ON SEMILINEAR CAUCHY PROBLEMS WITH NON-DENSE DOMAIN

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1 Advances in Differential Equations Volume 14, Numbers (29), ON SEMILINEAR CAUCHY PROBLEMS WITH NON-DENSE DOMAIN Pierre Magal UMR CNRS 5251, IMB & INRIA sud-ouest Anubis University of Bordeaux, 146 rue Léo Saignat 3376 Bordeaux, France Shigui Ruan Department of Mathematics, University of Miami Coral Gables, FL , USA (Submitted by: Giuseppe Da Prato) Abstract. We are interested in studying semilinear Cauchy problems in which the closed linear operator is not Hille-Yosida and its domain is not densely defined. Using integrated semigroup theory, we study the positivity of solutions to the semilinear problem, the Lipschitz perturbation of the problem, differentiability of the solutions with respect to the state variable, time differentiability of the solutions, and the stability of equilibria. The obtained results can be used to study several types of differential equations, including delay differential equations, age-structure models in population dynamics, and evolution equations with nonlinear boundary conditions. 1. Introduction The main purpose of this paper is to present a comprehensive semilinear theory (Cazenave and Haraux [5], Davies and Pang [8], Hieber [15, 16], Xiao and Liang [45], etc.) that will allow us to study the dynamics of nondensely defined Cauchy problems, such as asymptotic behavior of solutions and bifurcations. Consider the Cauchy problem: du dt = Au + F (t, u), t, u() = u D(A), (1.1) Accepted for publication: December 28. AMS Subject Classifications: 35K57, 45D5, 47D6. Research was partially supported by NSF grant DMS

2 142 Pierre Magal and Shigui Ruan where A : D(A) X X is a linear operator in a Banach space X and F : [, + ) D(A) X is a continuous map. We are interested in studying the problem when D(A) is not dense in X and A is not a Hille-Yosida operator. Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, some partial differential equations, evolution equations with nonlinear boundary conditions, can be written as semilinear Cauchy problems with non-dense domain (see Da Prato and Sinestrari [7], Thieme [35, 36], Liu et al. [19], Magal and Ruan [22]). When A is a Hille-Yosida operator (i.e., if there exist two constants ω R and M 1 such that (ω, + ) ρ(a) and (λi A) k L(X) M (λ ω) k for all λ > ω, k 1, where L(X) is the space of bounded linear operators from X into X and ρ(a) is the resolvent set of A) and is densely defined (i.e., D(A) = X), the problem has been extensively studied (see Segal [33], Weissler [44], Martin [25], Pazy [3], Hirsch and Smith [17]). When A is a Hille-Yosida operator but its domain is non-densely defined, Da Prato and Sinestrari [7] investigated the existence of several types of solutions for (1.1). Thieme [35] investigated the semilinear Cauchy problem with a Lipschitz perturbation of the closed linear operator A which is non-densely defined but is Hille-Yosida. Integrated semigroup theory was used to obtain a variation of constants formula which allows one to transform the integrated solutions of the evolution equation into solutions of an abstract semilinear Volterra integral equation, which in turn was used to find integrated solutions to the Cauchy problem. Moreover, sufficient and necessary conditions for the invariance of closed convex sets under the solution flow were found. Conditions for the regularity of the solution flow in time and initial state were derived. The steady states of the solution flow were characterized and sufficient conditions for local stability and instability were given. See also Thieme [36, 38]. In this paper, we attempt to extend Thieme s results [35] to the case when the operator A is not Hille-Yosida. The perturbations of the operator are only Lipschitz on bounded sets. In order to do such extensions, we need to obtain an estimate for nonhomogeneous equations which is obtained in Proposition Then we are able to develop and extend most of his results (except the positivity result) in [35]. We obtain some weak and classical conditions for positivity of the solutions by considering the problem in a more general setting; that is, we only impose the condition that the integrated solutions exist when the nonlinear interaction term F is continuous.

3 On semilinear Cauchy problems with non-dense domain 143 We would like to make some comments about the assumption that the linear operator A is not a Hille-Yosida operator. We first assume that the resolvent set ρ(a) of A is non-empty and that A, the part of A in D(A), is the infinitesimal generator of a strongly continuous semigroup {T A (t)} t of bounded linear operators on D(A). Then A generates an integrated semigroup {S A (t)} t on X, defined by S A (t) = (λi A ) T A (l)dl (λi A) 1 for each λ ρ(a). Then (see Magal and Ruan [22] and Thieme [38]) we need to impose an additional condition to assure the existence of integrated solutions of the non-homogeneous Cauchy problem du(t) = Au(t) + f(t) for t and u() =. (1.2) dt Here we assume that A is not a Hille-Yosida operator (Assumption 2.1), but for each f C ([, τ], X) the Cauchy problem (1.2) has an integrated solution u f (t), and there exists a map δ : [, + ) [, + ) (independent of f) such that (Assumption 2.8 and Theorem 2.9) u f (t) δ(t) sup f(s), (1.3) s [,t] where δ(t) as t. In particular as in the example presented in Magal and Ruan [22], the Cauchy problem may not have an integrated solution for f L 1 ((, τ); X). The goal of this paper is to show that under such a condition on the Cauchy problem (1.2), we still can extend the results for the classical semi-linear Cauchy problems. In practice, it is relatively easy to verify that the linear operator A is not a Hille-Yosida operator (i.e., satisfying Assumption 2.1). Nevertheless, for a given example, it may require some work to verify the condition (1.3) (i.e., Assumption 2.8). In the context of age-structured models the condition (1.3) has been successfully verified by Magal and Ruan [22] using some characterization on the resolvent of A. For the same class of PDEs, Thieme [38] also successfully applied the notion of integrated semigroup with bounded p- semi-variation to verify condition (1.3). For parabolic systems, this question has been studied by Prevost [32] and Ducrot et al. [13]. In particular, it has been proved that if A is almost sectorial and A, the part of A in D(A), is a sectorial operator, then condition (1.3) is satisfied. We also refer to Prevost [32] for more examples in the context of parabolic equations.

