NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS

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1 25-Ouja International Conerence on Nonlinear Analysis. Electronic Journal o Dierential Equations, Conerence 14, 26, pp ISSN: URL: or tp eje.math.tstate.eu (login: tp) NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS MOHAMMED FILALI, BELHADJ KARIM Abstract. In this paper we prove eistence an uniqueness o classical solutions or the non-autonomous inhomogeneous Cauchy problem u(t) = A(t)u(t) + (t), s t T, t L(t)u(t) = Φ(t)u(t) + g(t), s t T, The solution to this problem is obtaine by a variation o constants ormula. 1. Introuction Consier the bounary Cauchy problem u(t) = A(t)u(t), t s t T, L(t)u(t) = Φ(t)u(t), s t T, (1.1) In the autonomous case (A(t) = A, L(t) = L), the Cauchy problem (1.1) was stuie by Greiner [3]. The author use the perturbation o omains o ininitesimal generators to stuy the homogeneous bounary Cauchy problem. He has also showe the eistence o classical solution o (1.1) via a variation o constants ormula. In the non-autonomous case, Kellerman [5] an Lan [6] showe the eistence o an evolution amily (U(t, s)) s t T which provies classical solutions o homogeneous bounary Cauchy problems. Filali an Moussi [2] showe the eistence an uniqueness o classical solutions to the problem u(t) = A(t)u(t), s t T, t L(t)u(t) = Φ(t)u(t) + g(t), s t T, (1.2) 2 Mathematics Subject Classiication. 34G1, 47D6. Key wors an phrases. Bounary Cauchy problem; evolution amilies; solution; well poseness; variation o constants ormula. c 26 Teas State University - San Marcos. Publishe??,

2 192 M. FILALI, B. KARIM EJDE/CONF/14 In this paper, we prove eistence an uniqueness o classical solutions to the problem u(t) = A(t)u(t) + (t), t s t T, L(t)u(t) = Φ(t)u(t) + g(t), s t T, (1.3) Our technique consists on transorming (1.3) into an orinary Cauchy problem an giving an equivalence between the two problems. The solution is eplicitly given by a variation o constants ormula. 2. Evolution Family Deinition 2.1. A amily o boune linear operators (U(t, s)) s t T on X is an evolution amily i (a) U(t, r)u(r, s) = U(t, s) an U(t, t) = I or all s r t T ; an (b) the mapping (t, s) U(t, s) is continuous on, or all X with = {(t, s) R 2 + : s t T }. Deinition 2.2. A amily o linear (unboune) operators (A(t)) t T on a Banach space X is a stable amily i there are constants M 1, ω R such that ]ω, + [ ρ(a(t)) or all t T an 1 R(λ, A(t i )) M (λ ω) m or λ > ω an any inite sequence t 1 t 2 t m T. Let D, X an Y be Banach spaces, D ensely an continuously embee in X. Consier amilies o operators A(t) L(D, X), L(t) L(D, Y ), Φ(t) L(X, Y ) or t T. In this section, we use the operator matrices metho to prove the eistence o classical solutions or the non-autonomous inhomogeneous bounary Cauchy problem (1.3). We use the ollowing theorem ue to Tanaka [9]. Theorem 2.3. Let (A(t)) t T be a stable amily o linear operators on a Banach space X such that (a) the omain D = (D(A(t),. D ) is a Banach space inepenent o t, (b) the mapping t A(t) is continuously ierentiable in X or every D. Then there is an evolution amily (U(t, s)) s t T on D. Moreover, we have the ollowing properties: (1) U(t, s)d(s) D(t) or all s t T, where D(r) = { D : A(r) D}, r T ; (2) the mapping t U(t, s) is continuously ierentiable in X on [s, T ] an U(t, s) = A(t)U(t, s) t or all D(s) an t [, T ]. We will assume that the ollowing hypotheses: (H1) The mapping t A(t) is continuously ierentiable or all D. (H2) The amily (A (t)) t T, A (t) = A(t)/ ker L(t) the restriction o A(t) to ker L(t), is stable, with M an ω constants o stability.

