Parabolic Systems Involving Sectorial Operators: Existence and Uniqueness of Global Solutions
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1 Parabol Syses Involvng Seoral Operaors: Exsene and Unqueness of Global Soluons Parabol Syses Involvng Seoral Operaors: Exsene andunqueness of Global Soluons Mguel Yangar ; Dego Salazar Esuela Poléna Naonal, Deparaeno de Maeáa, Ladrón de Guevara E11-253, Quo, Euador Absra: The a of hs paper s o sudy he exsene and unqueness of global soluons n e o syses ofequaons, whenhe dffuson ers are gven by seoral generaors. Keywor: Reaon dffuson syses, seoral operaors, nfnesal generaors. Sseas Parabólos que Involuran Operadores Seorales: Exsena y Undad de Soluones Globales Resuen: El objevo de ese aríulo es esudar la exsena y undad de soluones globales en epo parasseas deeuaones, uando los érnos de dfusón esán dados por operadores seorales. Palabras lave: Sseas de reaón dfusón, operadores seorales, operadores nfnesales. In hs paper, we sudy he global exsene and unqueness of seoral soluons o he syse { u A u f (u, (, u (, x u (x, where A are seoral operaors n he Banah spae (X,, u X and f :, X X a gven valued funon for all 1, : {1,..., }, where X s he Banah produ spae doed wh he nor u 1 u. In he ase 1, sne work (Byszewsk and Lakshkanha, 199, (Byszewsk, 1991, (Byszewsk, 1993, here has been nreasng neres n sudyng absra probles n Banah Spaes (f., e.g., (Azov and Mkbben, 2 and referenes heren. For aeral naely relaed o he presen paper, we refer o (Henry, 1981, where s suded he exsene of seoral soluons for he sngle equaons. Also n (Jakson, 1993, (Lang e al, 22, nonloal auonoous parabol guel.yangar@epn.edu.e Reeved:12/6/215 Aeped: 23/5/216 Publshed: 3/9/ INTRODUCTION probles are nvesgaed, wh f beng Lpshz onnuous. The exsene of ld and lassal soluons for reaon dffuson equaons nvolvng a parular lass of seoral operaors (fraonal Laplaans are suded n (Cabré and Roquejoffre, 213. See also he resul n (Azov and Mkbben, 2, n whh {A(} T s a faly of -areve operaors n X generang a opa evoluon faly, and he exsene of negral soluons o he assoaed nonloal proble s shown. In he ase > 1, we refer o (Yangar, 215 n whh s suded he exsene and unqueness of ld soluons of a reaon dffuson syse wh nfnesal generaors. In order o prove he noaon, we onsder he syse { u Au f (, u, (, u( u, where u (u 1, f ( f 1 and A dag(a 1,..., A. Moreover, we onsder he nor u α 1 u α, on he spae X α X α, where he spae Xα s defned n he nex seon. Throughou hs paper, we assue Ω soe open se n R X α and f : Ω X s loally Hölder onnuous n and loally Lpshz onnuous n u on Ω for all 1,. More (1 Revsa Poléna - Sepebre 216, Vol. 38, No. 1