4 144 Pierre Magal and Shigui Ruan The rest of the paper is organized as follows. In section 2, we recall some results on integrated semigroups and give an estimate for solutions to the nonhomogeneous equation (see Proposition 2.14) which is crucial for the stability of equilibria to the semilinear problem. Sections 3-7 are devoted to the study of the semilinear problem. In section 3, positivity of solutions to the semilinear problem is considered. Section 4 focuses on Lipschitz perturbations of the problem. Section 5 deals with differentiability of the solutions with respect to the state variable. In section 6 we are concerned with time differentiability of the solutions. The stability of equilibria is studied in section 7. In section 8, as applications we discuss transport equations with nonlinear boundary conditions and parabolic equations with nonlocal boundary conditions and show that our results apply. In particular, we verify that our main Assumptions 2.1 and 2.8 hold for these two types of equations. Notice that Magal and Ruan [22] presented some techniques and results for integrated semigroups when the generator is not a Hille-Yosida operator and is non-densely defined, obtained necessary and sufficient conditions for the existence of mild solutions for non-densely defined non-homogeneous Cauchy problems, and applied the results to study age structured models. Recently, Magal and Ruan [23] developed the center manifold theory for non-densely defined Cauchy problems and employed the theory to establish a Hopf bifurcation theorem for age structured models. This paper complements our previous articles [22, 23] in studying semilinear Cauchy problems with non-dense domain. 2. Integrated Semigroups In this section we recall some results about integrated semigroups. We refer to Arendt [2, 3], Neubrander [28], Kellermann and Hieber [18], Thieme [36, 38], Arendt et al. [4], and Magal and Ruan [22, 23] for more detailed results on the subject. Let X and Z be two Banach spaces. Let L (X, Z) denote the space of bounded linear operators from X into Z and by L (X) the space L (X, X). Let A : D(A) X X be a linear operator. If A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on X, let {T A (t)} t denote this semigroup. The resolvent set of A is denoted by ρ (A) = {λ C : λi A is invertible}. Set X := D(A) and A the part of A in X, which is a linear operator on X defined by A x = Ax, x D(A ) := {y D(A) : Ay X }.

5 On semilinear Cauchy problems with non-dense domain 145 Assume that (ω, + ) ρ(a). Then it is easy to check that, for each λ > ω, D (A ) = (λi A) 1 X and (λi A ) 1 = (λi A) 1 X. Recall that A is a Hille-Yosida operator if there exist two constants, ω R and M 1, such that (ω, + ) ρ(a) and (λi A) k M L(X), λ > ω, k 1. k (λ ω) In the following, we assume that A satisfies some weaker conditions. Assumption 2.1. Assume that A : D(A) X X is a linear operator on a Banach space (X,. ) satisfying the following properties: (a) There exist two constants, ω A R and M A 1, such that (ω A, + ) ρ(a) and (λi A) k L(X ) (b) lim λ + (λi A) 1 x =, x X. M A (λ ω A ) k, λ > ω A, k 1; By using Lemma 2.1 in Magal and Ruan [22] and Assumption 2.1-(b) we deduce that D(A ) = X. By the Hille-Yosida theorem (see Pazy [3], Theorem 5.3 on page 2) and the fact that if ρ(a) then ρ(a) = ρ(a ) (see Magal and Ruan [23, Lemma 2.4]), one obtains the following lemma. Lemma 2.2. Assumption 2.1 is satisfied if and only if ρ(a), A is the infinitesimal generator of a linear C -semigroup {T A (t)} t on X, and T A (t) M A e ω At, t. Now we give the definition of an integrated semigroup. Definition 2.3. Let (X,. ) be a Banach space. A family of bounded linear operators {S(t)} t on X is called an integrated semigroup if the following hold. (i) S() =. (ii) The map t S(t)x is continuous on [, + ) for each x X. (iii) S(t) satisfies S(s)S(t) = s (S(r + t) S(r)) dr, t, s. (2.2)

6 146 Pierre Magal and Shigui Ruan An integrated semigroup {S(t)} t is said to be non-degenerate if S(t)x = for all t, then x =. According to Thieme [36], a linear operator A : D(A) X X is the generator of a non-degenerate integrated semigroup {S(t)} t on X if and only if x D(A), y = Ax S(t)x tx = S(s)yds, t. (2.3) From [36, Lemma 2.5], we know that, if A generates {S A (t)} t, then for each x X and t, S A (s)xds D(A) and S(t)x = A S A (s)xds + tx. An integrated semigroup {S(t)} t is said to be exponentially bounded if there exist two constants, M > and ω >, such that S(t) L(X) Me bωt, t. When we restrict ourselves to the class of non-degenerate exponentially bounded integrated semigroups, Thieme s notion of generator is equivalent to the one introduced by Arendt [3]. More precisely, combining Theorem 3.1 in Arendt [3] and Proposition 3.1 in Thieme [36], one has the following result. Theorem 2.4. Let {S(t)} t be a strongly continuous exponentially bounded family of bounded linear operators on a Banach space (X,. ) and A : D(A) X X be a linear operator. Then {S(t)} t is a non-degenerate integrated semigroup and A its generator if and only if there exists some ω > such that ( ω, + ) ρ (A) and (λi A) 1 x = λ e λs S(s)xds, λ > ˆω. The following result is well known in the context of integrated semigroups. Proposition 2.5. Let Assumption 2.1 be satisfied. Then A generates a uniquely determined non-degenerate exponentially bounded integrated semigroup {S A (t)} t. Moreover, for each x X, each t, and each µ > ω A, S A (t)x is given by S A (t)x = µ T A (s) (µi A) 1 xds + [I T A (t)] (µi A) 1 x. (2.4)

7 On semilinear Cauchy problems with non-dense domain 147 Furthermore, the map t S A (t)x is continuously differentiable if and only if x X and ds A (t)x = T A (t)x, t, x X. dt From now on we define (S A f) (t) = S A (t s)f(s)ds, t [, τ], whenever f L 1 ((, τ), X). We now consider the non-homogeneous Cauchy problem du = Au(t) + f(t), t [, τ], u() = x D(A) (2.5) dt and assume that f belongs to some appropriate subspace of L 1 ((, τ), X). Definition 2.6. A continuous map u C ([, τ], X) is called an integrated solution (or mild solution) of (2.5) if and only if and u(t) = x + A u(s)ds D(A), t [, τ] u(s)ds + f(s)ds, t [, τ]. Since A generates a non-degenerate integrated semigroup on X, we can apply Theorem 3.7 in Thieme [36] and obtain the following result. Lemma 2.7. Let Assumption 2.1 be satisfied. Then for each x D(A) and each f L 1 ((, τ ), X), (2.5) has at most one integrated solution. Denote (S A f) (t) = d dt (S A f) (t) whenever the map t (S A f) (t) is continuously differentiable. We will say that {S A (t)} t has a bounded semi-variation on [, t] if { V n ( (S A,, t) := sup SA (t i ) S A (t i 1 ) ) } x i < +, i=1 where the supremum is taken over all partitions = t <.. < t n = t of the interval [a, b] and over any (x 1,.., x n ) X n with x i X 1, for all i = 1,.., n. In the sequel, we will only assume that (2.5) has an integrated solution whenever f C ([, τ], X), so we will make the following assumption.

