Matsumoto, Toshitaka

Nakamura, Gen

Oharu, Shinnosuke

English

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 41, Issue 1 1999 Article 1JANUARY 1999Nonlinear Semigroups Analytic on SectorsGen Nakamura∗ Toshitaka Matsumoto†Shinnosuke Oharu‡∗Matsue National College of Technology†Hiroshima University‡Hiroshima UniversityCopyright c©1999 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 41 (1999), 137{147NONLINEAR SEMIGROUPS ANALYTIC ON SECTORSGEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU0. Introduction.Of concern in this paper is a typical class of nonlinear evolution equa-tions of the form(EE) (d=d»)u(») = Au(»); » 2 § ½ Cin a complex Banach space (X; j¢j). Here the Banach space X stands for anappropriate space of complex-valued functions and A represents a possiblynonlinear di®erential operator in X. Also, § denotes an open sector in thecomplex plain C§ = fseiÁ + teiÃ : s; t > 0g;where ¡¼=2 < Á < 0 < Ã < ¼=2. The closure of § is denoted by §¤,namely,§¤ = fseiÁ + teiÃ : s; t ¸ 0g:We here discuss the generation and characterization of a semigroup of thesolution operators to (EE) which provide solutions analytic in §.In this paper the totality of admissible initial data is denoted byD andis classi¯ed in terms of a lower semicontinuous functional © : X ! [0;+1]in such a way that D ½ D(©) = fv 2 X : ©(v) < +1g and D = S®>0D®,whereD® = fv 2 D : ©(v) · ®g; ® > 0:Evolution equation (EE) is considered under the initial condition(IC) lim»2§; »!0u(») = v 2 Dand the solution u(»; v) to the evolution problem (EE)-(IC) is sought as ananalytic function in §. Now the growth of the analytic solution u(»; v) isrestricted by means of the nonnegative-valued function ©(u(»; v)) on §¤.Also, the solution operators W (»), » 2 §¤, are constructed on D in such1991 Mathematics Subject Classi¯cation. 47H20.Key words and phrases. analytic semigroups, boundary semigroups, Morera'stheorem.137 1Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999138 GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARUa way that to each ® > 0 and each ¿ > 0 there corresponds a constantM(®; ¿) > 0 such that(LL) jS(»)v ¡ S(»)wj ·M(®; ¿)jv ¡ wjfor v; w 2 D® and » 2 §¤ with j»j · ¿ .The closed sector §¤ is an additive semigroup in the sense that 0 isthe apex of §¤ and ´; » 2 §¤ imply ´ + » 2 §¤. Therefore the solutionoperators fW (») : » 2 §¤g forms a nonlinear operator semigroup on Dsuch that u(»; v) =W (»)v is a solution of (EE)-(IC) and analytic in §.Nonlinear semigroups analytic on sectors as mentioned above was¯rst discussed by Hayden and Massey III [3]. Our results are concernedwith solutions analytic on sectors. These results can be extended to thecase of solutions analytic in neighborhoods of the real half line. Early workin this direction was done in Massey III [5], ¹Ouchi [8] and Promislow [9].An extension in this direction and generation of real analytic semigroupswill be treated in the forthcoming paper [6].This paper is organized as follows: Section 1 is devoted to the studyof semigroups analytic in §. In this section a class of analytic semigroupson § is introduced and the characterization of those analytic semigroups isdiscussed in terms of three di®erent types of conditions. Section 2 is con-cerned with the