeter fmily W {W (ξ) : ξ Σ } of nonliner opertors from D into itself is lled semigroup in the lss H (D, Σ ), if the following onditions hold: (W1) in Σ. For v D, W (0)v = v, W ( )v C(Σ, D), nd W ( )v is nlyti (W2) For v D nd η, ξ Σ, (W3) W (η + ξ)v = W (η)w (ξ)v. For α, τ > 0 there exist M α,τ > 0 nd β > 0 suh tht W (ξ)v W (ξ)w M α,τ v w nd Φ(W (ξ)v) β for v, w D α nd ξ Σ with ξ τ. Let W be semigroup in the lss H (D, Σ ). The opertors on the boundry lines Γ φ nd Γ ψ defined by (3) S(s) W (se iφ ) nd T (t) W (te iψ ) for s, t 0 ply n importnt role in this pper. The one-prmeter fmilies S {S(s) : s 0} nd T {T (t) : t 0} form semigroups of possibly nonliner opertors on D stisfying the three onditions below: (L1) For eh v D, S(0)v = T (0)v = v nd S( )v, T ( )v C([0, ); D). (L2) For v D nd s, t 0, S(s + t)v = S(t)S(s)v, nd T (s + t)v = T (t)t (s)v. Produed by The Berkeley Eletroni Press,

5 Mthemtil Journl of Okym University, Vol. 41 [1999], Iss. 1, Art GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU (L3) For α, τ > 0 there exist L α,τ > 0 nd β > 0 suh tht both S(s) nd T (t) mp D into itself nd S(s)v S(s)w T (t)v T (t)w L α,τ v w Φ(S(s)v) Φ(T (t)v) β for v, w D α nd s, t [0, τ], where b = mx{, b}. Definition. Given semigroup W in the lss H (D, Σ ), two nonliner semigroups S nd T re defined by (3). These semigroups re lled the boundry semigroups of W. (4) We then define n opertor by A + v = lim r R,r 0 r 1 (W (r)v v), whenever the limit exists in X. Hene the domin D(A + ) is the set of ll elements v X for whih the limits (4) exist. Sine W ( )v is nlyti in Σ for eh v D, the omplex derivtive exists t eh ξ Σ nd the identity (5) (d/dξ)w (ξ)v = A + W (ξ)v holds for ξ Σ nd v D. This mens tht D {W (ξ)v : ξ Σ, v D} D(A + ). Now the set D is dense in D by (W1) nd hene A + is densely defined in D. In this sense the limit opertor A + my be lled the infinitesiml genertor of W. Likewise, the infinitesiml genertors of the boundry semigroups S nd T re defined by (6) A φ v = e iφ lim r 0 r 1 (S(r)v v), A ψ v = e iψ lim r 0 r 1 (T (r)v v), respetively. In view of the definitions of S nd T nd eqution (5), we mke the following ssumption: (A) D(A) = D(A φ ) = D(A ψ ) nd A = A φ = A ψ in X, where A is the possibly nonliner opertor in (EE). Condition (A) is essentil in the subsequent disussions. In order to hrterize the struture of nlyti semigroups in the lss H (D, Σ ) in terms of their boundry semigroups, we impose the following regulrity ssumptions on the boundry semigroups: 4