8 148 Pierre Magal and Shigui Ruan Assumption 2.8. Let τ > be fixed. Assume that {S A (t)} t has a bounded semi-variation on [, τ ] and lim V (S A,, t) =. t(>) The following theorem, proved by Thieme [38], provides an equivalent condition to Assumption 2.8 and is very helpful in applications. We refer to Magal and Ruan [22], Prevost [32], and Ducrot et al. [13] for verifying Assumption 2.8 for age-structured models and parabolic equations. Theorem 2.9. Let Assumption 2.1 be satisfied. Then Assumption 2.8 is satisfied if and only if for each f C ([, τ ], X), the map t (S A f) (t) is continuously differentiable on [, τ ] and (S A f) (t) δ (t) sup f(s), t [, τ ], s [,t] where δ : [, τ ] [, + ) is a non-decreasing map satisfying lim δ (t) =. t(>) Proof. Assume first that Assumption 2.8 is satisfied. By using Lemma 3.1 in [38], we have for each t [, τ ] and each f C ([, τ ], X) that (S A f)(t) V (S A,, t) sup f(s), s [,t] where (S A f)(t) is defined as a Stieltjes convolution (see [38]). Moreover, by Theorem 4.2 in [38], we deduce that for each f C ([, τ ], X), the map t (S A f)(t) is continuously differentiable and (S A f)(t) = (S A f)(t) := d dt (S A f)(t). So by fixing δ (t) = V (S A,, t) we obtain the desired estimate. Conversely, by using the same arguments as in the proof of Theorem 3.4 in [38], one deduces that V (S A (.),, t) δ (t) for all t [, τ ], and the result follows. We have S A (τ + h) = S A (τ) + T A (τ)s A (h) for all h, so by using Assumption 2.8 we deduce that t S A (t) has a bounded semi-variation on [, 2τ ]. Now, using induction arguments, we deduce that t S A (t) has a bounded semi-variation on [, τ] for each τ, and by using Theorem 2.9 we obtain the following result.

9 On semilinear Cauchy problems with non-dense domain 149 Theorem 2.1. Let Assumptions 2.1 and 2.8 be satisfied. Then, for each τ >, t S A (t) has a bounded semi-variation on [, τ]. Moreover, for each f C ([, τ], X), the map t (S A f) (t) is continuously differentiable, (S A f) (t) D(A) for all t [, τ], and u(t) = (S A f) (t) satisfies u(t) = A u(s)ds + f(s)ds, t [, τ] and u(t) V (S A,, t) sup f(s), t [, τ]. s [,t] Furthermore, for each λ (ω, + ), we have (λi A) 1 (S A f) (t) = T A (t s) (λi A) 1 f(s)ds. (2.6) Corollary Let Assumptions 2.1 and 2.8 be satisfied. Then, for each x X and each f C ([, τ], X ), the Cauchy problem (2.5) has a unique integrated solution u C ([, τ], X ) given by Moreover, we have u(t) = T A (t)x + (S A f) (t), t [, τ]. u(t) M A e ωt x + V (S A,, t) sup f(s), t [, τ]. s [,t] We now consider a bounded perturbation of A. The following result was proved in Magal and Ruan [22, Theorem 3.1], which is also closely related to Desch and Schappacher s theorem (see [1] or Engel and Nagel [14, Theorem 3.1, page 183]). Theorem Let Assumptions 2.1 and 2.8 hold. Let L L (X, X). Then A + L : D(A) X X also satisfies Assumptions 2.1 and 2.8. More precisely, if {S A+L (t)} t denotes the integrated semigroup generated by A + L, and τ 1 (, τ ] is chosen such that L L(X,X) V (S A,, τ 1 ) < 1, then, for each f C ([, τ 1 ], X), (S A+L f) (t) We have the following lemma. V (S A,, t) 1 L L(X,X) V (S A,, τ 1 ) sup f(s), t [, τ 1 ]. s [,t] Lemma Let Assumptions 2.1 and 2.8 be satisfied. Then t S A (t) is continuous from [, + ) into L (X) and lim (λi A) 1 L(X) =. λ +

10 15 Pierre Magal and Shigui Ruan Proof. By Proposition 3.1 in Thieme [36], we have for each λ > max(, ω A ) that Note that (λi A) 1 x = λ S A (t)x = d dt so, by Assumption 2.8, we have But + e λt S A (t)xdt. S A (t s)xds, S A (t)x V (S A,, t) x, t. S A (t + h) S A (t) = T A (t)s A (h); it follows that t S A (t) is operator norm continuous. Let ε > be fixed and let τ ε > be such that V (S A,, τ ε ) ε. Choose γ > max(, ω A ) and M γ > such that S A (t)x M γ e γt, t. Then we have for each λ > γ that + (λi A) 1 x λ [M γ Thus, lim sup (λi A) 1 L(X) ε. λ + This proves the lemma. τ ε τε ] e (γ λ)t dt + ε e λt dt x. By using the fact that (S A f) (t) X for all t [, τ], and formula (2.6), we have for each f C ([, τ], X) that (S A f) (t) = lim µ + T A (t l)µ (µi A) 1 f(l)dl, t [, τ]. (2.7) This approximation formula was already observed by Thieme [35] in the classical context of integrated semigroups generated by a Hille-Yosida operator. From this approximation formulation, we then deduce that for each pair t, s [, τ] with s t, and f C ([, τ], X), (S A f) (t) = T A (t s) (S A f) (s) + (S A f (s +.)) (t s). (2.8) Let. denote the norm on X defined by x = sup e ωal T A (l) x, x X. l