3 EJDE/CONF/14 CAUCHY PROBLEMS 193 (H3) The operator L(t) is surjective or every t [, T ] an the mapping t L(t) is continuously ierentiable or all D. (H4) The mapping t Φ(t) is continuously ierentiable or all X. (H5) There eist constants γ > an ω 1 R such that L(t) Y λ ω 1 X (2.1) γ or ker(λi A(t)), ω 1 < λ an t [, T ]. Note that uner the above hypotheses, Lan [6] has showe that A (t) generates an evolution amily (U(t, s)) s t T such that: (a) U(t, r)u(r, s) = U(t, s) an U(t, t) = I X or all s t T ; (b) (t, s) U(t, s) is continuously ierentiable on or all X with = {(t, s) R + 2 : s t T }; (c) there eists constants M 1 an ω R such that U(t, s) M e ω(t s). The ollowing results with will be use in this article. Lemma 2.4 ([3]). For t [, T ] an λ ρ(a (t)), ollowing properties are satisie: (1) D = D(A (t)) ker(λi A(t)) (2) L(t)/ ker(λi A(t)) is an isomorphism rom ker(λi A(t)) onto Y (3) t L λ,t := (L(t)/ ker(λi A(t))) 1 is strongly continuously ierentiable. As a consequence o this lemma, we have L(t)L λ,t = I Y, L λ,t L(t) an (I L λ,t L(t)) are the projections rom D onto ker(λi A(t)) an D(A (t)). 3. The Homogeneous Problem In this section, we consier the Cauchy problem (1.1). A unction u : [s, T ] X is calle classical solution i it is continuously ierentiable, u(t) D or all s t T an u satisies (1.1). We now introuce the Banach spaces Z = X Y, Z = X {} Z an we consier the projection o Z onto X: p 1 (, y) =. Let M(t) be the matri-value operator eine on Z by where ( M(t) = l(t) = A(t) L(t) + Φ(t) ( ) A(t), φ(t) = L(t) an D(M(t)) = D {}. Now, we consier the Cauchy problem ) = l(t) + φ(t), ( ), Φ(t) u(t) = M(t)u(t), s t T, t u(s) = (, ). We start by proving the ollowing lemma. (3.1) Lemma 3.1. Assume that hypothesis (H1) (H5) hol. Then, the amily o operators (M(t)) t T is stable.

4 194 M. FILALI, B. KARIM EJDE/CONF/14 Remark 3.2. Since L λ,t L(t) is the projection rom D onto ker(λi A(t)) an L λ,t L(t) D(A (t)), we have an R(λ, A (t))((λi A(t))) + L λ,t L(t) = R(λ, A (t))((λi A(t))( L λ,t L(t)) + L λ,t L(t) R(λ, A (t))((λi A(t))) + L λ,t L(t) =. (3.2) Proo o Lemma 3.1. Since M(t) is a perturbation o l(t) by a linear boune operator on E, hence, in view o the perturbation result [7, Theorem 5.2.3], it is suicient to show the stability o l(t). For λ > ω an λ, let ( ) R(λ, A (t)) L R(λ) = λ,t. We have D(l(t)) = D {} an ( (λi l(t)) = ) ( ) (λi A(t)) L(t) or ( ) D {}. By Remark 3.2, we obtain ( ( ) R(λ, A (t))((λi A(t))) + L R(λ)(λI l(t)) = λ,t L(t). ) So that ( ( R(λ)(λI l(t)) =. (3.3) ) ) On the other han, or (, y) X Y, we have ( ( ) ( ) ( λi A(t) R(λ, A (t)) + L (λi l(t))r(λ) = λ,t y = y) L(t) λ y) (3.4) rom (3.3) an (3.4), we obtain that the resolvent o l(t) is given by ( ) R(λ, A (t)) L R(λ, l(t)) = λ,t. (3.5) By a irect computation, we obtain ( m R(λ, l(t i )) = R(λ, A m 1 (t i )) R(λ, A ) (t i )L λ,tm or a inite sequence t 1 t 2 t m T an we have ( ( m R(λ, l(t i )) = y) R(λ, A (t)) + m 1 R(λ, A ) (t))l λ,tm y. From hypothesis (H5), we conclue that L λ,t an by using (H2), we obtain ( R(λ, l(t i )) y) m 1 R(λ, A (t)) + γ (λ ω) or all t [, T ] an λ > ω R(λ, A (t))l λ,tm y M (λ ω ) m + γm (λ ω ) m 1 1 λ ω 1 y. (3.6)