6 Nkmur et l.: Nonliner Semigroups Anlyti on Setors NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 141 (D) For v D(A), S( )v nd T ( )v re strongly bsolutely ontinuous on bounded losed subintervls of [0, ), S(s)v D(A), (d/ds) + S(s)v = e iφ AS(s)v for.e. s, T (t)v D(A), (d/dt) + T (t)v = e iψ AT (t)v for.e. t, where (d/ds) + S(s)v nd (d/dt) + T (t)v stnd for the right-hnd side strong derivtives of S( )v t s nd T ( )v t t, respetively. Our hrteriztion problem my be reformulted s follows: we first onsider two evolution problems: (EP;φ) + lim u(s) s 0 = y D(A), (d/ds) + u(s) = e iφ Au(s), s > 0, (EP;ψ) + lim v(t) t 0 = z D(A). (d/dt) + v(t) = e iψ Av(t), t > 0, By ondition (D) nd (6), D-vlued funtions u( ) S( )y nd v( ) T ( )z provide solutions to the problems (EP; φ) + nd (EP; ψ) +, respetively. Using the semigroups S nd T nd pplying onditions (L1) through (L3), (A) nd (D), we seek neessry nd suffiient onditions under whih the problem (EE)-(IC) dmits solutions nlyti in Σ nd the solutions depend ontinuously upon initil-dt. Although the reltion between the opertor A nd the infinitesiml genertor A + of W n not be expliitly obtined in generl, it is shown in the next setion tht given pir of semigroups S nd T on D stisfying (A) nd (D), n nlyti semigroup W in the lss H (D, Σ ) n be onstruted in suh wy tht A A +. In this se the nlyti semigroup W is nothing but the fmily of solution opertors to (EE)-(IC). We here need lemm onerning wek-str derivtives of strongly bsolutely ontinuous funtions. Lemm. Let u : [, b] X be strongly bsolutely ontinuous. Then: () There exists n X -vlued, wekly-str integrble funtion ν( ) on [, b] suh tht u(t) u(s), f = for s t b nd f X. t s ν(r), f dr Produed by The Berkeley Eletroni Press,

7 Mthemtil Journl of Okym University, Vol. 41 [1999], Iss. 1, Art GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU (b) ν( ) is integrble over [, b] nd the totl vrition of u is given by T V [u] = ν(r) dr. () If in prtiulr, u hs strong right-hnd side derivtive (d + /dt)u(t) t.e. t [, b], then (d + /dt)u(t) = ν(t) for.e. t [, b] nd T V [u] = (d+ /dt)u(t) dt. This result is obtined in more generl setting, s shown in Hshimoto nd Ohru [4]. The min point here is tht for strongly bsolutely ontinuous funtion u, the strong right-derivtive is integrble over [, b] nd its L 1 -norm gives the totl vrition of u. We re now in position to stte the min theorem of this pper. Theorem. Let S nd T stisfy (L1) through (L3), (A) nd (D). Then the following three onditions re equivlent: (I) There is n nlyti semigroup W suh tht its boundry semigroups re S nd T. (II) For v D(A), the funtion T (t)s(s)v is ontinuously differentible with respet to s, t > 0, nd T (t)s(s)v = S(s)T (t)v for s, t 0. (III) For v D(A) there exists null set N [0, ) suh tht for eh pir of numbers s 0, t 0 (0, ) \ N nd for eh pir of funtions g, h : [0, 1] D γ for some γ > 0 stisfying we hve nd g(η) = T (t 0 )v + e iψ ηat (t 0 )v + o(η) s η 0, h(µ) = S(s 0 )v + e iφ µas(s 0 )v + o(µ) s µ 0, S(s)g(η) = S(s)T (t 0 )v + e iψ ηas(s)t (t 0 )v + o(η) for s > 0, T (t)h(µ) = T (t)s(s 0 )v + e iφ µat (t)s(s 0 )v + o(µ) for t > 0. Remrk. () For liner nlyti semigroups, the orresponding result is found in reent book Engel nd Ngel [1], Theorem 4.6 on pge 101. (b) If D is onvex, nd if S(s) nd T (t) re omplex Fréhet differentible t v D(A) for s, t [0, ), then ondition (III) holds. See lso Ohru [7]. 6