11 Clearly, we have On semilinear Cauchy problems with non-dense domain 151 T A (t) L(X ) eω At, t, x x M A x, x X, and by Assumption 2.8, for each f C ([, τ], X) and each t [, τ], (S A f) (t) M A (S A f) (t) M A V (S A,, t) sup f(s). (2.9) s [,t] The following proposition is one of the main tools for studying semilinear problems (see the next section and also Magal and Ruan [23] for another class of applications of this result). Proposition Let Assumptions 2.1 and 2.8 be satisfied. Let ε > be fixed. Then, for each τ ε > satisfying M A V (S A,, τ ε ) ε, we have (S A f) (t) C (ε, γ) sup e γ(t s) f(s), t s [,t] whenever γ (ω A, + ), f C (R +, X), with C (ε, γ) := 2ε max (1, e γτε ) 1 e (ω A γ)τε. Proof. Let ε >, f C (R +, X), and γ > ω A be fixed. Let τ ε = τ ε (ε) (, τ] be such that M A V (S A,, τ ε ) ε. By (2.9), we have (S A f) (t) ε sup f(s), t [, τ ε ]. (2.1) s [,t] Let γ > ω A be fixed. Set ε 1 = ε max (1, e γτε ). Let k N and t [kτ ε, (k + 1)τ ε ] be fixed. First, notice that if γ, we have ε sup s [kτ ε,t] f(s) = ε sup s [kτ ε,t] Moreover, if γ <, we have ε sup s [kτ ε,t] f(s) = ε e γs e γs f(s) εe γt = ε 1 e γt sup e γs f(s). s [kτ ε,t] sup s [kτ ε,t] e γs e γs f(s) εe γkτε = εe γt e γt e γkτε sup e γs f(s) s [kτ ε,t] = e γt εe γ(t kτε) sup e γs f(s) s [kτ ε,t] e γt εe γτε sup s [kτ ε,t] e γs f(s) = e γt ε 1 sup s [kτ ε,t] e γs f(s) sup e γs f(s) s [kτ ε,t] sup s [kτ ε,t] e γs f(s).

12 152 Pierre Magal and Shigui Ruan Therefore, for each k N, each t [kτ ε, (k + 1) τ ε ], and each γ R, we obtain ε sup f(s) e γt ε 1 sup e γs f(s). (2.11) s [kτ ε,t] s [kτ ε,t] It follows from (2.1) and (2.11) that, for all t [, τ ε ], (S A f) (t) ε sup f(s) = e γt ε 1 s [,t] sup s [,t] Using (2.8) with s = τ ε, we have for all t [τ ε, 2τ ε ] that e γs f(s). (2.12) (S A f) (t) = T (t τ ε ) (S A f) (τ ε ) + (S A f (τ ε +.)) (t τ ε ). Using (2.1), (2.11), and (2.12), we have (S A f) (t) e ω A(t τ ε) (S A f) (τ ε ) + (S A f (τ ε +.)) (t τ ε ) e ω A(t τ ε) e γτε ε 1 e ω A(t τ ε) e γτε ε 1 sup s [,τ ε] sup s [,τ ε] ( ) ε 1 e γt e (ω A γ)(t τ ε) + 1 Similarly, for all t [2τ ε, 3τ ε ], and e γs f(s) + ε sup f(s) s [τ ε,t] e γs f(s) + e γt ε 1 sup s [,t] e γs f(s). sup s [τ ε,t] e γs f(s) (S A f) (t) = T A (t 2τ ε ) (S A f) (2τ ε ) + (S A f (2τ ε +.)) (t 2τ ε ) ) (S A f) (t) e ω A(t 2τ ε) ε 1 e (e γ2τε (ω A γ)τ ε + 1 sup e γs f(s) + ε sup s [,2τ ε] ) e ω A(t 2τ ε) ε 1 e (e γ2τε (ω A γ)τ ε + 1 s [2τ ε,t] f(s) sup e γs f(s) + ε 1 e γt sup e γs f(s) s [,2τ [ ε] ( s [2τ ε,t] ) ] ε 1 e γt e (ω A γ)(t 2τ ε) e (ω A γ)τ ε e γs f(s). sup s [,t] By induction, we obtain for all k N with k 1, t [kτ ε, (k + 1) τ ε ], and for each γ > ω A that (S A f)(t) ε 1 e γt sup s [,t] [ e γs f(s) k 1 e (ω A γ)(t kτ ε) ] (e (ω A γ)τ ε ) n + 1 n=

13 On semilinear Cauchy problems with non-dense domain 153 ε 1 e γt sup s [,t] Since γ > ω A, we have for each t that e γt (S A f) (t) e γt (S A f) (t) [ ) n ] e γs f(s) (e (ω A γ)τ ε + 1. n= 2ε 1 1 e (ω sup e γs f(s). A γ)τ ε s [,t] This completes the proof. and Let I [, + ) be an interval. Set s := inf I. For each γ, define { } BC γ (I, Y ) := ϕ C (I, Y ) : sup e γ(l s) ϕ(l) Y < + l I ϕ BC γ (I,Y ) := sup e γ(l s) ϕ(l) Y. l I It is well known that BC γ (I, Y ) endowed with the norm. BC γ (I,Y ) is a Banach space. By using Proposition 2.14 we obtain the following result. Lemma Let Assumptions 2.1 and 2.8 be satisfied. For each s and each σ (s, + ], define a linear operator L s : C ([s, σ), X) C ([s, σ), X ) by L s (ϕ) (t) = (S A ϕ(. + s)) (t s), t [s, σ), ϕ C ([s, σ), X). Then, for each γ > ω A, L s is a bounded linear operator from BC γ ([s, σ), X) into BC γ ([s, σ), X ). Moreover, for each ε > and each τ ε > such that M A V (S A,, τ ε ) ε, L s (ϕ) L(BC γ ([s,σ),x),bc γ ([s,σ),x )) C (ε, γ). Proof. Let ϕ BC γ ([s, σ), X) be fixed. By using Proposition 2.14, we have for t [s, σ) that e γ(t s) (S A ϕ(. + s)) (t s) C (ε, γ) sup l [,t s] e γl ϕ(l + s) = C (ε, γ) sup r [s,t] e γ(r s) ϕ(r) C (ε, γ) sup r [s,σ) e γ(r s) ϕ(r) and the result follows.

14 154 Pierre Magal and Shigui Ruan 3. Semilinear Problems Positivity Starting from this section, we consider the semilinear Cauchy problem du(t, s)x = AU(t, s)x + F (t, U(t, s)x), t s, U(s, s)x = x X. dt We shall investigate the properties of the non-autonomous semiflow generated by the following problem U(t, s)x = x + A or equivalently s U(l, s)xdl + s F (l, U(l, s)x)dl, t s, (3.1) U(t, s)x = T A (t s)x + (S A F (. + s, U(. + s, s)x)) (t s), t s. (3.2) Definition 3.1. Consider two maps χ : [, + ) X (, + ] and U : D χ X, where { } D χ = (t, s, x) [, + ) 2 X : s t < s + χ (s, x). We say that U is a maximal non-autonomous semiflow on X if U satisfies the following properties: (i) χ (r, U(r, s)x)+r = χ (s, x)+s, s, x X, r [s, s + χ (s, x)). (ii) U(s, s)x = x, s, x X. (iii) U(t, r)u(r, s)x = U(t, s)x, s, x X, t, r [s, s + χ (s, x)) with t r. (iv) If χ (s, x) < +, then Set D = lim U(t, s)x = +. t (s+χ(s,x)) { } (t, s, x) [, + ) 2 X : t s. In order to present a theorem on the existence and uniqueness of solutions to equation (3.1), we make the following assumption. Assumption 3.2. Assume that F : [, + ) D(A) X is a continuous map, and for each σ > and each ξ > there exists K(σ, ξ) > such that F (t, x) F (t, y) K(σ, ξ) x y whenever t [, σ], y, x X with x ξ and y ξ. The following theorem is proved in Magal and Ruan [22, Theorem 5.2].