2 Mguel Yangar; Dego Salazar presely, f ( 1, x 1 Ω, here exss a neghborhood V Ω suh ha for (, x V, (s, y V f (, x f (s, y L ( s θ x y α for soe onsans L > and whou loss of generaly we assue θ : θ > for all 1,. Moreover, a soluon of he nal value syse (1 on (, 1 s a onnuous funon u :, 1 X suh ha u( u and on (, 1 we have (, u( Ω, u( D(A, u ( exss, f (, u( s loally Hï œlder onnuous and ρ f (, u( d < for soe ρ > and he dfferenal equaon (1 s verfed. 2. SECTORIAL OPERATORS Takng 1, fxed hroughou hs seon, we all a lnear operaor A n a Banah spae X, a seoral operaor f s a losed densely defned operaor suh ha, for soe φ (, π/2, M 1 and a real ε, he seor S ε,φ {λ φ arg(λ ε π, λ ε} s n he resolven se of A and (λi A 1 M/ λ ε for all λ Sε,φ. Le us noe ha every bounded lnear operaor on a Banah spae s seoral. Also, f we defne e A 1 (λi A 2π 1 e λ dλ Γ where Γ s a onour n he resolven of A wh arg(λ ±θ as λ for soe θ (π/2, π, we have ha A s he nfnesal generaor of he analy segroup (e A (, oreover, f Reσ(A > b, hen for > e A e b, A e A e b I s poran o noe ha, f B s a bounded lnear operaor, hen e B as defned above exen o a group of lnear operaors and verfes e B e Bs e B(s, f or <, s <. In order o defne he fraonal power of a seoral operaor A, we assue Reσ(A >, so, for any α (, 1 A α 1 α 1 e A d. Γ(α (2. (3 Takng A α defned as above, we have ha hs operaor s a bounded lnear operaor on X whh s one-one and sasfes A α A β A (αβ. Furherore, A α represens he nverse operaor of A α wh D(A α R(A α and A s he deny on X. An poran resul onernng posve powers of seoral operaors s A α e A α α e b f or > (4 wh Reσ(A > b > and f u D(A α (e A Iu 1 α 1 α α A α u (5 also, A α Aβ A β Aα A αβ on D(A γ wh γ ax(α, β, α β. Now, we onsder he fraonal powers of B : A a I wh a R hosen so Reσ(B >, where σ(b s he speru of B. We defne he Banah spae X α D(B α wh he nor u α, B α u, where D(B α s he doan of he operaor B α. Fnally, akng α β, hen Xα s a dense subspae of X β wh onnuous nluson, also, X X. For ore nforaon abou seoral operaors we refer he reader o (Henry, In order o sae our frs resul, sne A s he nfnesal generaor of he analy segroup (e A ( for eah 1,, we defne he weak forulaon for he syse (1 gven by wh u( P( u P( dag(e A 1(,..., e A (. In wha follows, he onsan C > represens dfferen onsans. Lea 3.1 If u s he soluon of he syse (1 on (, 1, hen equaon (6 s sasfed. Inversely, f u s a onnuous funon of (, 1 no X α, ρ 3. MAIN RESULTS P( s f (s, u(s (6 f (s, u(s < for soe ρ > and equaon (6 s sasfed for < < 1, hen u s a soluon of he syse (1 on (, 1. Revsa Poléna - Sepebre 216, Vol. 38, No. 1
3 Parabol Syses Involvng Seoral Operaors: Exsene and Unqueness of Global Soluons Proof. Le assue ha u s he soluon of he syse (1 on (, 1, akng 1, fxed, we defne he auxlary funon g (, v f (, u 1,..., u 1, v, u 1,..., u n. Le see ha g (, u ( s loally Hï œlder onnuous n and ρ g (, u ( d <. Indeed, sne g : (, 1 X α X and g (, u ( g (s, u ( f (, u( f (s, u(s f (, u( f (s, u(s L s ν sne f (, u( s Hï œlder onnuous wh exponen ν (, 1. Furherore, ρ g (, u ( X d ρ ρ < f (, u( X d f (, u( d for soe ρ >. Now, sne u verfes he syse (1, we have ha { u A u g (, u, (7 u ( u. Therefore by he heore n (Henry, 1981, we have ha u s he unque soluon of he syse (7, whh an be wren as u ( e A ( u e A ( s g (s, u (s. Repeang he sae proedure for all 1,, we have u( P( u P( s f (s, u(s naely u sasfy he equaon (6. Reproally, and u C ( we suppose (, 1 ; X α now ha u sasfy he equaon (6. Besdes, for eah 1,, we have ha u : (, 1 X α s onnuous and verfes u ( e A ( u e A( s g (s, u (s. Frs, we wll prove ha u s loally Hï œlder onnuous fro (, 1 o X α. Thus, f, h, 1 (, 1 wh h > and δ (, 1 α, we la ha u ( h u ( α, C h δ (8 for soe posve onsan C. Indeed, u ( h u ( (e A h Ie A ( u (e Ah Ie A( s g (s, u (s h e A(h s g (s, u (s. (9 Now, for any z X, by Theore n (Henry, 1981, (e Ah Ie A( s z C( s (αδ h δ e b( s z. α, Moreover, due o eah f s loally Hï œlder n and loally Lpshz n u, we have ha f (, u( f (, u( L ( θ u( u( α or equvalenly g (, u ( g (, u ( L ( θ u( α u( α. (1 Bu for hypohess, we know ha u : (, 1 X α s onnuous, hen, we have C α u( α ax u α <. 1 Hene, by he nequaly (1 g(, u ( L( θ 2C α g (, u L( θ. We begn boundng he frs er of he equaon (9,hus (e A h Ie A ( u α, C( (αδ h δ e b ( 1 u Ch δ, sne, 1 (, 1. Now, le us bound he seond er of he equaon (9 (e Ah Ie A( s g (s, u (s α, C( s (αδ h δ e b ( s g (s, u (s Ch δ Ch δ ( s (αδ e b ( s (L θ ( s (αδ Revsa Poléna - Sepebre 216, Vol. 38, No. 1
4 Mguel Yangar; Dego Salazar Ch δ. Boundng now he hrd er of he equaon (9, akng Re σ(b > γ > and Re(σ( a I (a γ, usng nequales (3 and (4, we have h e A(h s g (s, u (s h h h h h h h 1 α Ch δ α, B α e B (h s e a I(h s g (s, u (s C α ( h s α e a (h s g (s, u (s C α ( h s α e a (h s (L s θ ( h s α e a (h s ( h s α e γ e γ 1 e γh ( h s α for soe large enough posve onsan C. Hene, nequaly (8 s sasfed. Moreover, g(, u ( s loally Hï œlder onnuous on (, 1. Indeed, g (, u ( g (s, u (s Also, ρ f (, u ( f (s, u (s L ( s θ u( u(s α L ( s θ L ( s θ C( s θ g (, u ( ρ u ( u (s α, C s δ s δ. f (, u( d <. Then, by Theore n (Henry, 1981, u solves he equaon { u A u g (, u ( f (, u( for all 1,. u ( u Now, we are n poson o sae our an resul n whh we esablsh he exsene and unqueness of soluons o he syse (1. Theore 3.1 If f verfes he hypohess (2 for eah 1,, hen for any (, u Ω here exss T T(, u > suh ha he syse (1 has an unque soluon u on (, T wh nal ondon u( u. Proof. By he prevous lea s enough o fnd a soluon u of he equaon (6. We hoose δ >, > suh ha he se V {(, x, x u α δ} s onaned n Ω and f (, x f (, y L x y α for any (, x, (, y V. Moreover, we la ha for all 1,, here exss a onsan M > suh ha B α e A M α e a for all >. Indeed, akng Re σ(b > γ > and Re(σ( a I (a γ, usng nequales (3 and (4 B α e A B α e A e a I e a I B α e B e a I B α e B e a I Furherore, we se and κ C α α e γ e ( a I C α α e γ Ce (a γ M α e a. ax f (, u, L ax L, 1, a ax 1, a, M ax 1, M. Hene, we an hoose T (, suh ha (e A h Iu α, for all 1, and δ 2, f or h T T M(κ Lδ u α e au du δ 2. (11 (12 Revsa Poléna - Sepebre 216, Vol. 38, No. 1