5 EJDE/CONF/14 CAUCHY PROBLEMS 195 For ω 2 = ma(ω, ω 1 ), we have R(λ, l(t i ) ( y) M ( + y ), (λ ω 2 ) m where M = ma(m, Mγ). On E = X Y equippe with the norm (, y) 1 = + y, we have: ( M R(λ, l(t i ) y) (λ ω 2 ) m ( (, y) 1). In the ollowing proposition we give the equivalence between the bounary problem (1.1) an the Cauchy problem (3.1). Proposition 3.3. Let (, ) D {}. (1) I the unction t U(t) = (u 1 (t), ) is a classical solution o (3.1) with an initial value (, ) then t u 1 (t) is a classical solution o (1.1) with the initial value. (2) Let u be a classical solution o (1.1) with the initial value. Then the unction t U(t) = (u(t), ) is a classical solution o (3.1) with the initial value (, ). Proo. (1) Since U(t) = (u 1 (t), ) is a classical solution o (3.1), u 1 is continuously ierentiable on [s, T ] an u 1 (t) D. Moreover, ( t U(t) = t u ) ( 1(t) = M(t)U(t) an U(s) =. (3.7) ) Thereore, t u 1(t) = A(t)u 1 (t), s t T, L(t)u 1 (t) = Φ(t)u 1 (t), s t T, u 1 (s) =. (3.8) This implies that u 1 is a classical solution o (1.1). (2) Let u is a classical solution o (1.1), then u is continuously ierentiable, u(t) D or t s an Hence u(t) = A(t)u(t), t s t T, L(t)u(t) = Φ(t)u(t), s t T, t u(t) ) ( ( = A(t) L(t) + Φ(t) ) ( u(t) with (u(s), ) = (, ). This implies that U(t) = (u(t), ) is a classical solution o (3.2) with the initial value (, ). The above proposition allows us to get the aim o this section by showing the well-poseness o the Cauchy problem (1.1). ),

6 196 M. FILALI, B. KARIM EJDE/CONF/14 Theorem 3.4. Assume that the hypotheses (H1) (H5) hol. Then or every D, such that L(s) + Φ(s) =, the problem ( (1.1) has a unique classical solution. Moreover, u is given by t p 1 (U(t, s), where U(t, s) is the evolution amily ) generate by (M(t) t T ). Proo. For the Cauchy problem (3.1), we have the ollowing: (1) D(M(t)) = ( D ) {} is inepenent o t. (2) t M(t) is continuously ierentiable or (, ) D {}. (3) The amily (M(t)) t T is stable. Then the amily M(t) satisies all conitions o Theorem 2.3. Thus, there eist an evolution amily (U(t, s)) s t generate by the amily (M(t)) t T such that (a) U(t, t) = I X {}, (b) U(t, r)u(r, s) = U(t, s), s r t T, (c) (t, s) U(t, s) is strongly( continuous, ) () the unction t U(t, s) is continuously ierentiable in X {} on [s, T ], an satisies ( ) t U(t, s) an where ( = M(t)U(t, s) ) or ( D(s), ) U(t, s)d(s) D(t), or all s t T, (3.9) ( ) ( ) D(s) = { D {} : M(s) X {}} = ker ( L(s) Φ(s) ) {}. Let U(t, s)(, ) = (u 1 (t), ). We have ( t u ) ( ) 1(t) u1 (t) = M(t), (3.1) an or u(t) = (u 1 (t), ), we have tu(t) = M(t)u(t), with u(s) = (, ), thus u(t) = (u 1 (t), ) is a classical solution o (3.1) an rom Proposition 3.3, we have u 1 is a classical solution o (1.1) an u 1 (t) = p 1 ( U(t, s) ( ) ). (3.11) 4. First Inhomogeneous Problem In this section, we consier the inhomogeneous Cauchy problem u(t) = A(t)u(t) + (t), s t T, t L(t)u(t) = Φ(t)u(t), s t T, (4.1)