15 On semilinear Cauchy problems with non-dense domain 155 Theorem 3.3. Let Assumptions and 3.2 be satisfied. Then there exist a map χ : [, + ) X (, + ] and a maximal non-autonomous semiflow U : D χ X, such that for each x X and each s, U(., s)x C ([s, s + χ (s, x)), X ) is a unique maximal solution of (3.1) (or equivalently a unique maximal solution of (3.2)). Moreover, D χ is open in D and the map (t, s, x) U(t, s)x is continuous from D χ into X. We are now interested in the positivity of the solutions of equation (3.1). Let X + X be a cone of X; that is, X + is a closed convex subset of X satisfying the following two properties: (i) λx X +, x X +, λ. (ii) x X + and x X + x =. It is clear that X + = X X + is also a cone of X. Recall that such a cone defines a partial order on the Banach space X which is defined by x y if and only if x y X +. We need the following assumption to prove the positivity of solutions of equation (3.1). Assumption 3.4. Assume that there exists a linear operator B L (X, X) such that (a) for each γ >, A γb is resolvent positive (i.e., (λi (A γb)) 1 X + X + for all λ > ω A large enough); (b) for each ξ > and each σ >, there exists γ = γ(ξ, σ) >, such that F (t, x) + γbx X + whenever x X +, x ξ and t [, σ]. Proposition 3.5. Let Assumptions and be satisfied. Then for each x X + and each s, we have U(t, s)x X + for all t [s, s + χ (s, x)). Proof. Without loss of generality we can assume that s = and x X +. Moreover, using the semiflow property, it is sufficient to prove that there exists ε (, χ (, x)) such that U(t, )x X + for all t [, ε]. Let x X + be fixed. We set ξ := 2 ( x + 1). Let γ > such that F (t, x) + γbx X + when x X +, x ξ and t [, 1]. Fix τ γ > such that γv (S A,, τ γ ) < 1. For each σ (, τ γ ), define E σ = {ϕ C ([, σ], X + ) : ϕ(t) ξ, t [, σ]}. Then it is sufficient to consider the fixed-point problem u(t) = T (A γb) (t)x+(s A γb F (., u(.))+γbu(.))(t) =: Ψ(u)(t), t [, σ].

16 156 Pierre Magal and Shigui Ruan Since A γb is resolvent positive, we have T (A γb) (t)x + X + for all t. Using the approximation formula (2.7), we have for each τ > that (S A γb ϕ) (t) X +, t [, τ], ϕ C ([, τ], X + ). Moreover, by using Theorem 2.12, for each ϕ E σ and each t [, σ], we deduce that Ψ(ϕ)(t) = T (A γb) (t)x + (S A γb F (., ϕ(.)) + γϕ(.)) (t) T (A γb) (t)x + sup T(A γb) (t)x t [,σ] + V (S A,, σ) 1 γv (S A,, τ γ ) V (S A,, t) 1 γv (S A,, τ γ ) sup F (s, ϕ(s)) + γϕ(s) s [,t] [ sup s [,σ] ] F (s, ) + [K(1, ξ) + γ] ξ. Hence, there exists σ 1 (, 1) such that Ψ(E σ ) E σ for all σ (, σ 1 ]. Therefore, for each σ (, σ 1 ] and each pair ϕ, ψ E σ, we have for t [, σ] that Ψ(ϕ)(t) Ψ(ψ)(t) = (SA γb [F (., ϕ(.)) F (., ψ(.)) + γ(ϕ ψ)(.)])(t) V (S A,, σ) 1 γv [K(1, ξ) + γ] sup (ϕ ψ) (s). (S A,, τ γ ) s [,σ] Thus, there exists σ 2 (, σ 1 ] such that Ψ(E σ 2 ) E σ 2 and Ψ is a strict contraction on E σ 2. The result then follows. Example 3.6. (1) We refer to Thieme [35] and Hirsch and Smith [17] for more results on the positivity of semiflows. (2) Usually Proposition 3.5 is applied with B = I. But the case B I can also be useful. Consider the following functional differential equation: { dx(t) = f (x t ), t, dt (3.3) x = ϕ C ([ τ, ], R n ), where f : C ([ τ, ], R n ) R n is Lipschitz continuous on bounded subsets of C ([ τ, ], R n ). In order to obtain the positivity of solutions, it is sufficient to assume that for each M there exists γ = γ (M) > such that f (ϕ) + γϕ () whenever ϕ M and ϕ C ( [ τ, ], R n +). It is well known that this condition is sufficient to obtain the positivity of solutions (see Martin and

17 On semilinear Cauchy problems with non-dense domain 157 Smith [26, 27]). In order to prove ( this, ) one may also apply Proposition 3.5. By identifying x t with u(t) =, the system (3.3) can be rewritten as xt a non-densely defined Cauchy problem (see Liu, Magal and Ruan [19] for more details) ( ) dv(t) = Av(t) + F (v(t)), t, and v() = dt ϕ with X = R n C ([ τ, ], R n ), D (A) = { R n} C 1 ([ τ, ], R n ), where A : D(A) X X is defined by ( ) ( ϕ A = ) () ϕ ϕ and F : D(A) X by ( ) F = ϕ Then Proposition 3.5 applies with ( ) B = ϕ ( ) f (ϕ). C ( ϕ () C ). Recall that a cone X + of a Banach space (X,. ) is normal if there exists a norm. 1 equivalent to., which is monotone; that is, x, y X, x y x 1 y 1. Corollary 3.7. Let Assumptions and be satisfied. Assume in addition that Then (a) X + is a normal cone of (X,. ); (b) there exist a continuous map G : [, + ) X + X + and two real numbers k 1 and k 2, such that for each t and each x X +, F (t, x) G(t, x) and G(t, x) k 1 x + k 2. χ (s, x) = +, s, x X +. Moreover, we have the following estimate: for each γ > large enough, there exist C 1 > and C 2 > such that U(t, s)x e γ(t s) [C 1 x + C 2 ].