proof of the above characterization theorem. A characteris-tic feature of this argument is to apply Morera's theorem for vector-valuedanalytic functions. Application of Morera's theorem to nonlinear analyticsemigroups was ¯rst made by K. Furuya [1]. It should be noted that theclass D of initial-data is subdivided by means of the lower semicontinuousfunctional ©, and that our analytic semigroup is Lipschitz continuous oneach D®.1. Analytic semigroups on sectors.In this section we introduce the notion of nonlinear semigroup on asubset D of X which provides analytic solutions of the evolution problem(EE)-(IC) and discuss the analyticity in time of the semigroups. Here A is apossibly nonlinear operator in a complex Banach space (X; j ¢ j), D ´ D(A)stands for a given class of initial data in X, and Á, Ã are an arbitrary but¯xed pair of angles satisfying¡¼=2 < Á < 0 < Ã < ¼=2:(1)The union of § and its boundary lines¡Á ´ fseiÁ : s ¸ 0g and ¡Ã ´ fteiÃ : t ¸ 0gis its closure §¤. In what follows, the space of continuous functions from asubset ¢ of C into D is denoted by C(¢;D).2Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/1NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 139In order to introduce a class of operators which are Lipschitz contin-uous in a local sense on D, we employ a lower semicontinuous functional© : X ! [0;+1] such that D ½ fv 2 X : ©(v) < +1g. For each ® > 0we putD® = fv 2 D : ©(v) · ®g(2)and the classD of initial data is subdivided in the sense thatD =S®>0D®.Throughout this paper we assume that for each ® > 0 there exists ¯ > 0such that D® ½ D(A) \D¯ .Our aim in this section is twofold. First, we introduce a class ofoperator semigroups W = fW (»)g on D such that given an initial-valuev 2 D, u(»; v) ´ W (»)v is a unique analytic solution of (EE)-(IC) on §.Secondly, we discuss the characterization of such semigroup on D by meansof the boundary semigroups de¯ned as below.Our class of nonlinear analytic semigroups is formulated as follows:De¯nition. A one-parameter family W ´ fW (») : » 2 §¤g ofnonlinear operators from D into itself is called a semigroup in the classH (D;§¤), if the following conditions hold:(W1) For v 2 D, W (0)v = v, W (¢)v 2 C(§¤; D), and W (¢)v is analyticin §.(W2) For v 2 D and ´; » 2 §¤,W (´ + »)v =W (´)W (»)v:(W3) For ®; ¿ > 0 there exist M®;¿ > 0 and ¯ > 0 such thatjW (»)v ¡W (»)wj ·M®;¿ jv ¡ wj and ©(W (»)v) · ¯for v; w 2 D® and » 2 §¤ with j»j · ¿ .Let W be a semigroup in the class H (D;§¤). The operators on theboundary lines ¡Á and ¡Ã de¯ned byS(s) ´W (seiÁ) and T (t) ´W (teiÃ) for s; t ¸ 0(3)play an important role in this paper. The one-parameter families S ´fS(s) : s ¸ 0g and T ´ fT (t) : t ¸ 0g form semigroups of possiblynonlinear operators on D satisfying the three conditions below:(L1) For each v 2 D, S(0)v = T (0)v = v and S(¢)v; T (¢)v 2 C([0;1);D).