7 EJDE/CONF/14 CAUCHY PROBLEMS 197 A unction u : [s, T ] X is calle classical solution i it is continuously ierentiable, u(t) D, t s an u satisies (4.1). Consier the Banach space E = X Y C 1 ([, T ], X), T >, where C 1 ([, T ], X) is the space o continuously ierentiable unctions rom [, T ] into X equippe with the norm = +, or C 1 ([, T ], X). Let B(t) be the operator matrices eine on E by A(t) δ t B(t) = L(t) + Φ(t) (4.2) with D(B(t)) = D {} C 1 ([, T ], X). Where δ t : C 1 ([, T ], X) X is the Dirac unction concentrate at the point t with δ t () = (t). To the amily B(t) we associate the homogeneous Cauchy problem u(t) = B(t)u(t), s t T, t (4.3) u(s) = (,, ). with (,, ) D {} C 1 ([, T ]. Lemma 4.1. Assume that hypothesis (H1) (H5) hol. Then the amily operators (B(t)) t T is stable. Proo. For t [, T ], we write the operator B(t) as B(t) = l(t) + φ(t), with l(t) = A(t) L(t) an φ(t) = δ t Φ(t). We must show that l(t) is stable an that R(λ, A (t)) L λ,t R(λ, l(t)) =. (4.4) 1/λ For λ > ω, λ, an t [, T ], let R(λ, A (t)) L λ,t R(λ) =. 1/λ For (, y, ) X Y C 1 ([, T ], X), we have R(λ, A (t) L λ,t R(λ, A (t)) + L λ,t y y =, 1/λ by the Remark 3.2, we obtain (λi l(t))r(λ) y = (λi A(t))[R(λ, A (t)) + L λ,t y] L(t)[R(λ, A (t)) + L λ,t y] = y. (4.5) On the other han, or (,, ) D {} C 1 ([, T ], X), we have (λi l(t)) (λi A(t)) = L(t), λ λ

8 198 M. FILALI, B. KARIM EJDE/CONF/14 an R(λ)(λI l(t)) = R(λ, A (t))((λi A(t))) + L λ,t L(t). From Remark 3.2, we have R(λ)(λI l(t)) =. (4.6) From (4.5) an (4.6), we obtain that the resolvent o l(t) is given by R(λ, l(t)) = R(λ, A (t)) L λ,t. 1/λ By recurrence we can obtain m R(λ, l(t i )) = R(λ, A m 1 (t i )) R(λ, A (t i ))L λ,tm. 1/λ m For a inite sequence t 1 t 2 t m T an or (, y, ) E, we have m R(λ, l(t i )) y = R(λ, A (t i )) + m 1 R(λ, A (t i ))L λ,tm y. /λ m Using (H5), we obtain R(λ, l(t i )) y m 1 R(λ, A (t i )) + R(λ, A (t i ))L λ,tm y + λ m M (λ ω ) m + M (λ ω ) m 1 γ λ ω 1 y + λ m. Deine ω 2 = ma(, ω, ω 1 ). Then R(λ, l(t i )) y M ( + y + ), (λ ω 2 ) m where M = ma(m, Mγ) an M R(λ, l(t i )) (λ ω 2 ) m. (4.7) This inequality shows that the amily l(t) is stable an by using [7, Theorem 5.2.3], the amily B(t) is stable. Proposition 4.2. Let (,, ) D {} C 1 ([, T ], X). (1) I the unction t u(t) = (u 1 (t),, u 2 (t)) is a classical solution o (4.3) with an initial value (,, ) then t u 1 (t) is a classical solution o (4.1) with the initial value. (2) Let u is a classical solution o (4.1) with the initial value.then, the unction t U(t) = (u(t),, ) is a classical solution o (4.3) with the initial value (,, ).