18 158 Pierre Magal and Shigui Ruan Proof. Without loss of generality, we can assume that s = and the norm 1. is monotone. Let ε (, 2k 1 ) and τ ε > such that M A V (S A,, τ ε ) ε. Let x X + be fixed. Then by Proposition 3.5, we have for each t [, χ (, x)) that U(t, )x = T A (t)x + (S A F (., U(., )x)) (t) T A (t)x + (S A G(., U(., )x)) (t). Hence, for each γ > max(ω A, ), we have for each t [, χ (, x)) that e γt U(t, )x e γt T A (t)x + e γt (S A G(., U(., )x)) (t) M A e ( γ+ωa)t x + C (ε, γ) sup e γs G(s, U(s, )x) s [,t] M A x + C (ε, γ) sup e γs [k 1 U(s, )x + k 2 ] s [,t] M A x + k 2 C (ε, γ) + k 1 C (ε, γ) sup e γs U(s, )x. s [,t] Since 2k 1 ε < 1, for γ > max(ω A, ) sufficiently large, we obtain k 1 C (ε, γ) = γ)τε < 1 and the result follows. 2k 1 ε 1 e (ω A 4. Global Lipschitz Perturbation We now consider the case where the map x F (t, x) is Lipschitz continuous. Let (Y,. Y ) and (Z,. Z ) be two Banach spaces. Let E be a subset of Y and G : Y Z be a map. Define G Lip(E,Z) = For each x Y and r >, set G(x) G(y) sup Z. x,y E:x y x y Y B Y (x, r) = {y Y : x y Y < r}, B Y (x, r) = {y Y : x y Y r}. The main results of this section are on the existence and uniqueness of a solution to the integral equation (3.1) and its estimate when x F (t, x)x is globally Lipschitz continuous. Proposition 4.1. Let Assumptions be satisfied. Let F : [, + ) D(A) X be a continuous map and σ (, + ] be a fixed constant. Assume that Γ F (σ) := sup F (t,.) Lip(X,X) < +. t [,σ)

19 On semilinear Cauchy problems with non-dense domain 159 Then, for each x X and each s [, σ), there exists a unique solution U(., s)x C ([s, σ), X ) of U(t, s)x = x + A s U(l, s)xdl + s F (l, U(l, s)x)dl, t [s, σ). Moreover, there exists γ > max (, ω A ) such that for each γ γ, each pair t, s [, σ) with t s, and each pair x, y X, we have U(t, s)x e γ(t s)[ ] 2M A x + sup e γ(l s) F (l, ) l [s,σ) and U(t, s)x U (t, s) y e γ(t s) 2M A x y. Proof. Fix s, t [, σ) with s < t. Let ε > such that ε max(γ F (σ), 1) < 1/8. Let τ ε > such that M A V (S A,, τ ε ) ε. Then by Lemma 2.15 we have for each γ > ω A that L s (ϕ) L(BC γ ([s,+ ),X),BC γ ([s,+ ),X )) C (γ, ε) = 2ε max (1, e γτε ) ( 1 e (ω A γ)τ ε ). Let γ max(, ω A ) such that 1 ( 1 e (ω A γ)τ ε ) < 2, γ γ. To prove the proposition it is sufficient to prove that the following fixed-point problem U(., s)x = T A (. s)x + L s Ψ (U(., s)x) (3.4) admits a solution U(., s)x BC γ ([s, σ), X ), where Ψ : BC γ ([s, σ), X ) BC γ ([s, σ), X) is a nonlinear operator defined by We have and Ψ (ϕ) (l) = F (l, ϕ(l)), l [s, t], ϕ BC γ ([s, σ), X ). From this we deduce that T A (. s) L(X,BC γ ([s,σ),x )) M A, L s L(BC γ ([s,σ),x),bc γ ([s,σ),x )) 4ε, Ψ Lip(BC γ ([s,σ),x ),BC γ ([s,σ),x)) Γ F (σ). L s Ψ Lip(BC γ ([s,σ),x ),BC γ ([s,σ),x )) 4εΓ F (σ) 1/2.

20 16 Pierre Magal and Shigui Ruan Thus, the fixed-point problem (3.4) has a unique solution. Moreover, for each x X, there exists a unique solution U(., s)x in BC γ ([s, σ), X ) such that U(., s)x BC γ ([s,t],x ) M A x + L s (Ψ ()) + L s (Ψ (U(., s)x) Ψ ()) which implies that M A x + 4ε Ψ () BC γ ([s,σ),x) U(., s)x BC γ ([s,σ),x ), U(., s)x BC γ ([s,σ),x ) 2M A x + 8ε Ψ () BC γ ([s,σ),x). Since by construction ε 1 8, we have sup e γ(l s) U(., s)x 2M A x + sup e γ(l s) F (l, ). l [s,σ) l [s,σ) Similarly, we have for each pair x, y X that U(., s)x U(., s)y = T A (. s)(x y) + L s [Ψ (U(., s)x) Ψ (U(., s)y)]. Therefore, U(., s)x U(., s)y BC γ ([s,σ),x ) M A x y U(., s)x U(., s)y BC γ ([s,σ),x ). This completes the proof. 5. Differentiability with Respect to the State Variable In this section we investigate the differentiability of solutions with respect to the state variable. Proposition 5.1. Let Assumptions and 3.2 be satisfied. Assume in addition that (a) for each t the map x F (t, x) is continuously differentiable from X into X; (b) the map (t, x) D x F (t, x) is continuous from [, + ) X into L (X, X). Let x X, s, τ [, χ (s, x )), and γ (, χ (s, x ) τ). Let η > (there exists such a constant since D χ is open in D) such that χ (s, y) > τ + γ, y B X (x, η).