(L2) For v 2 D and s; t ¸ 0, S(s + t)v = S(t)S(s)v, and T (s + t)v =T (t)T (s)v.3Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999140 GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU(L3) For ®; ¿ > 0 there exist L®;¿ > 0 and ¯ > 0 such that both S(s) andT (t) map D into itself andjS(s)v ¡ S(s)wj _ jT (t)v ¡ T (t)wj · L®;¿ jv ¡ wj©(S(s)v) _ ©(T (t)v) · ¯for v; w 2 D® and s; t 2 [0; ¿ ], where a _ b = maxfa; bg.De¯nition. Given a semigroup W in the class H (D;§¤), two non-linear semigroups S and T are de¯ned by (3). These semigroups are calledthe boundary semigroups of W.We then de¯ne an operator byA+v = limr2R;r#0r¡1(W (r)v ¡ v);(4)whenever the limit exists in X. Hence the domain D(A+) is the set of allelements v 2 X for which the limits (4) exist. Since W (¢)v is analytic in§ for each v 2 D, the complex derivative exists at each » 2 § and theidentity(d=d»)W (»)v = A+W (»)v(5)holds for » 2 § and v 2 D. This means that D± ´ fW (»)v : » 2 §; v 2Dg ½ D(A+). Now the set D± is dense in D by (W1) and hence A+ isdensely de¯ned in D. In this sense the limit operator A+ may be calledthe in¯nitesimal generator of W. Likewise, the in¯nitesimal generators ofthe boundary semigroups S and T are de¯ned byAÁv = e¡iÁ limr#0r¡1(S(r)v ¡ v);AÃv = e¡iÃ limr#0r¡1(T (r)v ¡ v);(6)respectively. In view of the de¯nitions of S and T and equation (5), wemake the following assumption:(A) D(A) = D(AÁ) = D(AÃ) 6= ; andA = AÁ = AÃ in X;where A is the possibly nonlinear operator in (EE).Condition (A) is essential in the subsequent discussions. In order tocharacterize the structure of analytic semigroups in the class H (D;§¤) interms of their boundary semigroups, we impose the following regularityassumptions on the boundary semigroups:4Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/1NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 141(D) For v 2 D(A), S(¢)v and T (¢)v are strongly absolutely continuous onbounded closed subintervals of [0;1),S(s)v 2 D(A); (d=ds)+S(s)v = eiÁAS(s)v for a.e. s;T (t)v 2 D(A); (d=dt)+T (t)v = eiÃAT (t)v for a.e. t;where (d=ds)+S(s)v and (d=dt)+T (t)v stand for the right-hand side strongderivatives of S(¢)v at s and T (¢)v at t, respectively.Our characterization problem may be reformulated as follows: we ¯rstconsider two evolution problems:(EP;Á)+(d=ds)+u(s) = eiÁAu(s); s > 0;lims#0u(s) = y 2 D(A);(EP;Ã)+(d=dt)+v(t) = eiÃAv(t); t > 0;limt#0v(t) = z 2 D(A):By condition (D) and (6), D-valued functions u(¢) ´ S(¢)y and v(¢) ´ T (¢)zprovide solutions to the problems (EP;Á)+ and (EP;Ã)+, respectively. Us-ing the semigroups S and T and applying conditions (L1) through (L3),(A) and (D), we seek necessary and su±cient conditions under which theproblem (EE)-(IC) admits solutions analytic in § and the solutions dependcontinuously upon initial-data. Although the relation between the operatorA and the in¯nitesimal generator A+ of W can not be explicitly obtainedin general, it is shown in the next section that given a pair of semigroups Sand T on D satisfying (A) and (D), an analytic semigroup W in the classH (D;§¤) can be constructed in such a way that A ½ A+. In this case theanalytic semigroup W is nothing but the family of a solution operators to(EE)-(IC).We here need a lemma concerning weak-star derivatives of stronglyabsolutely continuous functions.Lemma. Let u : [a; b]! X be strongly absolutely continuous. Then:(a) There exists an X¤¤-valued, weakly-star integrable function º(¢) on[a; b] such thathu(t)¡ u(s); fi =Z tshº(r); fi drfor a · s · t · b and f 2 X¤.5Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999142 GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU(b) jº(¢)j is integrable over [a; b] and the total variation of u is given byTV [u] =R ba jº(r)j dr.