Semigroups and Linear Partial Differential Equations with Delay

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1 Journal of Mathematical Analysis and Applications 264, 1 2 (21 doi:1.16/jmaa , available online at on Semigroups and Linear Partial Differential Equations with Delay András Bátkai 1 and Susanna Piazzera 2 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 1, 7276 Tübingen, Germany anba@michelangelo.mathematik.uni-tuebingen.de; supi@michelangelo.mathematik.uni-tuebingen.de Submitted byirena Lasiecka Received June 23, 1999 We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, eponential stability and norm continuity of the solutions. Applications to a reaction diffusion equation with delay are given. 21 Elsevier Science KeyWords: C -semigroups; delay equations; eponential stability. 1. INTRODUCTION Partial differential equations with delay have been studied for many years. In an abstract way and using the standard notation (see [13, 26], they can be written as u t =Au t + u t t (DE u = u = f 1 Supported by the Konrad-Adenauer-Stiftung. 2 Supported by the Consiglio Nazionale delle Ricerche X/1 $ Elsevier Science All rights reserved

2 2 bátkai and piazzera on a Banach space X, where A D A is a (unbounded linear operator on X and the delay operator belongs to L W 1 p 1 X X. J. Hale [1] and G. Webb [24] were among the first who applied semigroup methods to the study of such equations; we refer to [21, 26] for more recent references. As a first step one needs to choose an appropriate state space. One possibility is to work in the space of continuous X-valued functions. In this case, the relationship between solutions of the delay equations (DE and a corresponding semigroup has been widely studied (see, for eample, [7, Sect. VI.6] and is well understood. On the other hand, the state space E = X L p 1 X turns out to be a better choice with regard to certain applications (e.g., to control theory, see [16] and will be the one used in this paper. We show in Section 2 that the linear partial differential equation with delay is equivalent to an abstract Cauchy problem, (ACP t = t = ( f t on the space E. In particular, the one-to-one correspondence between solutions of the delay equation (DE and solutions of the abstract Cauchy problem (ACP is shown. In the case where the operator generates a strongly continuous semigroup on E, the fact that the trajectories of the semigroup give solutions of the DE has been proved by G. Webb [25] in the nonlinear case with X a Hilbert space, and by J. A. Burns et al. [2] for neutral equations when X = n. The result in [2] was then improved by F. Kappel and K. Zhang [11], who proved the equivalence of the well-posedness of (DE and (ACP by using Laplace transform techniques (always in the case of neutral equations and X = n. In the third section we give sufficient conditions which ensure that the operator D generates a strongly continuous semigroup on the space E. Assuming that A D A is the generator of a strongly continuous semigroup on X, we show that the operator is of the form + B, where D generates a strongly continuous semigroup on E and B is - bounded. Moreover, we give a condition on the operators A and which implies that B is a Miyadera perturbation of and thus, generates a strongly continuous semigroup on E. Finally, for 1 < p <, we show that if the operator is given by the Riemann Stieltjes integral of a function of bounded variation

3 partial differential equations with delay 3 η 1 L X, then B is a Miyadera perturbation of for every generator A D A and = + B generates a strongly continuous semigroup on E. These results were inspired, on the one side, by a paper of K. Kunisch and W. Schappacher [13] containing necessary conditions for to generate a strongly continuous semigroup on E and, on the other side, by a work of K.-J. Engel [6] who used the theory of unbounded operator matrices and perturbation results for strongly continuous semigroups to show that in the case of a bounded operator A, the abstract Cauchy problem (ACP is well-posed for all L W 1 p 1 X X, i.e., always generates a strongly continuous semigroup on the space E. Finally, we remark that it is also possible to transform a partial differential equation with delay into an evolutionary integral equation (see J. Prüß [2, Corollary I.1.4]. In the fourth section we prove that the stability of the equation without delay persists in the equation with delay. We characterize the resolvent and use the theorem of Gearhart. This part is a generalization of a recent result of A. Fischer and J. van Neerven [8]. In [4, Theorem 5.1.7] one finds a similar stability result for the finite dimensional case. The resolvent characterization obtained turns out to be the same as in the well-known case of the state space of continuous functions. In the case where certain spectral mapping theorems are available, for eample, if the delay semigroup is eventally norm continuous, then we may use characteristic equation techniques as developed in [26, Chap. 3.1] where A D A is the generator of a compact semigroup. This, however, turns out to be etremly complicated, ecept in some special applications. In the case where no spectral mapping theorems are known, we cannot use characteristic equation techniques. We demonstrate the efficiency of our stability result in an eample where the delay is etremly complicated. Other applications, for eample, to wave equations, are the subject of further research by the authors. In the last section we give a sufficient condition for the eventual norm continuity of the semigroup generated by the operator D. We show that if A D A generates an immediately norm continuous semigroup on X and satisfies a certain technical condition, then D generates a strongly continuous semigroup which is norm continuous for t>1. We use these results to obtain eponential stability of the solutions. Finally, we illustrate our results with a reaction diffusion equation with delay. The authors anticipate that this approach can be etended with appropriate modifications to a large class of equations, where the operator in the delay term is not bounded. Etensions to nonlinear equations seem to be impossible at the present time due to the incompletness of the theory of nonlinear semigroups.