9 EJDE/CONF/14 CAUCHY PROBLEMS 199 Proo. (1) I u(t) = (u 1 (t),, u 2 (t)) is a classical solution o (4.3), then u 1 is continuously ierentiable on [s, T ], u 1 D an we have t u(t) = t u 1(t) = B(t)u(t), t u 2(t) which implies t u 1(t) A(t) δ t = L(t) + Φ(t) u 1(t), t u 2(t) u 2 (t) an t u 1(t) = A(t)u 1(t) + δ t u 2 (t) L(t)u 1 (t) + Φ(t)u 1 (t), t u 2(t) with u 1 (s) u(s) = =. u 2 (s) One has t u 2(t) =. This implies u 2 (t) = u 2 (s) = ; thereore, δ t u 2 (t) = δ t = (t) an we have t u 1(t) = A(t)u 1 (t) + (t), s t T, L(t)u 1 (t) = Φ(t)u 1 (t), s t T, u 1 (s) =. Thereore, u 1 is a classical solution o (4.1) with the initial value. (2) I u is a classical solution o (4.1), then u is continuously ierentiable, u(t) D an u(t) = A(t)u(t) + (t), s t T, t L(t)u(t) = Φ(t)u(t), s t T, Moreover, t u(t) A(t) δ t = L(t) + Φ(t) u(t). With u(s) =, U(t) = (u(t),, ) is continuously ierentiable, U(t) D(B(t)) = D {} C 1 ([, T ], X) then it is a classical solution o (4.3) with the initial value (,, ). Theorem 4.3. Let C 1 ([, T ], X). Assume that the hypothesis (H1) (H5) hol. Then or all D, such that L(s) + Φ(s) =, problem (4.1) has a unique classical solution solution u. Moreover, u is given by u(t) = U Φ (t, s) + t s U Φ (t, s)(r)r, (4.8) where U Φ (t, s) is an evolution amily solution o the problem (3.1)

10 2 M. FILALI, B. KARIM EJDE/CONF/14 Proo. Consier the problem u(t) = B(t)u(t), s t T, t u(s) = (,, ). We have showe that (B(t)) t T is a stable amily an the unction t B(t)y is continuously ierentiable, or all y D(B(t)) = D {} C 1 ([, T ], X) an that D(B(t)) is inepenent o t. Then there eist an evolution system U(t, s) on X {} C 1 ([, T ], X) such that U(t, s) = u 1(t) = u(t) u 2 (t) is a classical solution o (4.3) an rom the Proposition 4.2, u 1 is a classical solution o (4.1), or (,, ) ker(l(s) Φ(s)) {} C 1 ([, T ], X). Let v(r) = U Φ (t, r)u 1 (r). Then v is ierentiable an r v(r) = U Φ(t, r)a Φ (r)u 1 (r) + U Φ (t, r)[a Φ (r)u 1 (r) + (r)], where A Φ (t) = A(t)/ ker(l(t) Φ(t)); thereore, Integrating (4.9) rom s to t, we obtain which completes the proo. r v(r) = U Φ(t, r)(r). (4.9) u 1 (t) = U Φ (t, s) + t s U Φ (t, r)(r)r, 5. Secon Inhomogeneous Problem In this section, we consier the Inhomogeneous Cauchy problem u(t) = A(t)u(t) + (t), t s t T, L(t)u(t) = Φ(t)u(t) + g(t), s t T, (5.1) A unction u : [s, T ] X is a classical solution i it is continuously ierentiable, u(t) D, or all t s an u satisies (5.1). Consier the Banach space E = X Y C 1 ([, T ], X) C 1 ([, T ], Y ), where C 1 ([, T ], X) an C 1 ([, T ], Y ) are equippe with the norm = + or in C 1 ([, T ], X) or in C 1 ([, T ], Y ). Consier the operator matrices A(t) δ t B(t) = L(t) + Φ(t) δ t, (5.2) with D(B(t)) = D {} C 1 ([, T ], X) C 1 ([, T ], Y )

11 EJDE/CONF/14 CAUCHY PROBLEMS 21 where δ t : C 1 ([, T ], X) X such that δ t () = (t) an δ t : C 1 ([, T ], Y ) Y such that δ t (g) = g(t). To the amily B(t),we associate the homogeneous Cauchy problem u(t) = B(t)u(t), s t T, t (5.3) u(s) = (,,, g) or (,,, g) D {} C 1 ([, T ], X) C 1 ([, T ], Y ) = D 1. Lemma 5.1. Assume that the hypothesis (H1) (H5) hol. Then the amily operators B(t) is stable. Proo. For t [, T ], we write the B(t) eine in (5.2) as B(t) = l(t) + φ(t), where A(t) δ t l(t) = L(t) an φ(t) = Φ(t) δ t, we must show that the amily l(t) is stable. Let R(λ, A (t)) L λ,t R(λ) = 1/λ. 1/λ For λ > ω, λ an t [, T ] we show that R(λ, l(t)) = R(λ). For (, y,, g) X Y C 1 ([, T ], X) C 1 ([, T ], Y ), we have R(λ, A (t)) + L λ,t y R(λ) y = /λ, (5.4) g g/λ by the Remark 3.2 an with the same proo as Lemma 4.1 we obtain (λi l(t))r(λ) y = y. (5.5) g g On the other han, or (,,, g) D {} C 1 ([, T ], X) C 1 ([, T ], Y ), we have (λi A(t)) (λi l(t)) = L(t) λ, g λg an R(λ, A (t))((λi A(t))) + L λ,t L(t) R(λ)(λI l(t)) = =, (5.6) g g g