(c) If in particular, u has a strong right-hand side derivative (d+=dt)u(t)at a.e. t 2 [a; b], then j(d+=dt)u(t)j = jº(t)j for a.e. t 2 [a; b] and TV [u] =R ba j(d+=dt)u(t)j dt.This result is obtained in a more general setting, as shown in Hashimotoand Oharu [4]. The main point here is that for a strongly absolutely con-tinuous function u, the strong right-derivative is integrable over [a; b] andits L1-norm gives the total variation of u. We are now in a position tostate the main theorem of this paper.Theorem. Let S and T satisfy (L1) through (L3), (A) and (D).Then the following three conditions are equivalent:(I) There is an analytic semigroup W such that its boundary semigroupsare S and T.(II) For v 2 D(A), the function T (t)S(s)v is continuously di®erentiablewith respect to s, t > 0, and T (t)S(s)v = S(s)T (t)v for s; t ¸ 0.(III) For v 2 D(A) there exists a null set N ½ [0;1) such that for eachpair of numbers s0; t0 2 (0;1) n N and for each pair of functions g; h :[0; 1]! D° for some ° > 0 satisfyingg(´) = T (t0)v + eiÃ´AT (t0)v + o(´) as ´ # 0;h(¹) = S(s0)v + eiÁ¹AS(s0)v + o(¹) as ¹ # 0;we haveS(s)g(´) = S(s)T (t0)v + eiÃ´AS(s)T (t0)v + o(´) for s > 0;andT (t)h(¹) = T (t)S(s0)v + eiÁ¹AT (t)S(s0)v + o(¹) for t > 0:Remark. (a) For linear analytic semigroups, the correspondingresult is found in a recent book Engel and Nagel [1], Theorem 4.6 on page101.(b) If D is convex, and if S(s) and T (t) are complex Fr¶echet di®erentiableat v 2 D(A) for s; t 2 [0;1), then condition (III) holds. See also Oharu [7].6Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/1NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 1432. Morera's theorem and analyticity.A proof of the main theorem is given in ¯ve steps.Step 1. We ¯rst demonstrate that for ® > 0 the functions (s; v) 7!S(s)v and (t; v) 7! T (t)v are continuous from [0;1)£D® into D. In fact,let ¿ > 0 and v 2 D®. Then, by (L1), S(¢)v is strongly continuous on[0;1). HencejS(s)v ¡ S(s^)v^j · jS(s)v ¡ S(s^)vj+ jS(s^)v ¡ S(s^)v^j· L®;¿ jv ¡ v^j+ jS(s^)v ¡ S(s)vjso far as v; v^ 2 D®; s; s^ 2 [0; ¿ ].Step 2. Implication (III) ) (I): Set W (»)v = T (t)S(s)v for v 2 Dand » = seiÁ + teiÃ 2 §¤. Then W (¢)v is strongly continuous in » 2 §¤.Now it is seen from Step 1 that (W1) and (W3) follow from conditions (L1)and (L3). Let v be an arbitrary element of D(A). Then, by condition (D),there exists a null set N1 such thatS(s+ r)v = S(s)v + eiÁrAS(s)v + o(r)(7)for s 2 (0;1) n N1 and r ¸ 0. For each s 2 (0;1) n N1 there exists a nullset N2(s) such that@+t T (t)S(s)v = eiÃAT (t)S(s)v(8)for t 2 [0;1) nN2(s). By (III) and (7) we haveT (t)S(s+ r)v = T (t)S(s)v + eiÁrAT (t)S(s)v + o(r)for t > 0. Hencee¡iÁ@+s T (t)S(s)v = AT (t)S(s)v(9)for (s; t) 2 ((0;1) n N1) £ (0;1) and AT (t)S(s)v is strongly measurablewith respect to (s; t) 2 (0;1)£ (0;1).Let 0 < a < b and 0 < c < d. We wish to show the identityeiÁ·Z baT (d)S(s)v ds¡Z baT (c)S(s)v ds¸= eiÃ·Z dcT (t)S(b)v dt¡Z dcT (t)S(a)v dt¸:(10)Let a · s · b and c · t · d. It follows from the Lemma that @+s T (t)S(s)vis integrable with respect to (s; t) 2 [a; b]£ [c; d]. We infer from (8) and (9)7Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999144 GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARUthatT (t)S(b)v ¡ T (t)S(a)v =Z ba@+s T (t)S(s)v ds= eiÁZ baAT (t)S(s)v ds;(11)T (d)S(s)v ¡ T (c)S(s)v =Z dc@+t T (t)S(s)v dt= eiÃZ dcAT (t)S(s)v dt:(12)By (L3) and the assertion of Step 1, T (t)S(s)v is strongly continuous over[0;1)£ [0;1). The relations (11) and (12) then imply the identitiesZ dcT (t)S(b)v dt¡Z dcT (t)S(a)v dt = eiÁZ dcZ baAT (t)S(s)v ds dt;Z baT (d)S(s)v ds¡Z baT (c)S(s)v ds = eiÃZ baZ dcAT (t)S(s)v dt ds:Using Fubini's theorem and combining the above relations, we get thedesired identity (10). This impliesZ bahW (seiÁ + deiÃ)v ¡W (seiÁ + ceiÃ)vid(seiÁ)=Z dchW (beiÁ + teiÃ)v ¡W (aeiÁ + teiÃ)vid(teiÃ):Now one can apply Morera's theorem to the continuous function W (¢)von §¤ to conclude that W (¢)v is analytic on the connected domain §.Next, let ® > 0 and v 2 D®. Then, there exist ¯ > 0 and a sequencefvng ½ D(A) \ D¯ such that vn ! v as n ! 1. Hence (W3) impliesthat for any ¿ > 0 W (»)vn converges to W (»)v uniformly for » 2 §¤ withj»j · ¿ . Hence W (¢)v is analytic in § and S and T are its boundarysemigroups. It now remains to show that W ´ fW (») : » 2 §¤g formsa D-valued semigroup on §¤. To this end, we set V (»)v = S(s)T (t)v for» = seiÁ + teiÃ 2 §¤. Then it is seen in the same way as above that V (¢)vis continuous on §¤ and analytic on §. Moreover,W (seiÁ)v = V (seiÁ)v = S(s)v for s ¸ 0 and v 2 D:Put U(»)v = W (»)v ¡ V (»)v for » 2 §¤. Then U(seiÁ)v = 0 for s ¸ 0.By the re°ection principle and f(U(seiÁ)v) = 0 for s ¸ 0 and f 2 X¤, X¤being the dual space of X, we have U(»)v = 0 for » 2 §¤. This means8Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/1NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 145that S(s)T (t)v = T (t)S(s)v for s; t ¸ 0 and v 2 D. Hence W satis¯es(W2).Step 3. Implication (I) ) (II): (II) follows directly from (W2) andthe de¯nition of the boundary semigroups.Step 4. Implication (I) ) (III): By (W2) and the de¯nition ofboundary semigroups, we haveW (»)v = T (t)S(s)v = S(s)T (t)vfor » = seiÁ + teiÃ 2 §¤ and v 2 D. Let v 2 D(A). Then, by (D),there exists a null set N3 such that T (t)v is di®erentiable at any pointt = t0 2 (0;1) n N3. Let t0 2 (0;1) n N3 and let g : [0; 1] ! D° be suchthat g(t) = T (t0)v + eiÃtAT (t0)v + o(t) as t ! 0. Since T (t + t0)v =T (t0)v + eiÃtAT (t0)v + o(t) as t # 0, (L3) implies thatS(s)g(t)¡ S(s)T (t+ t0)v = o(t) as t # 0:(13)Since (I) implies that S(s)T (t + t0)v = T (t + t0)S(s)v and that it is con-tinuously di®erentiable with respect to t, we have@+t S(s)T (t+ t0)vjt=0 = @tT (t+ t0)S(s)vjt=0= eiÃAT (t0)S(s)v= eiÃAS(s)T (t0)v;and soS(s)T (t+ t0)v = S(s)T (t0)v + teiÃAS(s)T (t0)v + o(t) as t # 0.(14)Combining (13) with (14) givesS(s)g(t) = S(s)T (t0)v + teiÃAS(s)T (t0)v + o(t) as t # 0.Thus the property of S in (III) is obtained. The corresponding property ofT is obtained in the same way.Step 5. Implication (II) ) (I): We de¯neW (»)v = T (t)S(s)v for » = seiÁ + teiÃ 2 §¤ and v 2 D:(15)By de¯nition, (II) implies (W2). (W1) follows from (L1) and the assertionof Step 1. (W3) follows from (15) and (L3). Thus W = fW (») : » 2 §¤gforms a D-valued semigroup. Next, let v 2 D(A) and s; t > 0. Then, by9Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999146 GEN NAKAMURA, TOSHITAKA MATSUMOTO AND SHINNOSUKE OHARU(II), we have @tT (t)S(s)v = eiÃAT (t)S(s)v and@sT (t)S(s)v = @sS(s)T (t)v= eiÁAS(s)T®(t)v= eiÁAT (t)S(s)v:Therefore we havee¡iÁ@sT (t)S(s)v = e¡iÃ@tT (t)S(s)v = AT (t)S(s)v:Once this is obtained, one can show in the same way as in Step 2 that W isan analytic semigroup with the boundary semigroups S and T. The proofof the main theorem is now complete.References[1] K.