QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES. Citation Journal of Evolution Equations. 4(3.

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1 Title QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES Author() Favini, Angelo; Yagi, Atuhi Citation Journal of Evolution Equation. 4(3 Iue 24-9 Date Text Verion publiher URL DOI 1.17/ Right Oaka Univerity

2 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) Abtract. The quailinear degenerate evolution equation of parabolic type d(mv) + L(Mv)v = F (Mv), < t T conidered in a Banach pace X i dt written, putting Mv = u, in the form du + A(u)u F (u), < t T, where dt A(u) = L(u)M 1 are multivalued linear operator in X for u K, K being a bounded ball u Z < R in another Banach pace Z continuouly embedded in X. Exitence and uniquene of the local olution for the related Cauchy problem are given. The reult are applied to quailinear elliptic-parabolic equation and ytem. 1. Introduction We are concerned with the Cauchy problem of a degenerate abtract evolution equation of parabolic type dmv + L(Mv)v = F (Mv), < t T, (D.E) dt Mv() = u in a Banach pace X. Here, L(u) are cloed linear operator in X with ome contant domain D(L(u)) D L for u K = {u Z; u Z R}, R >, where Z i another Banach pace uch that Z X with continuou embedding. M i a cloed linear operator in X with the domain D(M) D L uch that M(D L ) Z. F ( ) i a nonlinear operator from K into X. u K i an initial value. v = v(t) i an unknown function. Cauchy problem of many concrete equation are formulated a thoe of abtract equation of the form (D.E), uch a elliptic-parabolic equation, elliptic-parabolic ytem, nonlinear equation of Sobolev type [14], emiconductor equation [15], and o on. Like in our previou paper [3] (cf. alo [11]) for linear problem, we rewrite the degenerate equation in (D.E) in the form du dt + A(u)u F (u) by changing unknown function from v = v(t) to u = M v(t) and introducing multivalued linear operator A(u) = L(u)M 1, u K, which act in X with a contant domain D(A(u)) = M(D L ). In thi way we have the Cauchy problem for a quailinear (*) Partially upported M. I. U. R. (Fund ex 6 % ) and by Univerity of Bologna Fund for elected reearch topic, the author i a menber of G. N. A. M. P. A. of INdAM. (**) Partially upported by Grant-in-Aid for Scientific Reearch (No ) by Japan Society for the Promotion of Science. 1

3 2 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) multivalued equation but of nondegenerate type du + A(u)u F (u), < t T, (E) dt u() = u. Sobolevkii [8] (cf. alo [12, Part 2] and [17]) ha firt tudied uch an abtract problem in the cae where the coefficient operator A(u) are all ingle valued and are the generator of analytic emigroup, that i A(u) atify (1.1) (λ A(u)) 1 L(X) C ( λ + 1) 1 κ, λ Σ, u K with the optimal exponent κ = and with ome ectorial domain Σ = {λ C; arg λ < φ}, < φ < π 2. He in fact contructed, under uitable aumption on A(u), F (u) and u, a unique X-valued C 1 local olution. We remark however that even if L(u) are the generator of analytic emigroup, A(u) = L(u)M 1 do not necearily atify (1.1) with κ =. The firt half of thi paper i then devoted to tudying the problem (E) with multivalued operator atifying (1.1). We hall prove exitence and uniquene of X-valued C 1 local olution by generalizing Sobolevkii reult on the bai of the previou work on multivalued linear evolution equation in [3]. In the econd half we hall apply our abtract reult to elliptic-parabolic equation and elliptic-parabolic ytem. There i an enormou literature on the ubject. We refer to the recent monograph by Showalter [16], ee alo [1]. In fact, mot available reult until now develop the baic approach by Brezi [2], where one ee the left hand ide of (D.E) a the um of two operator, the former being linear, the latter being (poibly nonlinear) monotone, and further aumption allow to apply the theory by Bardo and Brezi [1]. On the other hand, uch an approach force to tudy the equation in ome particular functional etting a L p (, T ; W ), where W i either the dual pace of a reflexive Banach pace or a weighted pace (depending on the operator M), 1 p + 1 p = 1, p 2. The mot recent main reult on quailinear degenerate evolution equation in [16, pp ] (ee in particular Corollary 6.2 and Corollary 6.3) are concerned with the equation d (S) (Bu) + A(t, u) = f(t), a. e. t (, T ). dt It i uppoed that B i a continuou, linear, ymmetric and monotone operator from the reflexive eparable Banach pace V to it dual V and A : [, T ] V V atifie ome appropriate continuity, monotonicity and coercivity aumption ([16, p. 129]). The application of thee reult to quailinear elliptic-parabolic equation i detailed in [16, Example 6.3]. For other reult, we quote Kuttler [5] and [6], too. Here it will be hown that our approach allow to conider problem of thi type having a nonlinearity (in u) in the right hand ide of (S), too, with a bit more retrictive aumption on the data, due to the greater regularity in time of our olution.

4 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 3 The plan of the paper i a follow. In Section 2 we recall baic reult on multivalued linear operator depending on time from [3] and [11], and prove new etimate for the evolution operator to be ued in the ubequent ection. Section 3 i devoted to the problem (E). Some condition given there are inpired by [8] and [9] too, and they guarantee the exitence and uniquene of the olution to (E). In ection 4 we apply uch reult to olve (D.E). Of coure, if L(u) L i independent of u, we can olve a emilinear degenerate differential equation. Section 5 and 6 contain example from partial differential equation of ellipticparabolic type to which our abtract theory applie. It i to be oberved that unlike all previou literature, where the ambient pace i a Sobolev pace of negative exponent, here we can take X = L 2 (Ω) (or, a hown in [4], L p (Ω), 1 < p < ), when Ω i a bounded region in R n with a mooth boundary. Notation. Throughout the paper, X denote a complex Banach pace whoe norm i denoted by X. If Y i another Banach pace, L(X, Y ) i the pace of all bounded linear operator from X to Y and L(X,Y ) denote the uniform operator norm. L(X) i ued for intead of L(X, X). An operator A : X 2 Y having the two propertie: Au + Av A(u + v), u, v X and λau A(λu), λ C, u X i called a multivalued linear operator from D(A) = {u X; Au } to Y. For u D(A), Au X = inf{ f Y ; f Au}. If A : D(A) Y i a multivalued linear operator, a ingle valued operator A : D(A) Y uch that A A in the graph ene i called a ection of A. With an arbitrary ection A, it hold that Au = A + A u, u D(A). If I i a nonempty interval in R and k i a nonnegative integer, C k (I; X) denote the pace of all k-time continuouly differentiable function with value in X defined on I, where C (I; X) = C(I; X). For < µ < 1, C µ (I; X) i the pace of µ-hölder continuou function with value in X defined on I. B(I; X) denote the pace of all bounded function on I with value in X. 2. Multivalued linear equation We conider a family of multivalued linear operator A(t), t T, acting in a Banach pace X which have a domain D(A(t)) D independent of t. In the previou paper [3], we have already contructed the evolution operator U(t, ) under the Aumption (L.A.i,ii) and (L.Ex) below. The purpoe of thi ection i then to review the baic propertie of U(t, ) and verify more refined one which will be required in tudying the multivalued quailinear equation. We make the following aumption. For every t T, the pectral et σ(a(t)) of A(t) i contained in a fixed open ectorial domain Σ, σ(a(t)) Σ = {λ C; arg λ < φ}, where < φ < π 2. And the reolvent atifie the etimate (L.A.i) (λ A(t)) 1 L(X) M, λ Σ, t T, ( λ + 1) 1 κ

5 4 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) with ome exponent κ < 1 and a contant M >. A( ) atifie a Hölder condition of the form (L.A.ii) A(t){A(t) 1 A() 1 }f X N t µ f X, f X,, t T with ome exponent < µ 1 and a contant N >. The exponent atify the relation (L.Ex) 3κ < µ 1. Before introducing the evolution operator, let u firt notice that (L.A.i) and (L.A.ii) imply the etimate (2.1) A(t) (λ A(t)) 1 {A(t) 1 A() 1 }f X Here, MN t µ ( λ + 1) 1 κ f X, λ Σ, f X. A(t) (λ A(t)) 1 = λ(λ A(t)) 1 1 A(t)(λ A(t)) 1 denote the linear ection of A(t)(λ A(t)) 1 introduced in [3, Theorem 2.7]. It i known that A(t) (λ A(t)) 1 = (λ A(t)) 1 A(t) on D, where A(t) in the right hand ide denote an arbitrary ection of A(t) not necearily linear. In fact, (2.1) i verified a follow. For f X, A(t) (λ A(t)) 1 {A(t) 1 A() 1 }f = (λ A(t)) 1 A(t) {A(t) 1 A() 1 }f. In addition, ince (λ A(t)) 1 A(t) {A(t) 1 A() 1 }f = (λ A(t)) 1 g with any g A(t){A(t) 1 A() 1 }f, it follow that (2.2) (λ A(t)) 1 A(t) {A(t) 1 A() 1 }f X (λ A(t)) 1 L(X) A(t){A(t) 1 A() 1 }f X, f X. Therefore, (L.A.i) and (L.A.ii) imply (2.1). In thi theory we hall make an eential ue of the Yoida approximation A n (t) = A(t) J n (t) = n{1 J n (t)}, n = 1, 2, 3,..., J n (t) = (1 + n 1 A(t)) 1 of A(t). A n (t) are ingle valued bounded linear operator on X with A n (t) L(X) Cn 1+κ. Since A n (t) 1 = A(t) 1 + n 1, we have A n (t){a n (t) 1 A n () 1 }f = J n (t)a(t) {A(t) 1 A() 1 }f. Therefore, by (2.2), A n (t){a n (t) 1 A n () 1 }f X N J n (t) L(X) t µ f X, f X. Thi how that, a J n (t) L(X) Cn κ, the Hölder condition (L.A.ii) may not imply that of the Yoida approximation in any uniform ene. Such a difficulty i

6 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 5 however overcome by the fact that (2.1) implie the ame one for A n (t). In fact, ince A n (t)(λ A n (t)) 1 {A n (t) 1 A n () 1 }f (2.1) yield that = n n λ ( n n λ A(t) (2.3) A n (t)(λ A n (t)) 1 {A n (t) 1 A n () 1 } L(X) ) 1 A(t) {A(t) 1 A() 1 }f, C t µ, λ Σ, ( λ + 1) 1 κ ee alo [3, (4.8)]. We hall alo ue the fractional power of A(t). For θ > κ, the fractional power A(t) θ i defined by the Dunford integral A(t) θ = 1 2πi Γ λ θ (λ A(t)) 1 dλ in L(X), where Γ i an integral contour lying in C σ(a(t)). A(t) θ, θ > κ, i then a multivalued linear operator in X. In particular, A(t) 1 = A(t). A n, A n (t) θ converge to A(t) θ in L(X). (L.A.i) yield that for each t T, A(t) generate an infinitely differential emigroup e τa(t), τ, on X, ee [3, Section 3]. For θ, a bounded linear operator on X given by the integral {A(t) θ } e τa(t) = 1 2πi Γ λ θ e τλ (λ A(t)) 1 dλ, τ >, i introduced. Obviouly, thi operator i alo obtained a a limit of A n (t) θ e τa n(t) alo. The following etimate (2.4) A n (t) θ e τa n(t) J n (t) k L(X) C θ τ θ κ, τ >, k =, 1, {A(t) θ } e τa(t) L(X) C θ τ θ κ, τ >, are verified. If θ > κ, {A(t) θ } e τa(t) i really a linear ection of the multivalued operator A(t) θ e τa(t). For θ > κ, τ (2.5) {e τa(t) 1}A(t) θ L(X) = A(t) e σa(t) dσa(t) θ L(X) τ = {A(t) 1 θ } e σa(t) dσ C θ τ θ κ, τ >. L(X) According to [3, Theorem 4.1] (cf. alo [11, Section 4.1]), under (L.A.i), (2.1), and (L.Ex), there exit an evolution operator U(t, ), t T, for A(t). U(t, ) i in fact obtained a a limit of U n (t, ), where U n (t, ) i an evolution operator for A n (t). Moreover, U n (t, )J n () ha the ame limit a U n (t, ), that i,

7 6 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) U n (t, )J n () alo converge to U(t, ). The etimate U n (t, )J n () k L(X) C(t ) κ, < t T, k =, 1, U(t, ) L(X) C(t ) κ, < t T hold. The convergence of A n (t)u n (t, ) i alo etablihed, it limit being denoted by A(t) U(t, ). A(t) U(t, ) i a linear ection of A(t)U(t, ). The etimate A n (t)u n (t, ) L(X) C(t ) 1 κ, < t T, A(t) U(t, ) L(X) C(t ) 1 κ, < t T hold. Similarly, for < θ < 1, a bounded linear operator {A(t) θ } U(t, ) i defined a a limit of A n (t) θ U n (t, ). The etimate (2.6) A n (t) θ U n (t, ) L(X) C(t ) θ κ, < t T, θ 1, {A(t) θ } U(t, ) L(X) C(t ) θ κ, < t T, θ 1 are verified by the moment inequality of the fractional power. In addition, it i verified that (2.7) A n (t) θ U n (t, )J n () L(X) A n (t) θ U n (t, t+ 2 ) L(X) U n ( t+ 2, )J n() L(X) C(t ) θ 2κ, < t T, < θ 1. To obtain (2.8) below, we notice from [3, (4.1)] that U n (t, )A n () θ = A n () θ e (t )A n() + U n (t, τ)a n (τ){a n (τ) 1 A n () 1 }A n () θ+1 e (τ )A n() dτ. By the ame argument a for (2.2), we oberve that U n (t, τ)a n (τ){a n (τ) 1 A n () 1 }f X Hence, for θ < µ κ, = U n (t, τ)j n (τ)a(τ) {A(τ) 1 A() 1 }f X U n (t, τ)j n (τ) L(X) A(τ){A(τ) 1 A() 1 }f X, f X. (2.8) U n (t, )A n () θ L(X) C(t ) θ κ + C (t τ) κ (τ ) θ 1 κ+µ dτ C θ (t ) θ κ, < t T. We now prove ome new etimate of U n (t, ) and U(t, ). Propoition 2.1. For κ < ϕ 1, (2.9) A n (t)u n (t, )A n () ϕ L(X) C ϕ (t ) ϕ 1 κ, < t T, A(t) U(t, )A() ϕ L(X) C ϕ (t ) ϕ 1 κ, < t T.