4 4 bátkai and piazzera Consider the equation where 2. THE SEMIGROUP APPROACH (DE u t =Au t + u t u = u = f X, X is a Banach space, t A D A X X is a closed and densely defined linear operator, f L p 1 X, 1 p<, W 1 p 1 X X is a bounded linear operator, u 1 X and u t 1 X is defined by u t σ = u t + σ for σ 1. We say that a function u 1 X is a (classical solution of the DE if (i u C 1 X C 1 X, (ii u t D A and u t W 1 p 1 X for all t, (iii u satisfies (DE for all t. It is now our purpose to investigate the eistence and uniqueness of the solutions of (DE. To do this we introduce the Banach space E = X L p 1 X and the operator A = d (1 dσ with domain { ( D = f D A W 1 p 1 X f =} (2 Lemma 2.1. The operator D is closed and densely defined. The proof of Lemma 2.1 is straightforward from the definition and is omitted. Remark 2.1. A necessary condition for (DE to have a solution is that u = f W 1 p 1 X and u = D A, i.e., ( f D.

5 partial differential equations with delay 5 In the following, we will see that equation (DE and the abstract Cauchy problem associated to the operator D t = t t (ACP = ( f are equivalent, i.e., (DE has a unique solution for every ( f D and the solutions depend continuously on the initial values if and only if (ACP is well-posed (in the sense of [7, Definition II.6.8]. Proposition 2.1. Let ( f D and let u 1 X be a solution of the DE. Then the map + t ( u t u E t is a classical solution of the abstract Cauchy problem ( ACP associated to the operator D with initial value ( f. Proof. Since the function u is a solution of (DE, we have u C 1 X, u t D A, u t W 1 p 1 X and u t =Au t + u t for all t, u =, and finally u = f. It remains to show that the map t u t is continuously differentiable for t and that d dt u t = d dσ u t for all t. So let T>t. Then the function u 1 T can be etended to a function v W 1 p X, i.e., v is in the domain of the first derivative, which is the generator of the left shift semigroup on L p X. From the definition of the generator we have d d v t + = v t + dt dσ in L p X for t. This implies that d dt u t = d d d u t + = u t + = dt dσ dσ u t in L p 1 X and that the map + t d dσ u t L p 1 X is continuous. We now prove the converse to Proposition 2.1. Proposition 2.2. Let ( ( f D and E, t = z t v t, be a classical solution of the ACP with initial value ( f associated to the operator D. Let u 1 X be the function defined by { z t t u t = (3 f t t 1 Then u t = v t for every t and u is a solution of the DE.

6 6 bátkai and piazzera Proof. Since is a classical solution of (ACP, v C 1 + L p 1 X solves the Cauchy problem d dt v t = d dσ v t t v t =z t t (4 v =f in the space L p 1 X. We now observe that the map + t u t L p 1 X solves (4 in the space L p 1 X. Soletw t = u t v t for t. Then w is a classical solution of the Cauchy problem d dt w t = d dσ w t t w t = t (5 w = Since (5 is the abstract Cauchy problem associated to the generator of the nilpotent left shift semigroup on L p 1 X with initial value, we have that w t = for all t. Therefore t = ( u t u for all t and t u is a solution of (DE. Let ( π 1 E X be the projection onto the first component of E, i.e., π ( 1 f = for all f E. Corollary 2.1. If D is the generator of a strongly continuous semigroup ( t t on E, then the DE has a unique solution u for every f D, which is given by ( ( π1 t u t ={ f t (6 f t t 1 Proof. Since D is the generator of the strongly continuous semigroup t t, the Cauchy problem (ACP has a unique classical solution for all ( f D and t = t ( f t Then the function u defined in (6 is a solution of (DE by Proposition 2.2. Uniqueness follows from Proposition 2.1. Corollary 2.2. If D is the generator of a strongly continuous semigroup t t, the function u 1 X defined by (6 for a given ( f E satisfies the integral equation u t = + A u s ds + u s ds f t t a.e. t 1 (7