21 On semilinear Cauchy problems with non-dense domain 161 Then for each t [s, s + τ + γ], the map x U(t, s)x is defined from B X (x, η) into X and is differentiable at x. Moreover, if we set V (t, s)y = D x U(t, s)(x)(y), y X, then t V (t, s)y is an integrated solution of the Cauchy problem dv (t, s)y dt = AV (t, s)y + D x F (t, U(t, s)x )V (t, s)y, t [s, s + χ (s, x )), V (s, s)y = y or equivalently, for all t [s, s + χ(s, x )), t V (t, s)y is a solution of V (t, s)y = T A (t s)y + (S A D x F (., U(., s)x )V (., s)y)(t s). Proof. First by using the result in Section 4 about the Lipschitz case, it clear that W (t, s) is well defined. Set Then But where R(t)(y) = U(t, s) (x + y) U(t, s)(x ) V (t, s)y. R(t)(y) = (S A [F (., U(., s) (x + y)) F (., U(., s) (x )) D x F (., U(., s)x )V (., s)y])(t s). F (t, U(t, s) (x + y)) F (t, U(t, s) (x )) = = 1 D x F (t, ru(t, s) (x + y) + (1 r)u(t, s) (x ))(U(t, s) (x + y) U(t, s) (x ))dr 1 Ψ 1 (t, r, y) (U(t, s) (x + y) U(t, s) (x )) dr +D x F (t, U(t, s) (x )) (U(t, s) (x + y) U(t, s) (x )), Ψ 1 (t, r, y) = D x F (t, ru(t, s)(x +y)+(1 r)u(t, s)(x )) D x F (t, U(t, s)(x )). Thus, where and R(t)y = (S A [K(.) + D x F (., U(., s)x )R(.)y]) (t s), K(t) = 1 Ψ 2 (t, r, y) (U(t, s) (x + y) U(t, s) (x )) dr Ψ 2 (t, r, y) = D x F (t, ru(t, s)(x +y)+(1 r)u(t, s)(x )) D x F (t, U(t, s)(x )).

22 162 Pierre Magal and Shigui Ruan The result follows from Proposition 2.14 and the continuity of (t, x) U(t, s)x. 6. Time Differentiability In this section, we study the time differentiability of the solutions. Consider a solution u C([, τ], D(A)) of u(t) = x + A u(s)ds + F (s, u(s))ds, t [, τ]. Assume that x D(A) and F : [, T ] D(A) X is a C 1 map. When the domain of A is dense, it is well known (see Pazy [3], Theorem 6.1.5, page 187) that, for each x D(A), the map t u(t) is a classical solution. That is, the map t u(t) is continuously differentiable and u(t) D(A) for all t [, τ], and satisfies u (t) = Au(t) + f(t, u(t)), t [, τ], u() = x. In this section, we consider the same problem but in the the context of non-densely defined Cauchy problems. When A satisfies the Hille-Yosida condition, this problem has been studied by Thieme [35] and Magal [21]. So the goal is to extend these results to the non-hille-yosida case. This problem turns out to be more difficult. For each τ >, set { C 1,+ ([, τ], X) = f C([, τ], X) : d+ f d + } f C([, τ), X), lim dt t τ dt (t) <. The following lemma is a variant of a result due to Da Prato and Sinestrari [7]. Lemma 6.1. Let A : D(A) X X be a closed linear operator. Let τ >, f C([, τ], X), and x X be fixed. Assume that u C ([, τ], X) is a solution of u(t) = x + A u(s)ds + f(s)ds, t [, τ]. Assume in addition that u belongs to C 1,+ ([, τ], X) or C([, τ], D(A)). Then u C 1 ([, τ], X) C([, τ], D(A)) and u (t) = Au(t) + f(t), t [, τ]. Proof. If u C([, T ], D(A)), since A is closed, we have u(t) = x + Au(s)ds + f(s)ds, t [, τ].

23 On semilinear Cauchy problems with non-dense domain 163 Thus u C 1 ([, τ], X) and u (t) = Au(t) + f(t), t [, τ]. If u C 1,+ ([, τ], X), then we have for each t [, τ) and h > that +h +h u(t + h) u(t) t f(s)ds t u(s)ds = A. h h h Since A is closed, we deduce that u(t) D(A) and Au(t) = d+ u dt (t) + f(t) for all t [, τ). Since u C 1,+ ([, τ], X), we then deduce that u C([, τ], D(A)) and complete the proof. Lemma 6.2. Let Assumptions 2.1 and 2.8 be satisfied. Assume that g C 1 ([, T ], X) and g() D(A); then t (S A g) (t) is continuously differentiable and d dt (S A g) (t) = T A (t)g() + ( S A g ) (t), t [, T ]. Proof. Since g is continuously differentiable, the map t (S A g) (t) is continuously differentiable, with d dt (S A g) (t) = S A (t)g() + ( S A g ) (t), t [, T ]. Since g() D(A), we have S A (t)g() = T A (l)g()dl for all t [, T ], and the result follows. The following theorem is due to Vanderbauwhede [39, Theorem 3.5]. Lemma 6.3. (Fibre contraction theorem) Let M 1 and M 2 be two complete metric spaces and Ψ : M 1 M 2 M 1 M 2 a mapping of the form Ψ (x, y) = (Ψ 1 (x), Ψ 2 (x, y)), (x, y) M 1 M 2 satisfying the following properties: (i) Ψ 1 has a fixed point x M 1 such that for each x M 1, Ψ n 1 (x) x as n +. (ii) There exists k [, 1) such that for each x M 1 the map y Ψ 2 (x, y) is k-lipschitz continuous. (iii) The map x Ψ 2 (x, y) is continuous, where y M 2 is a fixed point of the map y Ψ 2 (x, y). Then, for each (x, y) M 1 M 2, Ψ n (x, y) (x, y) as n +. The key result of this section is the following lemma.

24 164 Pierre Magal and Shigui Ruan Lemma 6.4. Let Assumptions 2.1 and 2.8 be satisfied. Let τ > be fixed and F : [, τ] D(A) X be continuously differentiable. Assume that there exists an integrated solution u C ([, τ], X) of the Cauchy problem du(t) dt = Au(t) + F (t, u(t)), t [, τ], u() = x X. Assume in addition that x D(A ) and F (, x) D(A). Then there exists ε >, such that u C 1 ([, ε], X) C([, ε], D(A)) and u (t) = Au(t) + F (t, u(t)), t [, ε]. Proof. Since F is continuously differentiable, there exist ε >, K 1 >, and K 2 > such that t F (t, y) K 1 and x F (t, y) L(X,X) K 2 whenever x y ε and t ε. For each ε (, ε ], set M ε 1 = {ϕ C([, ε], X ) : ϕ() = x, ϕ(t) x ε, t [, ε]}, M ε 2 = {ϕ C([, ε], X ) : ϕ() = A x + F (, x), ϕ(t) A x + F (, x) ε, t [, ε]}. From now on, we assume that for each i = 1, 2, Mi ε is endowed with the metric d(ϕ, ϕ) = ϕ ϕ,[,ε] and M1 ε M 2 ε is endowed with the usual product distance d((ϕ, ψ), ( ϕ, ψ)) = d(ϕ, ϕ) + d(ψ, ψ). For each ε (, ε ], set { } E ε = (ϕ 1, ϕ 2 ) M1 ε M2 ε : ϕ 1 (t) = x + ϕ 2 (s)ds, t [, ε]. Then it is clear that E ε is a closed subset of M ε 1 M ε 2. We consider a map Ψ : M ε 1 M ε 2 C ([, ε], X ) C ([, ε], X ) defined by Ψ (ϕ 1, ϕ 2 ) = (Ψ 1 (ϕ 1 ), Ψ 2 (ϕ 1, ϕ 2 )), (ϕ 1, ϕ 2 ) M ε 1 M ε 2, where, for each t [, ε], Ψ 1 (ϕ 1 ) (t) = T A (t)x + (S A F (., ϕ 1 (.))) (t), Ψ 2 (ϕ 1, ϕ 2 ) (t) = T A (t) [A x + F (, x)] + (S A t F (., ϕ 1 (.)) + x F (., ϕ 1 (.))ϕ 2 (.)) (t). One can easily check that Ψ is a continuous map. We now prove that for some ε > small enough, Ψ (M ε 1 M ε 2 ) M ε 1 M ε 2, and Ψ 1 (ϕ 1 ) () = x, Ψ 2 (ϕ 1, ϕ 2 ) () = [A x + F (, x)].