8 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 7 Proof. From U n (t, ) = e (t )An(t) it follow that e (t τ)a n(t) A n (t){a n (t) 1 A n (τ) 1 }A n (τ)u n (τ, )dτ, (2.1) A n (t)u n (t, )A n () ϕ = A n (t)e (t )An(t) A n () ϕ A n (t)e (t τ)a n(t) J n (t)a(t) {A(t) 1 A(τ) 1 }A n (τ)u n (τ, )A n () ϕ dτ with any ection A(t) A(t). We here how the following lemma. Lemma 2.1. For θ 1 and κ < ϕ 1, {A n (t) θ e τan(t) A n () θ e τan() }A n () ϕ L(X) CΓ(θ ϕ + 2κ)τ ϕ θ 2κ t µ, if ϕ < θ + 2κ, C{log(τ 1 + 1) + 1} t µ, if ϕ = θ + 2κ, C(ϕ θ 2κ) 1 t µ, if ϕ > θ + 2κ, where Γ( ) denote the gamma function. Letting n, the ame etimate are verified for the family A(t), too. Proof. We ee that {A n (t) θ e τan(t) A n () θ e τan() }A n () ϕ = 1 λ θ e τλ {(λ A n (t)) 1 (λ A n ()) 1 }A n () ϕ dλ 2πi Γ = 1 λ θ e τλ A n (t)(λ A n (t)) 1 {A n (t) 1 A n ()) 1 } 2πi Γ A n () 1 ϕ (λ A n ()) 1 dλ, where Γ i an integral contour: λ = ρe ±φi, ρ <. Therefore, by (2.3), it follow that {A n (t) θ e τan(t) A n () θ e τan() }A n () ϕ L(X) C ( λ + 1) θ ϕ+2κ 1 e τreλ dλ t µ. If ϕ < θ + 2κ, then ( λ + 1) θ ϕ+2κ 1 e τreλ dλ Γ C Γ ρ θ ϕ+2κ 1 e τρ co φ dρ Cτ ϕ θ 2κ ρ θ ϕ+2κ 1 e ρ dρ.

9 8 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) If ϕ = θ + 2κ, then ( λ + 1) θ ϕ+2κ 1 e τreλ dλ C Γ C τ 1 (ρ + 1) 1 e τρ co φ dρ (ρ + 1) 1 dρ + C ρ 1 e τρ co φ dρ. τ 1 Hence the deired etimate i obtained. Similarly, if ϕ > θ + 2κ, then ( λ + 1) θ ϕ+2κ 1 e τreλ dλ C (ρ + 1) θ ϕ+2κ 1 dρ. Γ Uing thi lemma with θ = 1, we have A n (t)e (t )An(t) A n () ϕ L(X) A n () 1 ϕ e (t )An() L(X) + {A n (t)e (t )A n(t) A n ()e (t )A n() }A n () ϕ L(X) C ϕ [(t ) ϕ κ 1 + log{(t ) 1 + 1}(t ) ϕ 2κ+µ 1 ] C ϕ (t ) ϕ κ 1. Then, from (2.1), the following integral inequality A n (t)u n (t, )A n () ϕ L(X) C ϕ (t ) ϕ 1 κ + C (t τ) µ 1 κ A n (τ)u n (τ, )A n () ϕ L(X) dτ i obtained, which implie the firt etimate (2.9). Obviouly the econd etimate i an immediate conequence of the firt one. Hence the proof of the propoition ha been accomplihed. Propoition 2.2. For θ < 1 κ and θ + κ < ϕ 1, A n (t) θ U n (t, )A n () ϕ L(X) C θ,ϕ, < t T, {A(t) θ } U(t, )A() ϕ L(X) C θ,ϕ, < t T. Proof. From (2.1) we can write that A n (t) θ U n (t, )A n () ϕ = A n (t) θ e (t )An(t) A n () ϕ A n (t) θ e (t τ)an(t) J n (t)a(t) {A(t) 1 A(τ) 1 }A n (τ)u n (τ, )A n () ϕ dτ. In addition, by (L.Ex), (2.5) and Lemma 2.1, we can oberve that A n (t) θ e (t )An(t) A n () ϕ L(X) {A n (t) θ e (t )An(t) A n () θ e (t )An() }A n () ϕ L(X) + {e (t )An() 1}A n () θ ϕ L(X) + A n () θ ϕ L(X) C θ,ϕ.

10 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 9 Therefore, in view of (L.Ex), (2.4) and (2.9), we obtain that A n (t) θ U n (t, )A n () ϕ L(X) C θ,ϕ {1 + } (t τ) µ θ κ (τ ) ϕ κ 1 dτ C θ,ϕ {1 + (t ) ϕ θ 2κ+µ } C θ,ϕ. The econd etimate i an immediate conequence of thi. A to the difference of the evolution operator and the emigroup, we verify the following reult. Propoition 2.3. For θ < 1 2κ and κ < ϕ 1, A n (t) θ {U n (t, ) e (t )A n() }A n () ϕ L(X) For κ < θ < 1 2κ and κ < ϕ 1, A(t) θ {U(t, ) e (t )A() }A() ϕ f X Proof. Uing [3, (4.1)] with ρ = 1, we ee that (2.11) A n (t) θ {U n (t, ) e (t )An() }A n () ϕ = C θ,ϕ (t ) ϕ θ 3κ+µ, < t T. C θ,ϕ (t ) ϕ θ 3κ+µ f X, < t T, f X. A n (t) θ U n (t, τ)j n (τ)a(τ) {A(τ) 1 A() 1 }A n () 1 ϕ e (t )A n() dτ. Then, by (2.7) and (2.9), the norm of the right hand ide i etimated by C θ,ϕ (t τ) θ 2κ (τ ) ϕ 1 κ+µ dτ C θ,ϕ (t ) ϕ θ 3κ+µ. Let κ < θ < 1 2κ. Operating A n (t) θ to (2.11) and letting n in the reulting equality, we obtain that {U(t, ) e (t )A() }A() ϕ = A(t) θ {A(t) θ } U(t, τ)a(τ) {A(τ) 1 A() 1 }A() 1 ϕ e (t )A() dτ. From thi the econd etimate of the propoition i obtained. We finally how a formula which give a olution to the Cauchy problem of a multivalued linear equation du + A(t)u F (t), < t T, (L.E ) dt u() = u

11 1 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) in X. [, T ) i a fixed initial time. F i a given Hölder continuou function on [, T ] uch that (L.F) F C σ ([, T ]; X), σ > κ. u i an initial value in X uch that (L.In) u D(A() γ ), κ < γ 1. A proved by [3, Theorem 4.1], there exit a unique olution to (L.E ) in the function pace: u C([, T ]; X) C 1 1+κ γ du ((, T ]; X), (t ) dt B((, T ]; X). u i in fact given by u(t) = U(t, )u + U(t, τ)f (τ)dτ, t T. Moreover, we can verify the following etimate A (2.12) n(t) U n (t, τ)f (τ)dτ C σ (t ) κ F C σ ([,T ];X), < t T, X A(t) (2.13) U(t, τ)f (τ)dτ C σ (t ) κ F Cσ ([,T ];X), < t T. X Indeed, (2.14) A n (t) + U n (t, τ)f (τ)dτ = A n (t)u n (t, τ){f (τ) F (t)}dτ {A n (t)u n (t, τ) A n (t)e (t τ)an(t) }F (t)dτ + {1 e (t )An(t) }F (t). Uing the integral equation [3, (4.12)], it i een that A n (t)u n (t, ) A n (t)e (t )A n(t) L(X) C(t ) 1 3κ+µ, < t T. Then (2.12) i obtained directly from (2.14). Operating A n (t) 1 to (2.14) and letting n in the reulting equation, we obtain that U(t, τ)f (τ)dτ = A(t) 1[ A(t) U(t, τ){f (τ) F (t)}dτ + ] {A(t) U(t, τ) A(t) e (t τ)a(t) }F (t)dτ + {1 e (t )A(t) }F (t). From thi the etimate (2.13) i verified.