7 partial differential equations with delay 7 Proof. Let π 2 E ( L p 1 X be the projection onto the second component of E, i.e., π ( 2 f = f for all f E. Then, by Proposition 2.2 ( ( and using the density of D in E, we obtain u t = π 2 t f for all E and t. Take now the first component of the identity ( f to obtain (7. t ( ( f f = s ( f ds t A solution of the integral equation (7 is called mild solution in the literature, see for eample [17, Section 2.2]. At this point we need the appropriate terminology. Definition 2.1. We call the DE well-posed if (i for every ( f D there is a unique solution u f of (DE and (ii the solutions depend continuously on the initial values, i.e., if a sequence ( ( n f in D converges to n f D, then u n f n t converges to u f t uniformly for t in compact intervals. This well-posedness of (DE can now be characterized by the wellposedness of the abstract Cauchy problem (ACP for the operator D. Theorem 2.1. Let D be the operator defined by (1 and (2. Then the following assertions are equivalent: (i (DE is well-posed. (ii D is the generator of a strongly continuous semigroup on E. Proof. We first show (i (ii. Assume that for every ( f D equation (DE has a unique solution u. Then Proposition 2.1 yields that for every ( f D the abstract Cauchy problem (ACP has a classical solution which is unique by Proposition 2.2. It is easy to see that these solutions depend continuously on the initial values. Finally, by Lemma 2.1, D is a closed and densely defined operator. So D generates a strongly continuous semigroup on E by [7, Theorem II.6.7]. Conversely, if is a generator, we have by Corollary 2.1 that for every initial value ( f D there is a unique solution u of (DE which is given by (6. This implies that the solutions depend continuously on the initial values.

8 8 bátkai and piazzera 3. THE GENERATOR PROPERTY In the previous section we transformed the problem of solving the partial differential equation with delay (DE into the functional analytical problem, when does generate a strongly continuous semigroup on E? In the following, we will give sufficient conditions on A and such that this is true. First, we observe that we can write as the sum + B, where A = d (8 dσ with domain and D =D = { ( f D A W 1 p 1 X f =} ( B = L D E The idea now is to show first that under appropriate conditions becomes a generator and then apply perturbation results to show that the sum + B is a generator as well. The first step is quite easy. Proposition 3.1. Let A D A be the generator of a strongly continuous semigroup S t t on X. Then D generates the strongly continuous semigroup t t on E given by ( S t t = (1 T t S t where T t t is the nilpotent left shift semigroup on L p 1 X and S t X L p 1 X is defined by { S t + τ t <τ S t τ = 1 τ t Therefore, we will always assume that A D A generates a stronglycontinuous semigroup S t t on X We will see later in Remark 3.1 that this condition is necessary for the well-posedness in the case of many applications. To the perturbation B we will apply the theorem of Miyadera and Voigt (see [14, 22], which we quote from [7, Corollary III.3.16]. (9

9 partial differential equations with delay 9 Theorem 3.1. Let G D G be the generator of a strongly continuous semigroup T t t on a Banach space X and let C L D G G X satisfy CT r dr q for all D G (11 and some t >, q<1. Then G + C D G generates a strongly continuous semigroup U t t on X which satisfies U t = T t + T t s CU s ds and CU t dt q for D G and t 1 q In our situation, D generates a strongly continuous semigroup on E if there eist t > and q<1 such that B r ( ( f dr q f (12 for all ( f D. However, since B r ( t ( f dr = ( S r T r S r ( f dr = S r + T r f dr we conclude that (12 holds if and only if there eist t > and q<1 such that S r + T r f dr q ( M for all ( f D. We therefore obtain the following result. Theorem 3.2. Let A D A be the generator of a strongly continuous semigroup on X and let condition (M be satisfied. Then the operator D is the generator of a strongly continuous semigroup on E. Thus, the DE is well-posed. We now give some important eamples of to satisfy this condition (M. Eample 3.1. (a Let be bounded from L p 1 X to X. Then the perturbation B is bounded, and D is a generator on E. f