5 Parabol Syses Involvng Seoral Operaors: Exsene and Unqueness of Global Soluons If S denoe he se of onnuous funons y :, T X α suh ha y( u α δ on T, provded by he nor y T sup{ y ( α,, T} 1 hen S s a oplee er spae sne s he produ of oplee er spaes wh he produ nor. So, for y S, we defne H(y :, T X gven by H(y( P( u P( s f (s, y(s. We la ha H : S S s a onraon. Indeed, f Y S and T, we have ha H(y( u α (P( Iu α P( s f (s, y(s α (e A ( α, Iu 1 1 e A ( s f (s, y(s α, α 2 1 (A α 1 e A ( s f (s, y(s δ 2 M ( s α e a( s (L y(s u α κ δ 2 M(κ Lδ u α e au du δ. We prove now ha H(y :, T X α s onnuous. Indeed, whou loss of generaly, we suppose ha z <, hus H(y( H(y(z α H (y( H (y(z α, (e A ( e A (z u α, z (e A( s e A(z s f (s, y(s e A( s f (s, y(s α, z (I 1 I 2 I 3. α, Le ɛ >, we need o fnd δ > suh ha z < δ ples H(y( H(y(z α < ɛ. Thus, we bound eah er of he las nequaly I 1 (e A ( e A (z α, u B α e A (z (e A( z u u B α e A (z e A ( z u u B α e b (z e A ( z u u C e A ( z u u < ɛ 3 he las nequaly s sasfed f z < δ 1 for soe δ 1 >. Now, boundng I 2 I 2 z (e A( s e A(z s f (s, y(s α, z B α (e A( s e A(z s f (s, y(s z B α (e A ( z Ie A(z s f (s, y(s z B α A C α e A(z s f (s, y(s ( z C( z ɛ 3 z C α (z s α e b (z s he las nequaly s sasfed f z < δ 2 for soe δ 2 >. Now, proeedng wh I 3, I 3 z e A( s f (s, y(s α, B α e A ( s f (s, y(s z C e b( s ɛ z 3 wh z < δ 3 for soe δ 3 >. Therefore, akng δ ín{δ 1, δ 2, δ 3 } we onlude ha H(y C (, T; X α and hen H(y S. We now prove ha H s a onraon. Le y, z S and H (y( H (z( α, B α e A ( s f (s, y(s f (s, z(s ML M ( s α e a ( s L y(s z(s α ( s α e a ( s y z T Revsa Poléna - Sepebre 216, Vol. 38, No. 1
6 Mguel Yangar; Dego Salazar T ML u α e au y z T herefore, wh I, T sup H (y( H (z( α, ML u α e au y z T I for all 1,. Hene T H(y( H(z( T ML u α e au y z T 1 2 y z T. For he fxed pon heore, H has an unque soluon u S, where u s a onnuous funon u :, T (x α whh verfes he equaon (6. Also, f sasfes hus, f (, u f (, u L ( θ u u α f (, u L ( θ δ f (, u and hen, for any fxed ρ > ρ f (s, u(s L(T θ δ B C ρ f (s, u(s 1 C 2 ρ <. Therefore, for he Lea 3.1, u s soluon of he syse (1 on (, T. As a onsequene of he prevous heore, we presen a resul abou he behavor of he soluon. Theore 3.2 If f verfes he hypohess (2 for eah 1, and for all losed and bounded subse V Ω he age of f (V s bounded n X. Then u s a soluon of he syse (1 on (, 1 and 1 s axal,.e., here s no soluon of he syse (1 on (, 2 f 2 > 1, hen eher 1 or else here exss a sequene n 1 as n suh ha ( n, u( n Ω. If Ω s unbounded, he pon a nfny s nluded n Ω. Proof. Proeedng by onradon, we suppose ha 1 <, bu (, u( s no n a neghborhood N of Ω for < 1, we an ake N of he for N Ω\B, where B s a losed and bounded se of Ω and (, u( B for all < 1. We wll prove ha here