12 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES Quailinear evolution equation Let X be a Banach pace. We conider the Cauchy problem of a multivalued abtract evolution equation du + A(u)u F (u), < t T, (E) dt u() = u in X. Let Z be a econd Banach pace continuouly embedded in X and let K be an open ball of Z uch that K = {u Z; u Z < R}, < R <. For each u K, A(u) i a multivalued linear operator of X with domain D(A(u)) D which i contant in u. F i a nonlinear operator from K to X. u i an initial value in K. We make the following aumption. The pectral et σ(a(u)) i contained in a fixed open ectorial region σ(a(u)) Σ = {λ C; arg λ < φ}, where < φ < π 2, and the reolvent atifie (A.i) (λ A(u)) 1 M L(X) ( λ + 1) 1 κ, λ Σ, u K with ome exponent κ < 1 and a contant M > which are independent of u. A( ) atifie a Lipchitz condition of the form (Aii) A(u){A(u) 1 A(v) 1 }f X N u v Z f X, f X, u, v K with a contant N >. F atifie the Lipchitz condition (F) F (u) F (v) X L u v Z, u, v K with a contant L >. The pace X and Z are a follow (Sp.i) Z X with continuou embedding. There i ome exponent β (κ, 1) uch that, for every u K, D(A(u) β ) Z with the etimate (Sp.ii) ũ Z D A(u) β ũ X, ũ D(A(u) β ), u K, D > being ome contant. u K atifie a compatibility condition of the form (In) u D(A(u ) γ ) with ome exponent γ uch that β < γ 1. Finally, the exponent atify the relation (Ex) κ < β < γ 1 and 5κ + β < γ. A a matter of fact, (Ex) how that κ mut be le than 1 6. Then, the following reult i proved.

13 12 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) Theorem 3.1. Under (A.i,ii), (F), (Sp.i,ii), (In), and (Ex), there exit a unique local olution to (E) in the function pace: { u(t) D(A(u(t)) for < t Tu and u C([, T u ]; Z), (3.1) u C 1 1+κ γ du ((, T u ]; X) and t dt B((, T u ]; X), where T u > i determined by the norm u Z and A(u ) γ u X. Proof. The proof conit of everal tep. C denote a univeral contant which i determined by the exponent and the initial contant. f tand for an arbitrary element in A(u ) γ u. Step 1. For S uch that < S T, we et a Banach pace Z(S) = C([, S]; Z) and a ubet of Z(S) uch that K(S) = {u C µ ([, S]; Z); u() = u, u(t) u() up Z t 1 and up u(t) <t S µ Z D u }. t S Here, µ i ome fixed exponent o that 3κ < µ < γ β 2κ, (Ex) how that uch a µ really exit. The contant D u i fixed o that (3.2) u Z < D u < R. Clearly, K(S) i a nonempty cloed ubet of Z(S). Step 2. For each v K(S), let u conider a linear problem du (3.3) dt + A v(t)u F v (t), < t S, u() = u, where A v (t) = A(v(t)) and F v (t) = F (v(t)) for t S. It i eay to oberve that A v (t) atifie (L.A.i,ii) and (L.Ex) in Section 2 and that F v C µ ([, S]; X) and u atify (L.F) and (L.In), repectively. Therefore, there exit a unique olution to (3.3) in the pace u C([, S]; X) C 1 1+κ γ du ((, S]; X), t dt B((, S]; X), and the olution u i given by u(t) = U v (t, )u + U v (t, )F v ()d, t S, where U v (t, ) denote the evolution operator for the family of multivalued linear operator A v (t) = A(v(t)). We then arrive at defining a correpondence Φ from K(S) to Z(S) by etting Φ(v)(t) = u(t), t S, for each v K(S). Step 3. If S > i ufficiently mall, then Φ map the et K(S) into itelf. Indeed, for u = Φ(v), we write that u(t) = u + {e ta(u) 1}u + {U v (t, ) e tav() }u + U v (t, )F v ()d.

14 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 13 Then, ince u = A(u ) γ f, it i een by (2,5) that {e ta(u ) 1}u Z D A(u ) β {e ta(u ) 1}A(u ) γ f X Similarly, by Propoition 2.3, C {e ta(u ) 1}A(u ) β γ f X Ct γ β κ f X. {U v (t, ) e ta v() }u Z D A v (t) β {U v (t, ) e ta v() } Finally, by (2.6), U v (t, )F v ()d D Z Therefore we obtain by definition (3.2) that A v () γ f X Ct γ β 3κ+µ f X. {A v (t) β } U v (t, )F v ()d Ct 1 β κ. X u(t) Z u Z + C(S γ β 3κ+µ + S 1 β κ )( A(u ) γ u X + 1) D u, provided that S > i ufficiently mall. Note that f denote an arbitrary element of A(u ) γ u. We next fix an exponent ϕ o that and notice that β + 3κ < β + κ + µ < ϕ < γ κ 1 κ, A v (t) ϕ u(t) {A v (t) ϕ } U v (t, )A v () γ f + {A v (t) ϕ } U v (t, )F v ()d = g v (t). By (2.6) and Propoition 2.2, g v (t) i hown to be uniformly bounded with (3.4) g v (t) X C ϕ ( A(u ) γ u X + 1), t S. Uing g v (), we can write that u(t) u() = {U v (t, ) 1}u() + U v (t, τ)f v (τ)dτ = [ {U v (t, ) e (t )Av() }A v () ϕ + {e (t )Av() 1}A v () ϕ] g v () Then, by Propoition 2.3, it i een that + U v (t, τ)f v (τ)dτ, < t S. {A v (t) β } {U v (t, ) e (t )Av() }A v () ϕ L(X) C ϕ (t ) ϕ β 3κ+µ. Similarly, by (2.5) and (2.6), A v () β {e (t )Av() 1}A v () ϕ L(X) C ϕ (t ) ϕ β κ, {A v (t) β } U v (t, τ)f v (τ)dτ C(t ) 1 β κ. X