8 Nkmur et l.: Nonliner Semigroups Anlyti on Setors NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS Morer s theorem nd nlytiity. A proof of the min theorem is given in five steps. Step 1. We first demonstrte tht for α > 0 the funtions (s, v) S(s)v nd (t, v) T (t)v re ontinuous from [0, ) D α into D. In ft, let τ > 0 nd v D α. Then, by (L1), S( )v is strongly ontinuous on [0, ). Hene S(s)v S(ŝ)ˆv so fr s v, ˆv D α, s, ŝ [0, τ]. S(s)v S(ŝ)v + S(ŝ)v S(ŝ)ˆv L α,τ v ˆv + S(ŝ)v S(s)v Step 2. Implition (III) (I): Set W (ξ)v = T (t)s(s)v for v D nd ξ = se iφ + te iψ Σ. Then W ( )v is strongly ontinuous in ξ Σ. Now it is seen from Step 1 tht (W1) nd (W3) follow from onditions (L1) nd (L3). Let v be n rbitrry element of D(A). Then, by ondition (D), there exists null set N 1 suh tht (7) S(s + r)v = S(s)v + e iφ ras(s)v + o(r) for s (0, ) \ N 1 nd r 0. For eh s (0, ) \ N 1 there exists null set N 2 (s) suh tht (8) + t T (t)s(s)v = eiψ AT (t)s(s)v for t [0, ) \ N 2 (s). By (III) nd (7) we hve for t > 0. Hene (9) T (t)s(s + r)v = T (t)s(s)v + e iφ rat (t)s(s)v + o(r) e iφ + s T (t)s(s)v = AT (t)s(s)v for (s, t) ((0, ) \ N 1 ) (0, ) nd AT (t)s(s)v is strongly mesurble with respet to (s, t) (0, ) (0, ). (10) Let 0 < < b nd 0 < < d. We wish to show the identity [ ] e iφ T (d)s(s)v ds T ()S(s)v ds [ = e iψ T (t)s(b)v dt ] T (t)s()v dt. Let s b nd t d. It follows from the Lemm tht + s T (t)s(s)v is integrble with respet to (s, t) [, b] [, d]. We infer from (8) nd (9) Produed by The Berkeley Eletroni Press,

9 Mthemtil Journl of Okym University, Vol. 41 [1999], Iss. 1, Art GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU tht (11) T (t)s(b)v T (t)s()v = + s T (t)s(s)v ds = e iφ AT (t)s(s)v ds, (12) T (d)s(s)v T ()S(s)v = t + T (t)s(s)v dt = e iψ AT (t)s(s)v dt. By (L3) nd the ssertion of Step 1, T (t)s(s)v is strongly ontinuous over [0, ) [0, ). The reltions (11) nd (12) then imply the identities T (t)s(b)v dt T (d)s(s)v ds T (t)s()v dt = e iφ T ()S(s)v ds = e iψ AT (t)s(s)v ds dt, AT (t)s(s)v dt ds. Using Fubini s theorem nd ombining the bove reltions, we get the desired identity (10). This implies [ ] W (se iφ + de iψ )v W (se iφ + e iψ )v d(se iφ ) = [ ] W (be iφ + te iψ )v W (e iφ + te iψ )v d(te iψ ). Now one n pply Morer s theorem to the ontinuous funtion W ( )v on Σ to onlude tht W ( )v is nlyti on the onneted domin Σ. Next, let α > 0 nd v D α. Then, there exist β > 0 nd sequene {v n } D(A) D β suh tht v n v s n. Hene (W3) implies tht for ny τ > 0 W (ξ)v n onverges to W (ξ)v uniformly for ξ Σ with ξ τ. Hene W ( )v is nlyti in Σ nd S nd T re its boundry semigroups. It now remins to show tht W {W (ξ) : ξ Σ } forms D-vlued semigroup on Σ. To this end, we set V (ξ)v = S(s)T (t)v for ξ = se iφ + te iψ Σ. Then it is seen in the sme wy s bove tht V ( )v is ontinuous on Σ nd nlyti on Σ. Moreover, W (se iφ )v = V (se iφ )v = S(s)v for s 0 nd v D. Put U(ξ)v = W (ξ)v V (ξ)v for ξ Σ. Then U(se iφ )v = 0 for s 0. By the refletion priniple nd f(u(se iφ )v) = 0 for s 0 nd f X, X being the dul spe of X, we hve U(ξ)v = 0 for ξ Σ. This mens 8