exss x 1 X α suh ha ( 1, x 1 B wh u( x 1 n X α when 1. Even ore u( 1 x 1, whh eans ha he soluon ould be exended unl 1. Indeed, le C sup{ f (, u, (, u B}, so C < beause B s bounded and losed and by hypohess f (B s bounded n X. Frsly, we an see ha f α β < 1 and < 1, hen u( β C. u ( x β e A ( u β, e A ( s f (s, u(s β, B α Bβ α e A ( u 1 B β e A ( s f (s, u(s B α B β α e A ( u α, B β e A ( s f (s, u(s M ( (β α e a ( u α, M ( s β e a( s f (s, u(s ( s β ( (β α Thus, u( β reans bounded when 1. Now, we suppose ha < < 1, so u( u( (P( Iu( P( s f (s, u(s, hen u( u( α, (e A( Iu ( 1 1 β, e A( s f (s, u(s α, B α B β α (e A( Iu ( B α e A( s f (s, u(s. Boundng he frs er, le ɛ 1 > an arbrary nuber. Sne D(A β α s dense n X, we ake v D(A β α suh ha u ( v < η, hus Revsa Poléna - Sepebre 216, Vol. 38, No. 1
7 Parabol Syses Involvng Seoral Operaors: Exsene and Unqueness of Global Soluons (e A( Iu ( B β α (e A( Iu ( (e A( I(u ( v (e A( Iv e A ( (u ( v u ( v (e A( Iv B β α e b( u ( v u ( v (e A( Iv e b 1 η η ( β α β α A v ɛ 1 C( β α. When he las bound s gven when η s sall enough suh ha (e a ( η η < ɛ 1. Now, boundng he seond er B α e A( s f (s, u(s B α e A ( s f (s, u(s M ( s α e a( s C( s 1 α. Thus, we have u( u( α C ɛ 1 ( β α ( 1 α. Now, we onsder n 1 and le defne u n u( n, hus, akng ɛ >, we have u n u α C ɛ 1 ( n β α ( n 1 α < ɛ. f n, are large enough. Therefore, (u n s a Cauhy sequene n he oplee spae X α, hus, here exss x 1 X α suh ha u n x 1, eans ( n, u( n ( 1, x 1 and sne ( n, u( n B wh B losed, we an onlude ha ( 1, x 1 B. Also, sne u : (, 1 X s onnuous and n 1, we have ha ( n, u( n ( 1, u( 1 B, hus, we an onlude ha u( 1 x 1. To fnsh he proof, usng he Theore 3.1 and onsderng ( 1, x 1 B Ω, we an fnd an unque soluon v on ( 1, 1 T( 1 for soe T( 1 > of he syse (1 wh nal ondon v(1 x 1. Hene, akng { u( f, z( 1 v( f 1, 1 T( 1 we noe ha z s onnuous n, 1 T( 1. So, we onlude ha z s a soluon of he syse (1 wh z( x on (, 1 T( 1 whh onrad he axaly of 1. Fnally, we sae ha under soe exra ondons he unque soluon s global n e. Theore 3.3 Le us suppose ha Ω (, X α and f (, x sasfes hypohess (2 for eah 1,. Furher- ore, here exss k( a onnuous funon on (, ha verfes f (, u k((1 u α for all (, u Ω. If >, u X α, he unque soluon of he syse (1 wh u( u exss for all >. Proof. Frsly, we an noe ha hypohess of he Theore 3.2 are sasfed. Proeedng by onradon, we ake > and assue ha here exss an unque soluon of he syse (1 defned n (, 1 where 1 s axal, so, for he las resul exss a sequene n 1 suh ha u( n α. However, sne β < α ples X α < X β for all 1,, akng (, 1, by a slar proedure o he prevous heore and sne K( s onnuous on (,,.e., bounded on, 1, we have u( α C ( (α β ( s α f (s, u(s ( (α β ( s α k(s(1 u(s α ( (α β ( s α (1 u(s α ( s α u(s α for he Bellan-Gronwall heore, we an onlude ha ( s u( α Ce α Revsa Poléna - Sepebre 216, Vol. 38, No. 1