15 14 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) Therefore, in view of (3.4), we oberve that u(t) u() Z C ϕ (S ϕ β 3κ + S ϕ β κ µ + S 1 β κ µ )( A(u ) γ u ) X + 1)(t ) µ. Therefore, in view of the definition of ϕ, we conclude that u(t) u() Z (t ) µ, provided S > i ufficiently mall. Step 4. If S > i ufficiently mall, then the mapping Φ : K(S) K(S) i a contraction with repect to the norm Z(S). Indeed, for u i = Φ(v i ), v i K(S), i = 1, 2, we have u 1 (t) u 2 (t) = {U v1 (t, ) U v2 (t, )}u + {U v1 (t, ) U v2 (t, )}F v2 ()d + Here we etablih the following lemma. Lemma 3.1. We have A v1 (t) β {U v1 (t, ) U v2 (t, )}u X U v1 (t, ){F v1 () F v2 ()}d. Ct γ β 3κ A(u ) γ u X v 1 v 2 Z(S), t S, and A v 1 (t) β {U v1 (t, ) U v2 (t, )}F v2 ()d X Ct 1 β 3κ F v2 C µ ([,t]; X) v 1 v 2 Z(S), t S. Proof. In order to verify thee fundamental reult, we have to employ the evolution operator U vi,n(t, ) (i = 1, 2) for the familie of the Yoida approximation A vi,n(t) (i = 1, 2) of A vi (t). Indeed we oberve that (3.5) A v1,n(t) β {U v1,n(t, ) U v2,n(t, )}A v2,n() γ = A v1,n(t) β U v1,n(t, ) A v1,n(){a v1,n() 1 A v2,n() 1 }A v2,n()u v2,n(, )A v2,n() γ d. By the ame argument a for (2.2), we can how by (2.7) that A v1,n(t) β U v1,n(t, )J v1,n()a v1 () {A v1 () 1 A v2 () 1 }f X C A v1,n(t) β U v1,n(t, )J v1,n() L(X) A v1 (){A v1 () 1 A v2 () 1 f X C(t ) β 2κ v 1 () v 2 () Z f X, f X. where A v1 () A v1 () i an arbitrary ection. Therefore, (3.6) {A v1 (t) β } U v1 (t, )A v1 () {A v1 () 1 A v2 () 1 }f X C(t ) β 2κ v 1 () v 2 () Z f X, f X.

16 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 15 Operating A v1,n(t) β to (3.5) and letting n in the reulting equality, we obtain that {U v1 (t, ) U v2 (t, )}u = A v1 (t) β {A v1 (t) β } U v1 (t, ) Thi yield that A v1 () {A v1 () 1 A v2 () 1 }A v2 ()U v2 (, )A v2 () γ f d. A v1 (t) β {U v1 (t, ) U v2 (t, )}u X C (t ) β 2κ γ 1 κ d v 1 v 2 Z(S) f X. Since f A(u ) γ u i arbitrary, we obtain the firt etimation. Next, we can write that A v1,n(t) β {U v1,n(t, ) U v2,n(t, )}F v2 ()d = A v1,n(t) β U v1,n(t, τ) A v1,n(τ){a v1,n(τ) 1 A v2,n(τ) 1 }A v2,n(τ)u v2,n(τ, )F v2 ()dτd = A v1,n(t) β U v1,n(t, τ)a v1,n(τ){a v1,n(τ) 1 A v2,n(τ) 1 } A v2,n(τ) τ U v2,n(τ, )F v2 ()ddτ. From (2.12), A τ v2,n(τ) U v 2,n(τ, )F v2 ()d atifie the uniform etimate τ A v 2,n(τ) U v2,n(τ, )F v2 ()d Cτ κ F v2 C µ ([,S]; X), X and converge a n to a continuou function g(τ) on (, S]. Then, we obtain in the ame way a above that {U v1 (t, ) U v2 (t, )}F v2 ()d = A v1 (t) β {A v1 (t) β } U v1 (t, τ)a v1 (τ) {A v1 (τ) 1 A v2 (τ) 1 }g(τ)dτ. Therefore, A v 1 (t) β {U v1 (t, ) U v2 (t, )}F v2 ()d C X Hence we verify the econd etimate of the lemma. (t τ) β 2κ τ κ dτ F v2 Cµ ([,S]; X) v 1 v 2 Z(S). Let u now complete the proof of thi Step. It i eay to ee that A v 1 (t) β U v1 (t, ){F v1 () F v2 ()}d Ct 1 β κ v 1 v 2 Z(S). X

17 16 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) Thi together with the lemma then yield that u 1 (t) u 2 (t) Z CS γ β 3κ ( A(u ) γ u X + 1) v 1 v 2 Z(S), t S. Hence, we have verified that Φ i a contraction, provided S > i ufficiently mall. Step 5. Take a T u = S > in uch a way that the reult of Step 3 and 4 are valid. Then, there exit a unique fixed point u K(S) of Φ. Since u atifie the formula u(t) = U u (t, )u + U u (t, )F u ()d, t S, u i hown to be a olution to (E) which belong to the pace (3.1). Step 6. Finally we verify the uniquene of olution. Let u be the olution contructed above. We conider the Yoida approximation A u,n (t) of the operator A u (t) and the evolution operator U u,n (t, ) correponding to A u,n (t). Let ũ be any other olution to (E) in the pace (3.1). Then, for < t < S ( T u ), U u,n(t, )ũ() = U u,n (t, ){A u,n ()ũ() g()} + U u,n (t, )Fũ(), < < t, where g() Aũ()ũ() with dũ d + g() = F ũ(). Integrating thi identity in (, t) and operating A u,n (t) β to the reulting one yield that A u,n (t) β {ũ(t) u n (t)} = A u,n (t) β U u,n (t, ) A u,n (){Aũ() 1 A u,n () 1 } g()d + A u,n (t) β U u,n (t, ){Fũ() F u ()}d, where u n (t) = U u,n (t, )u + U u,n(t, )F u ()d. We are concerned with the limit a n. By the ame method a in Step 4, it i in fact verified that (3.7) ũ(t) u(t) = A u (t) β[ {A u (t) β } U u (t, )A u () {Aũ() 1 A u () 1 } g()d + Moreover, we oberve the following fact. ] {A u (t) β } U u (t, ){Fũ() F u ()}d lim n n 1 U u,n (t, )A u,n () g()d. Lemma 3.2. lim n n 1 U u,n (t, )A u,n () g()d =.

18 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 17 Proof. For any ρ (, µ κ), we have by (2.8) that U u,n (t, )A u,n () L(X) U u,n (t, )A u,n () ρ L(X) A u,n () 1 ρ L(X) C ρ (t ) ρ κ A u,n () 1 ρ L(X) C ρn (1+κ)(1 ρ) (t ) ρ κ. In addition, from (3.1), g atifie that g() X Cũ γ κ 1. Therefore, U u,n (t, )A u,n () g()d C ρ Cũn (1+κ)(1 ρ) X (t ) ρ κ γ κ 1 d C ρ Cũn (1+κ)(1 ρ) t γ ρ 2κ. κ It then uffice to take a ρ o that 1+κ < ρ < µ κ. Since κ 1+κ + κ 3κ (< µ) for κ < 1 6, it i clearly poible to take uch a ρ. In view of (3.6), we verify from (3.7) that A u (t) β {ũ(t) u(t)} X CũS γ β 3κ ũ u Z(S), t S. Thi in turn how that ũ(t) = u(t) for all t [, S] if S > i ufficiently mall. A a matter of fact, we have hown by thi argument that the et {S (, T U ]; ũ(t) = u(t) for all t [, S]} i nonempty and open in (, T U ]. On the other hand, it i clear that the et i cloed. Therefore, ũ(t) = u(t) for all t [, T U ]. Remark 3.1. A hown in the proof, T u i determined by the norm A(u ) γ u X = inf{ f X ; f A(u ) γ u }. Thi then mean that the global exitence of olution to (E) will be etablihed if we can verify a priori etimate u(t) Z < R and A(u(t)) γ u(t) X C for every local olution. 4. Degenerate Abtract Evolution Equation We conider the Cauchy problem of a degenerate abtract evolution equation dmv + L(Mv)v = F (Mv), < t T, (D.E) dt Mv() = u in a Banach pace X. Let Z X be the econd Banach pace continuouly embedded in X and K be a bounded ubet of Z uch that K = {u Z; u Z < R}, < R <. For each u K, L(u) i a denely defined cloed linear operator of X with contant domain D(L(u)) D L. M i a cloed linear operator of X with domain D(M) D L, and M map D L into Z. F i a nonlinear operator from K into X. u K i an initial value of the problem. v = v(t) i the unknown function.