12 22 M. FILALI, B. KARIM EJDE/CONF/14 then rom (5.5), (5.6) an Remark 3.2, we have R(λ) = R(λI, l(t)). By recurrence we obtain m R(λ, A m 1 (t i )) R(λ, A (t i ))L λ,tm R(λ, l(t i )) = 1/λ m, 1/λ m or a inite sequence t 1 t 2 t m T. Now on the space X Y C 1 ([, T ], X) C 1 ([, T ], Y ),we consier the norm (, y,, g) = ( + y + + g ). (5.7) For (, y,, g) X Y C 1 ([, T ], X) C 1 ([, T ], Y ), we have R(λ, l(t i )) y M (λ ω ) m + Mγ 1 (λ ω ) m 1 y + λ ω 1 λ m + g λ m g M ( + y + + g ), (λ ω 2 ) m where ω 2 = ma(, ω, ω 1 ) an M = ma(m, Mγ). Since B(t) is a perturbation o l(t), by a linear operator φ(t) on E; hence, in view o perturbation result [7, Theorem 5.2.3], B(t) is stable. Proposition 5.2. Let (,,, g) D {} C 1 ([, T ], X) C 1 ([, T ], Y ) (1) I the unction t u(t) = ( u 1 (t),, u 2 (t), u 3 (t) ) is a classical solution o (5.3) with an initial value (,,, g) then t u 1 (t) is a classical solution o (5.1) with the initial value. (2) Let u is a classical solution o (5.1) with the initial value. Then, the unction t U(t) = ( u(t),,, g ) is a classical solution o (5.3) with the initial value (,,, g). Proo. (1) I u(t) = ( u 1 (t),, u 2 (t), u 3 (t) ) is a classical solution o (5.3), then u 1 is continuously ierentiable on [s, T ] an we have t u 1(t) A(t) δ t u 1 (t) t u 2(t) = L(t) + Φ(t) δ t u 2 (t). t u 3(t) u 3 (t) This implies t u 1(t) = A(t)u 1 (t) + δ t u 2 (t), s t T, L(t)u 1 (t) = Φ(t)u 1 (t) + δ t u 3 (t), s t T, t u 2(t) =, t u 3(t) =. One has t u 3(t) = which implies u 3 (t) = u 3 (s) = g an L(t)u 1 (t) = Φ(t)u 1 (t) + g(t). Also t u 2(t) = implies u 2 (t) = u 2 (s) = an t u 1(t) = A(t)u 1 (t) + (t).

13 EJDE/CONF/14 CAUCHY PROBLEMS 23 Then t u 1(t) = A(t)u 1 (t) + (t), s t T, L(t)u 1 (t) = Φ(t)u 1 (t) + g(t), s t T, u 1 (s) =. Thus u 1 is a classical solution o (5.1) with the initial value. (2) Let u is a classical solution o (5.1). This implies that u is continuously ierentiable an u(t) D {} C 1 ([, T ], X) C 1 ([, T ], Y ). Moreover, This implies u(t) = A(t)u(t) + (t), t L(t)u(t) = Φ(t)u(t) + g(t), s t T s t T t u(t) A(t) δ t u(t) = L(t) + Φ(t) δ t, g with Then U(t) = (u(t),,, g) is continuously ierentiable, U(t) D {} C 1 ([, T ], X) C 1 ([, T ], Y ), or all t [s, T ] an U(t) is a classical solution o (5.1) with the initial value (,,, g). Theorem 5.3. Let C 1 ([, T ], X) an g C 1 ([, T ], Y ). Assume that the hypothesis (H1) (H5) hol. Then or every D such that L(s)+Φ(s)+g(s) =, problem (5.1) has a unique classical solution. Proo. Consier the homogenous Cauchy problem u(t) = B(t)u(t), s t T, t u(s) = (,,, g). By Lemma 5.1, B(t) is a stable amily an the unction t B(t)y is continuously ierentiable or all y D 1 = D(B(t)) inepenent o t. Then there eist an evolution amily U(t, s) on X {} C 1 ([, T ], X) C 1 ([, T ], Y ) such that u 1 (t) U(t, s) = u 2 (t) = u(t) g u 3 (t) is a classical solution o (5.3) an rom the Proposition 5.2, u 1 is a classical solution o (5.1). The uniqueness o u 1 comes rom the uniqueness o the solution o (5.3) an Proposition 5.2. Theorem 5.4. Let C 1 ([, T ], X) an g C 1 ([, T ], Y ). I u is a classical solution o (5.1) then u is given by the variation o constants ormula u(t) = U(t, s)(i L λ,s L(s))+g(t, u(t))+ t s U(t, r)[λg(r, u(r)) g(r, u(r)) +(r)]r, (5.8)