10 1 bátkai and piazzera (b Let 1 <p< and let η 1 L X be of bounded variation. Let C 1 X X be the bounded linear operator given by the Riemann Stieltjes integral f = 1 dη f for all f C 1 X (13 Since W 1 p 1 X is continuously embedded in C 1 X, defines a bounded operator from W 1 p 1 X to X. For <t<1we obtain that S r + T r f dr = r 1 r 1 t σ + t dη σ f r + σ + r dη σ S r + σ dr f r + σ d η σ dr + S r + σ d η σ dr r t +σ f s ds d η σ + f s ds d η σ 1 σ M η 1 dr σ 1/p f p d η σ + + tm η 1 1 t 1 t 1/p f p d η σ t 1/p f p d η σ +tm η 1 = ( t 1/p f p + tm η 1 where 1 p + 1 = 1, M = sup p r 1 S r and η is the positive Borel measure on 1 defined by the total variation of η. Finally we conclude that S r + T r f dr t 1/p M η 1 f p + (14 for all <t<1. Now choose t small enough such that t 1/p M η 1 < 1. Then condition (M is satisfied with q = t 1/p M η 1. (c An important special case of (b consists of the operators defined by f = n k= B kf h k, f W 1 p 1 X, where B k L X and h k 1 for k = n. Remark 3.1. We note that from the perturbation theorem of Miyadera and Voigt (see Theorem 3.1 it follows that the generator property of A D A

11 partial differential equations with delay 11 is necessary and sufficient for the well-posedness of (DE if is defined as in (13 because we can choose q small enough. Similar results for more special cases were proved by Kunisch and Schappacher (see [13, Proposition 4.2] and for a slightly different equation by Prüß (see [2, Corollary I.1.4]. 4. STABILITY In this section, we prove a stability result for the delay equation (DE etending a recent result of Fischer and van Neerven [8]. First, we recall some notation and definitions. Let = T t t be a C -semigroup of bounded linear operators on the Banach space X with generator G D G. The spectral bound of G is given by s G = Rλ λ σ G the abscissa of uniform boundedness of the resolvent of G is defined by { } s G =inf ω Rλ>ω ϱ G and sup R λ G < Rλ>ω and the uniform growth bound or type of the semigroup ω G =inf { ω M> such that T t Me ωt t } We say that is uniformlyeponentiallystable if ω G <. It is known (see [18, Sec. 1.2, 4.1] that s G s G ω G < The theorem of Gearhart (see [7, Theorem V.1.11] says that in a Hilbert space X s G =ω G (15 To find estimates for the above quantities, we first calculate the resolvent R λ and the resolvent set ρ of the operator. Given λ and ( y ( g E we are looking for f D such that ( ( ( λ A f y λ = f λf f = g Since f =, the second component of this identity is equivalent to f = ɛ λ + R λ A g (16 where ɛ λ s =e λs for s 1 and A D A is the infinitesimal generator of the nilpotent left shift semigroup T t t on L p 1 X whose spectrum is empty. Hence, has to satisfy the equation λ A ɛ λ Id = R λ A g + y (17 This leads to the following lemma (see also [6, 17].

12 12 bátkai and piazzera Lemma 4.1. For λ we have λ ρ if and only if λ ρ A + ɛ λ Id. Moreover, for λ ρ the resolvent R λ is given by ( R λ A + ɛ λ Id R λ A + ɛ λ Id R λ A ɛ λ R λ A + ɛ λ Id ɛ λ R λ A + ɛ λ Id + Id R λ A (18 Proof. Let λ ρ A + ɛ λ Id. Then the matri in (18 is a bounded operator from E to D defining the inverse of λ. Conversely, if λ ρ, then for every ( y g E there eists a unique D such that (16 and (17 hold. In particular, for g = and for ( f every y X, there eists a unique D A such that λ A ɛ λ Id = y This means that λ A ɛ λ Id is invertible, i.e., λ ρ A ɛ λ Id. We now assume the well-posedness conditions from the previous section. In particular, we will assume that A D A generates a strongly continuous semigroup on X, that p>1 and that satisfies the assumption (13 of Eample 3.1(b, i.e., f = 1 dη f where η 1 L X is a function of bounded variation. Hence, our matri D, defined in (1 and (2, is the generator of a strongly continuous semigroup t t on the Banach space E and the delay equation (DE is well posed. For this generator we can now estimate s, generalizing [8, Theorem 3.3] with a similar proof. Theorem 4.1. Assume that s A < and let α s A. If ɛ α+iω Id < sup ω then s <α. 1 sup ω R α + iω A (19 To prove this statement we recall from [8, Proposition 1.1] the following lemma in a modified form (compare also [12, Theorem IV.1.16]. Lemma 4.2. Let A be a closed linear operator on a Banach space X, and suppose λ ρ A. If L X satisfies 1 1 δ R λ A for some δ 1, then λ ρ A + and R λ A + 1 R λ A δ