19 18 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) We make the following aumption. For every u K, the M-modified pectral et σ M (L(u)) i contained in a fixed open ectorial region σ M (L(u)) Σ = {λ C; arg λ < φ}, where < φ < π 2, and the M-modified reolvent atifie (D.A.i) M(λM L(u)) 1 C L(X) ( λ + 1) 1 κ, λ Σ, u K with ome exponent κ < 1 and a contant C > which are independent of u. The M-modified reolvent alo atifie (D.A.ii) M(λM L(u)) 1 C L(X,Z) ( λ + 1) 1 ρ, λ, u K with ome exponent κ ρ < 1 and a contant C > independent of u. L(u) atifie the Lipchitz condition (D.A.iii) L(u){L(u) 1 L(ũ) 1 } L(X) C u ũ Z, u, ũ K with ome contant C >. F alo atifie the Lipchitz condition (D.F) F (u) F (ũ) X C u ũ Z, u, ũ K with ome contant C >. We et u(t) = Mv(t) and rewrite (D.E) in the form du (4.1) dt + L(u)M 1 u F (u), < t T, u() = u. Here, L(u)M 1 = A(u) i a multivalued linear operator defined for u K with the contant domain D(A(u)) = M(D L ). Our goal i then to apply the Theorem 3.1 to the preent Cauchy problem. According to [3, Theorem1.14], if λ σ M (L(u)), then λ σ(a(u)), and it hold that M(λM L(u)) 1 = (λ A(u)) 1, λ σ M (L(u)). Therefore, (D.A.i) yield that For u, ũ K, (λ A(u)) 1 L(X) C, λ Σ. ( λ + 1) 1 κ {A(u) 1 A(ũ) 1 }f = M{L(u) 1 L(ũ) 1 }f D(A(u)), f X. In addition, L(u){L(u) 1 L(ũ) 1 }f A(u){A(u) 1 A(ũ) 1 }f. Therefore, it follow from (D.A.iii) that A(u){A(u) 1 A(ũ) 1 }f X L(u){L(u) 1 L(ũ) 1 }f X C u ũ Z f X, f X. Hence, (A.i,ii) in the preceding ection have been verified.

20 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 19 For κ < β < 1 it i known that A(u) β f = From (D.A.ii) it i een that in πβ π (λ + A(u)) 1 L(X,Z) Therefore, for any β uch that ρ < β < 1, A(u) β f Z 1 π Setting ũ = A(u) β f, we oberve that λ β (λ + A(u)) 1 fdλ, f X. C, λ. (λ + 1) 1 ρ λ β (λ + 1) ρ 1 dλ f X C f X. ũ Z C f X. For a given ũ D(A(u) β ), thi i true for any f A(u) β ũ. Hence, (Sp.ii) i fulfilled with any β (ρ, 1). For (In), we aume that u K and u atifie a compatibility condition of the form (D.In) u D({L(u )M 1 } γ ) with ome exponent < γ 1. For the exponent we aume the relation (D.Ex) κ ρ < γ 1 and 5κ + ρ < γ. It i then poible to take the exponent β in uch a way that (Sp.ii) and (Ex) hold. We have thu found out the condition to be aumed to apply Theorem 3.1 and obtained the main reult of the paper. Theorem 4.1. Under (D.A.i,ii,iii), (D.F), (D.Ex), and (D.In), there exit a unique local olution to (D.E) in the function pace { Mv C([, Tu ]; Z) C 1 ((, T u ]; X), v C((, T u ]; D L ), t 1+κ γ L(Mv)v B((, T u ]; X), where T u > i determined by the norm u Z and {L(u )M 1 } γ u X. 5. Quailinear elliptic-parabolic equation A an application of our abtract reult, we hall conider the Cauchy problem of a quailinear elliptic-parabolic equation of the form m(x)v = {a(x, m(x)v) v} + f(x, m(x)v) in (, T ] Ω, t (5.1) v = on (, T ] Ω, m(x)v(x, ) = u (x) in Ω

21 2 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) in a bounded region Ω R n of C 2 cla. In thi paper we handle the cae where n = 1, 2, and 3. m(x) i a nonnegative function uch that (5.2) m(x) C 1 (Ω) when n = 1, m(x) C 2 (Ω) when n = 2, 3. a(x, u) i a real valued mooth function defined for (x, u) Ω (R + ir), and it i aumed for each < R < to atify (5.3) a(x, u) δ R > for all x Ω and u uch that u R with ome poitive contant δ R. f(x, u) i a mooth function of (x, u) Ω (R + ir). For the initial function, we aume that { H 1 (Ω) when n = 1, (5.4) u = m(x)v with ome v H 2 (Ω) H 1 (Ω) when n = 2, 3. Cae when n = 1. In thi cae we formulate (5.1) in the pace X = H 1 (Ω). We take a Z the Sobolev pace H 1 2 +ε (Ω), where ε i an exponent arbitrarily fixed o that ε (, 1 2 ). Let u Z and u Z < R <, then K i taken a K = {u Z; u Z < R}. By the embedding theorem, K i a bounded et of C(Ω). For u K, the equilinear form a(u; w 1, w 2 ) = a(x, u(x)) w 1 (x) w 2 (x) dx, w 1, w 2 H 1 (Ω) Ω i defined. According to the Lax-Milgram theorem (ee e. g. [13, Chap. 2,Thm. 9.1]), thi equilinear form determine under (5.3) a cloed linear operator L(u) in H 1 (Ω) with the domain D(L(u)) = H 1 (Ω) = D L which i in fact an iomorphim from D L to X. Implicitly, L(u) i the differential operator {a(x, u) w}. We define M a the multiplication operator of the function m(x), in view of (5.2), M i a bounded linear operator on both X and H 1 (Ω), that i M L(X) L(H 1 (Ω)). F (u) i defined by F (u) = f(x, u(x)), u K. In thi way, (5.1) i written a the Cauchy problem of an abtract equation of the form (D.E) in X. Let u now verify all the aumption (D.Ai,ii,iii) and (D.F) in Section 4. It i already known by [3, Example 6.3, (6.7)] that (D.A.i) i fulfilled with a uitable ectorial domain Σ and κ =. In order to verify (D.A.ii), we ue the interpolation property that H 1 2 +ε (Ω) = [L 2 (Ω), H 1 (Ω)] 1 2 +ε. Then, M(λM L(u)) 1 L(X,Z) C M(λM L(u)) ε L(X,H 1 ) M(λM L(u)) ε L(X,L 2 ). But, from [3, (6.6) and (6.8)] it i known that (λm L(u)) 1 L(X,H1 ) C, λ Σ, u K, M(λM L(u)) 1 C L(X,L 2 ), λ Σ, u K, ( λ + 1) 1 2