8 Mguel Yangar; Dego Salazar C (, 1. Whh s a onradon wh he fa ha u( n α when n CONCLUSIONS Slarly o he proble wh a sngle equaon, usng he properes and esaons of seoral operaors, we sae a general resul onernng he exsene and unqueness of soluons o syses of equaons, when he dffuson ers are gven by seoral generaors, also, assung addonal hypohess on he forng er, a resul of global exsene n e s presened. The opuaons saed n he paper are based n he applaon of he Banah Fxed Pon Theore. REFERENCES Azov S., and Mkbben M. (2. Exsene resuls for a lass of absra nonloal Cauhy probles. NonlnearAnalyss. 39, Benhohra, M and Nouyas, S. (21. Nonloal Cauhy probles for neural funonal dfferenal and negrodf- ferenal nlusons n Banah spaes. J. Mah. Anal. Appl. 258, Byszewsk L. and Lakshkanha V. (199. Theore abou he exsene and unqueness of a soluons of a non- loal Cauhuy proble n a Banah spae. Appl. Anal. 4, Byszewsk, L. (1991. Theores abou he exsene and unqueness of soluons of a selnear evoluon nonloal Cauhy proble. J. Mah. Anal. Appl. 162, Byszewsk, L. (1993. Exsene and unqueness of soluons of selnear evoluon nonloal Cauhy proble. Zesz. Nauk. Pol. Rzes. Ma. Fz. 18, Byszewsk, L. and Aka, H. (1998. Exsene of soluons of a selnear funonal-dfferenal evoluon non-loal proble. Nonlnear Analyss. 34, Fu, X. and Ezznb, K. (23. Exsene of soluons for neural funonal dfferenal evoluon equaons wh non- loal ondons. Nonlnear Analyss. 54, Henry, D. (1981. Geoer heory of selnear parabol equaons. Berln, Gerany: Sprnger-Verlag. Jakson, D. (1993. Exsene and unqueness of soluons o selnear nonloal parabol equaons.j.mah. Anal. Appl. 172, Lang, J., Van Caseren, J., and Xao, T. (22. Nonloal Cauhy probles for selnear evoluon equaons. Nonl- near Anal. Ser. A: Theory Meho. 5, Ln,Y. and Lu, J. (1996. Selnear negrodfferenal equaons wh nonloal Cauhy Probles. NonlnearAnalyss. 26, Nouyas, S. and Tsaoas, P. (1997. Global exsene for selnear evoluon equaons wh nonloal ondons. J. Mah. Anal. Appl. 21, Nouyas, S and Tsaoas, P. (1997. Global exsene for selnear negrodfferenal equaons wh delay and non- loal ondons. J. Mah. Anal. Appl. 64, Yangar, M. (215. Exsene and Unqueness of Global Mld Soluons for Nonloal Cauhy Syses n Banah Spaes. Revsa Poléna. 35(2, Mguel Angel Sosa Yangar. Maheaan graduaed fro Esuela Poléna Naonal, Euador. Door of Engneerng Senes, enon Maheaal Modelng a he Unversdad de Chle, Chle. Door n Appled Maheas a he Unversy of Toulouse, Frane. Currenly s Assoae Professor, Level 3, Grade 5 a Esuela Poléna Naonal. Dego Salazar Israel Orellana. Maheaan graduaed fro Esuela Poléna Naonal, Euador and graduae M.S. Pure Maheas, UCE-EPN-USFQ. He s urrenly Oasonal Professor II a Esuela Poléna Naonal. Cabré, X. and Roquejoffre, J. (213. The nfluene of fraonal dffuson n Fsher-KPP equaon. Coun. Mah. Phys. 32, Revsa Poléna - Sepebre 216, Vol. 38, No. 1
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