13 Proof of Theorem 4.1. sup ω partial differential equations with delay 13 Choose δ 1 such that 1 ɛ α+iω Id 1 δ sup ω R α + iω A We observe that ɛ λ Id is an analytic function and that the suprema of bounded analytic functions along vertical lines, Rλ = c, decrease as c increases (see [3, Chap. 6.4] for the details. For all λ with Rλ >αwe obtain ɛ λ Id sup ɛ α+iω Id ω 1 δ sup ω R α + iω A 1 δ R λ A Therefore, by Lemma 4.2, Rλ >α ρ A + ɛ λ Id, and for all λ with Rλ >αwe have Hence, by Lemma 4.1, R λ A + ɛ λ Id 1 R λ A δ Rλ >α ρ We show that R λ is bounded on this half-plane. We have shown above that R λ A + ɛ λ Id is bounded. The operator A generates the nilpotent left shift, so R λ A is bounded on this right half-plane. The function ɛ λ Id is bounded and analytic and R λ A is continuous and bounded, since R λ A f = dη σ e λ σ τ f τ dτ = 1 1 σ 1 σ 1 σ 1 1 σ e λ σ τ f τ dτ d η σ e Rλ σ τ f τ dτ d η σ e α f τ dτ d η σ e α f τ dτ d η σ e α η 1 f p for every λ with Rλ > α and every f L p 1 X. Therefore sup Rλ>α R λ <, and we conclude that s <α.

14 14 bátkai and piazzera Using Gearhart s theorem quoted above, this result can be improved for Hilbert spaces. Corollary 4.1. Take p = 2 and let X be a Hilbert space. If ω A < α and then ω <α. 1 sup ɛ α+iω Id < ω sup ω R α + iω A Eample 4.1. We consider the reaction diffusion equation with delay (see [26, Sect. 2.1] t w t = w t +c 1 w t + τ dg τ w t = w t =f t t t t 1 where c is a constant, n is a bounded domain, f t L 2 for all t, f W 1 2 W 2 2 and the map 1 t f t L 2 belongs to W ( L 2. The function g 1 1 is the Cantor function (see [9, Eample I.8.15], which is singular and has total variation 1. We consider X = L 2, A = D the Dirichlet Laplacian with usual domain, and η = c g Id. The well-posedness follows from the calculations in Eample 3.1(b. We have to verify the stability estimate. First, the epression with satisfies ɛ λ Id y = c y e λτ dg τ c y The other epression, using that A is a normal operator on a Hilbert space (see [12, Sect. V.3.8], can be computed as sup R iω A 1 = sup d iω σ A = 1 d σ A = 1 λ 1 ω ω where λ 1 is the first eigenvalue of the Laplacian. Thus the solutions decay eponentially if 1 c < λ 1 We refer for eample to [5, Chap. 6] for estimates on λ 1 and for further references. The same result holds for more general elliptic operators as considered in [5, Sect. 6.3].

15 partial differential equations with delay NORM CONTINUITY In this section, we show that if the operator A D A in (DE generates an immediately norm continuous semigroup on X, then, under an appropriate assumption, the operator matri associated to the delay equation generates an eventually norm continuous semigroup on E. This fact is important for the study of the asymptotic behavior of the solutions. For convenience, we repeat here the definitions from [7, Definition II.4.17]. A strongly continuous semigroup T t t on a Banach space Y is called eventuallynorm continuous, if there eists t such that the function t T t is norm continuous from t to L Y. The semigroup is called immediatelynorm continuous if t can be chosen to be t =. Let A D A be the generator of a strongly continuous semigroup T t t on a Banach space X and let C L D A A X. Moreover, let us assume that there eist ε> and a function q ε + such that lim t q t = and CT s ds q t (2 for every D A and every <t<ε. Estimates of type (2 were introduced by J. Voigt in [23]. From the perturbation theorem of Miyadera and Voigt we then know that A + C D A generates a strongly continuous semigroup U t t on X given by the Dyson Phillips series U t = V n T t t (21 n= where V is the abstract Volterra operator defined in [7, Theorem III.3.14] converging uniformly on compact intervals of + (see [7, Corollary III.3.15]. For D A we have VT t = T t s CT s ds Theorem 5.1. If T t t is norm continuous for t>αand there eists n such that V n T is norm continuous for t>, then the perturbed semigroup U t t is norm continuous for t>nα. The proof is a straightforward generalization of [15, Theorem 6.1]. We refer to [19, Corollary 2.7] for the details. We now apply this result to the operator D associated to the delay equation (DE. As before, we will assume that A D A generates a strongly continuous semigroup S t t on X and that the perturbation B satisfies the