-J. Engel and R. Nagel: One-Parameter Semigroups for Linear Evolution Equa-tions, Graduate Texts in Math. 194, Springer-Verlag, New York, Berlin, 2000.[2] K. Furuya: Analyticity of nonlinear semigroups, Israel J. Math. 68 (1989), 89{101.[3] T. L., Hayden and F. J. Massey III: Nonlinear analytic semigroups, Paci¯c J. Math.57 (1975), 423{439.[4] K. Hashimoto and S. Oharu: Gel'fand integrals and generalized derivatives of vectormeasures, Hiroshima Math. J. 13 (1983), 301{326.[5] F. J., Massey III: Analyticity of solutions of nonlinear evolution equations, J. Di®er-ential Equations 22 (1976), 416{427.[6] G. Nakamura, T. Matsumoto and S. Oharu: Analyticity in time of solutions ofnonlinear evolution systems, preprint.[7] S. Oharu: On the generation of semigroups of nonlinear contractions, J. Math. Sci.Japan 22 (1970), 526{550.[8] S. ¹Ouchi: On the analyticity in time of solutions of initial boundary value problemsfor semilinear parabolic di®erential equations with monotone nonlinearity, J. Fac.Sci. Univ. Tokyo 20 (1974), 19{41.[9] K. Promislow: Time analyticity and Gevrey regularity for solutions of a class ofdissipative partial di®erential equations, Nonlinear Anal. 16 (1991), 959{980.Gen NAKAMURADepartment of MathematicsMatsue National College of TechnologyMatsue 690-8518, Japane-mail address: nakamura@gs.matsue-ct.ac.jpToshitaka MATSUMOTODepartment of Mathematical and Life SciencesGraduate School of Science, Hiroshima UniversityHigashi-Hiroshima 739-8526, Japane-mail address: mats@math.sci.hiroshima-u.ac.jp10Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/1NONLINEAR SEMIGROUPS ANALYTIC ON SECTORS 147Shinnosuke OHARUDepartment of Mathematical and Life SciencesGraduate School of Science, Hiroshima UniversityHigashi-Hiroshima 739-8526, Japane-mail address: oharu@math.sci.hiroshima-u.ac.jp(Received June 28, 2000)11Nakamura et al.: Nonlinear Semigroups Analytic on SectorsProduced by The Berkeley Electronic Press, 1999

Analyticity in time of solutions of nonlinear evolution systems,

Analyticity of nonlinear semigroups,

Gel’fand integrals and generalized derivatives of vector measures,

III: Nonlinear analytic semigroups,

Massey III: Analyticity of solutions of nonlinear evolution equations,

On the analyticity in time of solutions of initial boundary value problems for semilinear parabolic diﬀerential equations with monotone nonlinearity,

On the generation of semigroups of nonlinear contractions,

One-Parameter Semigroups for Linear Evolution Equations,

Time analyticity and Gevrey regularity for solutions of a class of dissipative partial diﬀerential equations, Nonlinear Anal.

Nonlinear Semigroups Analytic on Sectors

Okayama University Digital Information Repository

Nonlinear Semigroups Analytic on Sectors

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository

Okayama University Digital Information Repository