10 Nkmur et l.: Nonliner Semigroups Anlyti on Setors NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 145 tht S(s)T (t)v = T (t)s(s)v for s, t 0 nd v D. Hene W stisfies (W2). Step 3. Implition (I) (II): (II) follows diretly from (W2) nd the definition of the boundry semigroups. Step 4. Implition (I) (III): By (W2) nd the definition of boundry semigroups, we hve W (ξ)v = T (t)s(s)v = S(s)T (t)v for ξ = se iφ + te iψ Σ nd v D. Let v D(A). Then, by (D), there exists null set N 3 suh tht T (t)v is differentible t ny point t = t 0 (0, ) \ N 3. Let t 0 (0, ) \ N 3 nd let g : [0, 1] D γ be suh tht g(t) = T (t 0 )v + e iψ tat (t 0 )v + o(t) s t 0. Sine T (t + t 0 )v = T (t 0 )v + e iψ tat (t 0 )v + o(t) s t 0, (L3) implies tht (13) S(s)g(t) S(s)T (t + t 0 )v = o(t) s t 0. Sine (I) implies tht S(s)T (t + t 0 )v = T (t + t 0 )S(s)v nd tht it is ontinuously differentible with respet to t, we hve nd so + t S(s)T (t + t 0)v t=0 = t T (t + t 0 )S(s)v t=0 = e iψ AT (t 0 )S(s)v = e iψ AS(s)T (t 0 )v, (14) S(s)T (t + t 0 )v = S(s)T (t 0 )v + te iψ AS(s)T (t 0 )v + o(t) s t 0. Combining (13) with (14) gives S(s)g(t) = S(s)T (t 0 )v + te iψ AS(s)T (t 0 )v + o(t) s t 0. Thus the property of S in (III) is obtined. The orresponding property of T is obtined in the sme wy. Step 5. Implition (II) (I): We define (15) W (ξ)v = T (t)s(s)v for ξ = se iφ + te iψ Σ nd v D. By definition, (II) implies (W2). (W1) follows from (L1) nd the ssertion of Step 1. (W3) follows from (15) nd (L3). Thus W = {W (ξ) : ξ Σ } forms D-vlued semigroup. Next, let v D(A) nd s, t > 0. Then, by Produed by The Berkeley Eletroni Press,

11 Mthemtil Journl of Okym University, Vol. 41 [1999], Iss. 1, Art GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU (II), we hve t T (t)s(s)v = e iψ AT (t)s(s)v nd Therefore we hve s T (t)s(s)v = s S(s)T (t)v = e iφ AS(s)T α (t)v = e iφ AT (t)s(s)v. e iφ s T (t)s(s)v = e iψ t T (t)s(s)v = AT (t)s(s)v. One this is obtined, one n show in the sme wy s in Step 2 tht W is n nlyti semigroup with the boundry semigroups S nd T. The proof of the min theorem is now omplete. Referenes [1] K.-J. Engel nd R. Ngel: One-Prmeter Semigroups for Liner Evolution Equtions, Grdute Texts in Mth. 194, Springer-Verlg, New York, Berlin, [2] K. Furuy: Anlytiity of nonliner semigroups, Isrel J. Mth. 68 (1989), [3] T. L., Hyden nd F. J. Mssey III: Nonliner nlyti semigroups, Pifi J. Mth. 57 (1975), [4] K. Hshimoto nd S. Ohru: Gel fnd integrls nd generlized derivtives of vetor mesures, Hiroshim Mth. J. 13 (1983), [5] F. J., Mssey III: Anlytiity of solutions of nonliner evolution equtions, J. Differentil Equtions 22 (1976), [6] G. Nkmur, T. Mtsumoto nd S. Ohru: Anlytiity in time of solutions of nonliner evolution systems, preprint. [7] S. Ohru: On the genertion of semigroups of nonliner ontrtions, J. Mth. Si. Jpn 22 (1970), [8] S. Ōuhi: On the nlytiity in time of solutions of initil boundry vlue problems for semiliner prboli differentil equtions with monotone nonlinerity, J. F. Si. Univ. Tokyo 20 (1974), [9] K. Promislow: Time nlytiity nd Gevrey regulrity for solutions of lss of dissiptive prtil differentil equtions, Nonliner Anl. 16 (1991), Gen NAKAMURA Deprtment of Mthemtis Mtsue Ntionl College of Tehnology Mtsue , Jpn e-mil ddress: nkmur@gs.mtsue-t..jp Toshitk MATSUMOTO Deprtment of Mthemtil nd Life Sienes Grdute Shool of Siene, Hiroshim University Higshi-Hiroshim , Jpn e-mil ddress: mts@mth.si.hiroshim-u..jp 10

12 Nkmur et l.: Nonliner Semigroups Anlyti on Setors NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 147 Shinnosuke OHARU Deprtment of Mthemtil nd Life Sienes Grdute Shool of Siene, Hiroshim University Higshi-Hiroshim , Jpn e-mil ddress: (Reeived June 28, 2000) Produed by The Berkeley Eletroni Press,