16 16 bátkai and piazzera following condition being slightly stronger than condition (M. There eists q + + with lim t + q t = and S s + T s f ds q t ( K for all ( f D and t>. We then have that D is a generator by Theorem 3.1. Moreover, from (14 it follows that all of the cases in Eamples 3.1 satisfy this condition. We recall that D is the operator defined in (8 and (9 generating the strongly continuous semigroup t t given by (1. Proposition 5.1. If S t t is immediately norm continuous, then t t is norm continuous for t 1. Proof. Let t 1. Then T t = and t = ( S t S t So it suffices to show that the map t S t from 1 to L X L p 1 X is norm continuous. For s t 1 we have lim S s S t =lim s t sup s t 1 ( S s + σ S t + σ p dσ 1 f 1 p ( 1/p lim sup S s + σ S t + σ dσ p s t 1 1 ( 1 = lim S s + σ S t + σ p p dσ s t 1 which converges to as s tends to t since S t t is immediately norm continuous and therefore uniformly norm continuous on compact intervals. We are now ready to apply Theorem 5.1. Proposition 5.2. If the semigroup S t t generated by A D A is immediately norm continuous, then t t is norm continuous for t 1. The same result was already proved by a different technique, and for the special case = n k= B kδ hk, by Fischer and van Neerven [8, Proposition 3.5]. Proof of Proposition 5.2. From Proposition 5.1, we have that t t is norm continuous for t 1. We now show that V is norm continuous

17 partial differential equations with delay 17 for t. In fact, for t and ( f D we have V t ( f = = = = t s B s ( f ds ( ( t s ( S t s S t s T t s ( S t s Ss + T s f S t s S s + T s f S s ds S s + T s f ( Ss + T s f ds We prove norm continuity of the two components separately. +h ds 1. Let t and 1 >h>. Then we have S t + h s S s + T s f ds S t s S s + T s f ds +h S t + h s S s + T s f ds t + S t + h s S t s S s + T s f ds h + S h s S s+t + T s + t f ds sup r 1 + S t + h s S t s S s + T s f ds S r q h t ( f S t + h s S t s S s + T s f ds By condition (K, the Lebesgue dominated convergence theorem, and by the immediate norm continuity of S t t, we have that S r q h t ( sup r 1 f + S t + h s S t s S s + T s f ds D, 1. The proof for tends to as h + uniformly in ( f ( f h is analogous. Since D A is dense in E, the first component of V is immediately norm continuous.

18 18 bátkai and piazzera 2. To prove immediate norm continuity of the second component of V one proceeds in a similar way. We only have to use the norm continuity of the map t S t, which was proved in Proposition 5.1. Hence, the map t V t is norm continuous on + and by Theorem 5.1 we have that t t is norm continuous for t 1. Using the stability results obtained in the previous section and the spectral mapping theorem for eventually norm continuous semigroups (see [7, Theorem IV.3.9] for the details, we can prove the following stability result. Corollary 5.1. Assume that A generates an immediately norm continuous semigroup, ω A < and let α ω A. If then ω <α. 1 sup ɛ α+iω Id < (22 ω sup ω R α + iω A Eample 5.1. We consider the reaction-diffusion equation from Eample 4.1 in the state space E = L r L p 1 L r for 1 r<, 1 <p<. The well-posedness follows again from the calculations in Eamples 3.1(b. We show again that the solutions decay eponentially if c < λ 1 (23 etending our result from the Hilbert space case. Unfortunately, Corollary 5.1 is not optimal for this problem. If we estimate the resolvent for the stability condition in Corollary 5.1, we obtain that it is sufficient for the eponential stability if c <c r where c r λ 1 is a constant depending on r. To obtain estimate (23, we etend the result obtained for the Hilbert space case in two steps. Consider first the case r = 2. We know from Lemma 4.1 that σ, the spectrum of D, does not depend on p, and that s =ω since by Proposition 5.2 D generates an eventually norm continuous semigroup on E. Forp = 2 we had a condition to obtain eponential stability, i.e., ω <. Hence, as in the Hilbert space case, we obtain that the solutions decay eponentially if (23 holds. For the general case E = L r L p 1 L r, we observe first that the operator A D A has compact resolvent. From [7, Proposition

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