22 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 21 therefore we obtain that (5.5) M(λM L(u)) 1 L(X,Z) C( λ + 1) ε 2 1 4, λ Σ, u K. Thi how that (D.A.ii) i valid with ρ = ε 2. Verification of (D.A.iii) i very eay. Indeed, we have L(u){L(u) 1 L(ũ) 1 }f, w H 1 H 1 = {L(ũ) L(u)}L(ũ) 1 f, w H 1 H 1 = {a(x, ũ(x)) a(x, u(x))} L(ũ) 1 f w dx, f X, w D L. Therefore, ince Ω a(x, ũ) a(x, u) C C ũ u C C ũ u Z, ũ, u K, (D.A.iii) follow immediately. Verification of (D.F) i alo very eay. Finally, (5.4) implie that u belong to D(L(u )M 1 ), that i u atifie (D.In) with γ = 1. (D.Ex) i then fulfilled with κ =, ρ = ε 2, and γ = 1. We have thu hown that, under (5.2), (5.3) and (5.4), we can apply the Theorem 4.1 to the problem (5.1). Cae when n = 2, 3. In thi cae, we take a X the pace L 2 (Ω) and a Z the Sobolev pace H n 2 +ε (Ω), where ε i an exponent arbitrarily fixed o that ε (, 1 2 ). Let u Z and u Z < R <. Then K i taken a K = {u Z; u Z < R}. K i a bounded et of C(Ω). For u K, the linear operator L(u) i defined by L(u)w = {a(x, u(x)) w} + cw with D(L(u)) = H 2 (Ω) H 1 (Ω), where c i ome ufficiently large contant for which all the argument below are true. L(u) i a poitive definite elf-adjoint operator of X, the domain D(L(u)) D L being independent of u. The following etimate alo hold w H 2 C L(u)w L 2, w D L, u K. M i a multiplication operator of m(x); in view of (5.2), M i een to be a bounded linear operator on both X and H 2 (Ω). F (u) i defined by F (u) = cu+f(x, u), u K. Then (5.1) i formulated a the Cauchy problem of an abtract equation of the form (D.E) in X. In the preent cae we have to aume in addition to (5.2) the following order condition (5.6) m(x) Cm(x) ζ, x Ω with ome uitable exponent ζ [, 1) which will be pecified below. A hown in [3, Example 6.3], condition (5.6) yield that M(λM L(u)) 1 L(X) C( λ + 1) 1 2 ζ, λ Σ, u K with ome uitable ectorial domain Σ, < φ < π 1 ζ 2. Therefore, with κ = 2 ζ, (5.7) (5.8) M(λM L(U)) 1 L(X) C( λ + 1) κ 1, λ Σ, u K, (λm L(u)) 1 L(X,H 2 ) C( λ + 1) κ, λ Σ, u K.

23 22 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) By the interpolation property that Z = H n 2 +ε (Ω) = [L 2 (Ω), H 2 (Ω)] n 4 + ε 2, it follow that M(λM L(u)) 1 L(X,Z) (5.7) and (5.8) yield that C M(λM L(u)) 1 1 n 4 ε 2 L(X,X) M(λM L(u)) 1 n 4 + ε 2 L(X,H 2 ). M(λM L(u)) 1 L(X,Z) C( λ + 1) 1+ n 4 + ε 2 +κ, λ Σ, u K. Therefore, (D.A.ii) i fulfilled with ρ = n 4 + ε 2 + κ. (D.A.iii) i verified directly from L(u){L(u) 1 L(ũ) 1 }f = {L(ũ) L(u)}L(ũ) 1 f = {a(x, u) a(x, ũ)} L(ũ) 1 f + {a(x, u) a(x, ũ)} L(ũ) 1 f. Note that the following etimate a(x, u) a(x, ũ) Z C u ũ Z, u, ũ K i verified by uing the theory of Sobolev pace (cf. [7]). (D.F) i alo verified immediately. (5.4) implie that u D(L(u )M 1 ), that i (D.In) i valid with γ = 1. Therefore, by imple calculation, (D.Ex) i hown to be valid, provided that (5.9) 16+2n+4ε 2+n+2ε < ζ < 1, n = 2, 3. Thu, under (5.2), (5.3), (5.4), (5.6), and (5.9), Theorem 4.1 i applicable to the problem (5.1). Remark 5.1. According to Favini et al. [4], (D.A.i) i valid even in the pace L p (Ω), 1 < p <. If we utilize thee reult, it i equally poible to handle the problem (5.1) in L p pace. 6. Quailinear Elliptic-Parabolic Sytem In thi ection let u conider an elliptic-parabolic ytem of the form u t = { a(x, u) } u + b(x, u) x x x v + f(x, u) in (, T ] Ω, (6.1) = { c(x, u) } u + d(x, u) x x x v + g(x, u) in (, T ] Ω, u = v = on (, T ] Ω, u(x, ) = u (x) in Ω in a bounded open interval Ω = (, l). a(x, u), b(x, u), c(x, u), and d(x, u) are all real valued mooth function of variable (x, u) Ω (R + ir). It i aumed that, for each < R <, there exit

24 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 23 ome poitive contant δ R > dependent on R uch that the following etimate hold: (6.2) a(x, u)ξ 2 + (b(x, u) + c(x, u))ξη + d(x, u)η 2 δ R (ξ 2 + η 2 ), ξ, η R for all x Ω and u uch that u R. f(x, u) and g(x, u) are given mooth function of variable (x, u) Ω (R + ir). Initial value u i aumed to atify (6.3) u H 1 (Ω). We intend to formulate the problem (6.1) in a product pace {( ) } f X = ; f, g H 1 (Ω). g A Z we take the pace Z = {( ) } u ; u H 1 2 +ε (Ω), where ε i an arbitrarily fixed exponent o that ε (, 1 2 ), in view of the fact that H 1 2 +ε (Ω) C(Ω). Then K i taken a K = {( u ) Z; u 1 H +ε 2 } < R with ome fixed < R < uch that u 1 H +ε 2 < R, K being a bounded ubet ( u of C(Ω). For U = K, a linear operator L(U) acting in X i defined by ) ( ) (ũ ) (ũ ) Dx {a(x, u)d L(U)V = x } D x {b(x, u)d x }, V =, D x {c(x, u)d x } D x {d(x, u)d x } ṽ ṽ where D x = x, with the domain F (U) : K X i defined by {(ũ D(L(U)) D L = ṽ F (U) = ( ) f(x, u), U = g(x, u) ) ; ũ, ṽ H 1 (Ω) ( ) u K. Finally, M i defined a the projection on X uch that ( ) ( ) ( ) f f f (6.4) M =, X. g g (6.5) }. Obviouly, M map D L into Z. In thi way we are led to the following abtract formulation of (6.1) + L(MV )V = F (MV ), < t T, dmv dt MV () = U = ( u )