14 24 M. FILALI, B. KARIM EJDE/CONF/14 where U(t, s) is the evolution amily generate by A (t) an g(t, u(t)) = L λ,t (Φ(t)u(t) + g(t)). Proo. Let now u be a classical solution o (5.1). Take Then the unctions u 2 (t) = L λ,t L(t)u(t) an u 1 (t) = (I L λ,t L(t))u(t). u 2 (t) = g(t, u(t)) = L λ,t (Φ(t)u(t) + g(t)) an u 1 (t) are ierentiable. Since u 2 (t) ker(λi A(t)), we have A(t)u 2 (t) = λu 2 (t) an t u 1(t) = t u(t) t u 2(t) = A(t)u(t) (g(t, u(t))) + (t) = A(t)(u 1 (t) + u 2 (t)) + (t) (g(t, u(t))) = A(t)u 1 (t) + λ(g(t, u(t)) + (t) (g(t, u(t))). When we eine h(t) := λg(t, u(t)) + (t) (g(t, u(t))), we get u 1 (t) = U(t, s)u 1 (s) + t By replacing u 1 (s) by (I L λ,s L(s)), we obtain it ollows that u 1 (t) = U(t, s)(i L λ,s L(s)) + s U(t, r)h(r)r. (5.9) t u(t) = u(t, s)(i L λ,s L(s)) + g(t, u(t)) + t which completes the proo. s s U(t, r)h(r)r, (5.1) u(t, r)[λg(r, u(r)) (g(r, u(r))) + (r)]r, Reerences [1] K. J. Engel an R. Nagel, One-Parameter Semigroups or Linear Evolution Equations, Grauate Tets in mathematics,springer-verlag, 194(2). [2] M. Filali an M. Moussi, Non-autonomous Inhomogeneous Bounary Cauchy Problems an Retare Equations. Kyungpook Math. J. 44(24), [3] G. Greiner, Perturbing the bounary conitions o a generator, Houston J. math., 13(1987), [4] T. Kato, Linear evolution equations, o hyperbolic type, J. Fac. Sci. Tokyo, 17(197), [5] H. Kellerman, Linear evolution equations with time-epenent omain, semesterbericht Funktionalanalysis, Tübingen, Wintersemester, ( ). [6] N. T. Lan, On nonautonomous Functional Dierential Equations, J. Math. Anal. App., 239 (1999), [7] A. Pazy, Semigroups o linear operators an applications to partial ierential equations, New York Springer, [8] A. Rhani, Etrapolation methos to solve nonautonomous retare ierential equations, Stuia Math, 126 (1997), [9] N. Tanaka, Quasilinear Evolution Equations with non-ensely eine operators, Di. Int. Equat., 9 (1996),

15 EJDE/CONF/14 CAUCHY PROBLEMS 25 Mohamme Filali Département e Mathématiques et Inormatique, Faculté es Sciences, Université Mohamme 1er, Ouja, Maroc aress: ilali@sciences.univ-ouja.ac.ma Belhaj Karim Département e Mathématiques et Inormatique, Faculté es Sciences, Université Mohamme 1er, Ouja, Maroc aress, B. Karim: karim@sciences.univ-ouja.ac.ma