25 24 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) in the pace X. Let u how that the Theorem 4.1 i applicable to thi Cauchy problem. In order to verify (D.A.i) we have to conider equilinear form on D L : { A λ (U; V 1, V 2 ) = a(x, u)dx ũ 1 D x ũ 2 + b(x, u)d x ṽ 1 D x ũ 2 where U = Ω } + c(x, u)d x ũ 1 D x ṽ 2 + d(x, u)d x ṽ 1 D x ṽ 2 dx ) ) (ũ1 (ũ2 λ ũ 1 ũ 2 dx, V 1 =, V 2 = D L, Ω ( ) u K and λ C. It i immediate to ee that (6.6) A λ (U; V 1, V 2 ) C( λ + 1) V 1 DL V 2 DL, V 1, V 2 D L. In addition, we verify that ReA λ (U; V, V ) = Reλ ũ 2 { dx + a(x, u) Dx ũ 2 + d(x, u) D x ṽ 2 Ω Ω + (b(x, u) + c(x, u))(red x ũ ReD x ṽ + ImD x ũ ImD x ṽ) } dx and ImA λ (U, V, V ) = Imλ ũ 2 dx Ω + (b(x, u) c(x, u))(red x ũ ImD x ṽ ImD x ũ ReD x ṽ)dx. Ω Then, by (6.2) there exit δ > uch that { ReA λ (U; V, V ) δ( Dx ũ 2 + D x ṽ 2 ) Reλ ũ 2} dx, V D L, Ω { ImA λ (U; V, V ) Imλ ũ 2 C R ( D x ũ 2 + D x ṽ 2 ) } dx, V D L, here C R denote a contant Ω C R = ṽ 1 up b(x, u(x)) c(x, u(x)). x Ω, U K Let u introduce a parameter < θ < 1, and oberve that A λ (U; V, V ) (1 θ)rea λ (U; V, V ) + θ ImA λ (U; V, V ) ((1 θ)δ θc R )( D x ũ 2 L 2 + D xṽ 2 L 2 + (θ Imλ (1 θ)reλ) ũ 2 L 2. Then, if θ > i ufficiently mall o that (1 θ)δ θc R δ 2, and if λ i taken in uch a way that λ Σ = {λ C; arg λ < φ}, Tan 1 1 θ < φ < π θ 2, then (6.7) A λ (U; V, V ) δ ( V 2 D L + λ MV 2 L 2), V D L, U K ṽ 2

26 QUASILINEAR DEGENERATE EVOLUTION EQUATIONS IN BANACH SPACES 25 with ome poitive contant δ >. Here we ued the Poincaré inequality ũ L 2 C D x ũ L 2, ũ H 1 (Ω). In view of (6.6) and (6.7), we can now apply the Lax-Milgram theorem (ee e. g. [13, Chap. 2,Thm. 9.1,Rem. 9.3]). Let λ Σ, then for a given F X, the problem A λ (U; V, W ) = F, W X DL, W D L ha a unique olution V D L with the etimate (6.8) (6.9) Since V DL (1/δ ) F X, λ 1 2 MV L 2 (1/δ ) F X. A λ (U; V, W ) = (λm L(U))V, W X DL, W D L, it follow that for a given F X, the problem (λm L(U))V = F ha a unique olution with the etimate (6.8) and (6.9). Thi then mean that λ ρ M (L(U)) and the reolvent atifie the etimate (6.1) (6.11) (λm L(U)) 1 F DL (1/δ ) F X, F X, λ 1 2 (λm L(U)) 1 F L 2 (1/δ ) F X, F X. In thi way we have verified that σ M (L(U)) Σ, U K. Therefore it uffice to verify the etimate (D.A.i). But by (6.1) it i now een that λm(λm L(U)) 1 F X = F + L(U)(λM L(U)) 1 F X F X + C (λm L(U)) 1 F DL C(1 + (1/δ )) F X, F X. Hence, (D.A.i) i fulfilled with the domain Σ determined above and κ =. To verify (D.A.ii) we ue the interpolation property that H 1 2 +ε (Ω) = [L 2 (Ω), H 1 (Ω)] 1 2 +ε. Then, by the ame argument a for (5.5), (6.1) and (6.11) yield that (λm L(U)) 1 L(X,Z) C( λ + 1) ε 2 1 4, λ Σ. Hence, (D.A.ii) i valid with ρ = ε 2. (D.A.iii) i verified directly a in the previou ection by uing the expreion L(U 1 ){L(U 1 ) 1 L(U 2 ) 1 } = {L(U 2 ) L(U 1 )}L(U 2 ) 1 ( ) Dx {[a(x, u = 1 ) a(x, u 2 )]D x } D x {[b(x, u 1 ) b(x, u 2 )]D x } L(U D x {[c(x, u 1 ) c(x, u 2 )]D x } D x {[d(x, u 1 ) d(x, u 2 )]D 2 ) 1, x } ( ) ui U i = K (i = 1, 2).

27 26 ANGELO FAVINI ( ) AND ATSUSHI YAGI ( ) (D.F) i alo verified directly. Since D(L(U )M 1 ) = M(D L ), (6.3) together with (6.4) implie that the initial value U in (6.5) belong to D(L(U )M 1 ). Hence, (D.In) i valid with γ = 1. (D.Ex) i alo fulfilled a well. In thi way we conclude that, under (6.2) and (6.3), Theorem 4.1 i applied to (6.5) to obtain the exitence and uniquene of local olution. Reference [1] C. Bardo and H. Brezi, Sur une clae de problème d évolution nonlinéaire, J. Diff. Eq. 6, 1969, [2] H. Brezi, On ome degenerate nonlinear parabolic equation, Nonlinear Functional Analyi, Proc. Symp. Pure Math. 18, 197, [3] A. Favini and A. Yagi, Multivalued linear operator and degenerate evolution equation, Ann. Mat. Pura. Appl. (IV) 163, 1993, [4] A.Favini, A. Lorenzi, H. Tanabe and A. Yagi, An L p -approach to ingular linear parabolic equation in bounded domain, Preprint. [5] A. Kuttler, A degenerate nonlinear Cauchy problem, Applicable Analyi 13, 1982, [6] A. Kuttler, Implicit evolution equation, Appl. An. 16, 1983, [7] K. Oaki, T. Tujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxigrowth ytem of equation, Nonlinear Analyi 51, 22, [8] P. E. Sobolevkii, Equation of parabolic type in Banach pace, Amer. Math. Soc. Tran. Ser. 2 49, 1966, [9] A. Yagi, Abtract quailinear evolution equation of parabolic type in Banach pace, Boll. Un. Mat. Ital. 5-B, 1991, [1] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problem, 1976, Academic Pre, London, New York. [11] A. Favini and A. Yagi, Degenerate Differential Equation in Banach Space, 1999, Marcel Dekker, New York. [12] A. Friedman, Partial Differential Equation, 1969, Holt, Rinehart and Winton, New York. [13] J. L. Lion et E. Magene, Problème aux limite non homogène et Application, vol. 1, 1968, Dunod, Pari. [14] Yu. N. Rabotnov, Polzuchet Elementov Kontruktii (Creep of contructional element), 1967, Nauka, Mocow. [15] W. Shockley, Electron and Hole in Semiconductor, 195, D. Van Notrand, Princeton, New Jerey. [16] R. E. Showalter, Monotone Operator in Banach Space and Nonlinear Partial Differential Equation, Math. Survey & Monograph 49, 1997, AMS, Providence. [17] H. Tanabe, Functional Analytic Method for Partial Differential Equation, 1997, Marcel Dekker, New York. (*) Department of Mathematic, Univerity of Bologna, Piazza di Porta S. Donato 5, 4126 Bologna, Italia (**) Department of Applied Phyic, Oaka Univerity, Suita, Oaka , Japan