25 On semilinear Cauchy problems with non-dense domain 165 For each ε (, ε ], each t [, ε], and each ϕ M1 ε, we have Ψ 1 (ϕ) (t) x T A (t)x x + (S A F (., ϕ(.)) (t) T A (t)x x + V (S A,, t) sup s [,t] s [,t] F (s, ϕ(s)) ( T A (t)x x + V (S A,, ε) sup F (s, x) + K 2 sup sup T A (t)x x + V (S A,, ε) t [,ε] ( sup s [,ε] s [,t] F (s, x) + K 2 ε ). ) ϕ(s) x Thus, there exists ε 1 (, ε ] such that for each ε (, ε 1 ], Ψ 1 (M ε 1 ) M ε 1. Moreover, for each ε (, ε 1 ], each t [, ε], and each (ϕ 1, ϕ 2 ) M ε 1 M ε 2, we have Ψ 2 (ϕ 1, ϕ 2 ) (t) [A x + F (, x)] T A (t) [A x + F (, x)] [A x + F (, x)] + (S A t F (., ϕ 1 (.)) + x F (., ϕ 1 (.))ϕ 2 (.)) (t) sup T A (t) [A x + F (, x)] [A x + F (, x)] t [,ε] +V (S A,, ε) sup t F (s, ϕ 1 (s)) s [,ε] +V (S A,, ε) sup x F (s, ϕ 1 (s)) ϕ 2 (.) s [,ε] sup T A (t) [A x + F (, x)] [A x + F (, x)] t [,ε] +V (S A,, ε) {K 1 + K 2 [ A x + F (, x) + ε ]}. Therefore, there exists ε 2 (, ε 1 ] such that for each ε (, ε 2 ], Ψ 2 (M1 ε M2 ε) M 2 ε. Similarly, for each ε (, ε 2 ], Ψ (M1 ε M 2 ε) M 1 ε M 2 ε. Now we check that Ψ (E ε ) E ε. Let (ϕ 1, ϕ 2 ) E ε. Then ϕ 1 C 1 ([, ε], X ) and ϕ 1 (t) = ϕ 2 (t) for all t [, ε]. Notice that Ψ 1 (ϕ 1 ) (t) = T A (t)x + (S A F (., ϕ 1 (.)) (t); using Lemma 6.2 and the fact that x D(A ) and F (, x) D(A), we have dψ 1 (ϕ 1 ) (t) ( = A T A (t)x + T A (t)f (, x) + S A d ) dt dt F (., ϕ 1(.) (t) = T A (t) [A x + F (, x)] + (S A t F (., ϕ 1 (.)) + x F (., ϕ 1 (.))ϕ 2 (.)) (t).

26 166 Pierre Magal and Shigui Ruan Thus, dψ 1 (ϕ 1 ) (t) = Ψ 2 (ϕ 1, ϕ 2 ) (t) dt and Ψ (E ε ) E ε. Next we apply Lemma 6.3. It remains to verify i) and ii) for some ε (, ε 2 ] small enough. Let (ϕ 1, ϕ 2 ), ( ϕ 1, ϕ 2 ) M1 ε Mε 2 be fixed. We have for each ε (, ε 2 ] that Ψ 1 (ϕ 1 ) (t) Ψ 1 ( ϕ 1 ) (t) = (S A F (., ϕ 1 (.)) F (., ϕ 1 (.))) (t) V (S A,, ε) F (s, ϕ 1 (s)) F (s, ϕ 1 (s)) V (S A,, ε)k 2 sup ϕ 1 (s) ϕ 1 (s). s [,ε] Thus there exists ε 3 (, ε 2 ] such that δ 1 := V (S A,, ε 3 )K 2 (, 1) ; we have for each ε (, ε 3 ] that Moreover, Ψ 1 (ϕ 1 ) Ψ 1 ( ϕ 1 ),[,ε] δ 1 ϕ 1 ϕ 1,[,ε]. Ψ 2 (ϕ 1, ϕ 2 ) (t) Ψ 2 (ϕ 1, ϕ 2 ) (t) = (S A x F (., ϕ 1 (.)) (ϕ 2 (.) ϕ 2 )) (t) V (S A,, ε)k 2 which implies that sup s [,ε] ϕ 2 (s) ϕ 2 (s) δ 1 sup s [,ε] ϕ 2 (s) ϕ 2 (s), Ψ 2 (ϕ 1, ϕ 2 ) (t) Ψ 2 (ϕ 1, ϕ 2 ),[,ε] δ 1 ϕ 2 ϕ 2,[,ε]. Hence, for ε = ε 3 we have Ψ (M ε 1 M ε 2 ) M ε 1 M ε 2, Ψ (Eε ) E ε and Ψ satisfies the assumptions of Lemma 6.3. We deduce that there exists (u, v) M ε 1 M ε 2 such that for each (ϕ 1, ϕ 2 ) M ε 1 M ε 2, Ψ n (ϕ 1, ϕ 2 ) (u, v) as n +. Since Ψ (E ε ) E ε and E ε is closed, we deduce that (u, v) E ε. In particular, u C 1 ([, ε], X), and the result follows. Lemma 6.5. Let Assumptions 2.1 and 2.8 be satisfied. Let τ > be fixed and F : [, τ] D(A) X be continuously differentiable. Assume that there exists an integrated solution u C ([, τ], X) of the Cauchy problem du(t) dt = Au(t) + F (t, u(t)), t [, τ], u() = x X. Assume in addition that x D(A ) and F (, x) D(A). Then u C 1 ([, τ], X) C([, τ], D(A))

27 On semilinear Cauchy problems with non-dense domain 167 and u (t) = Au(t) + F (t, u(t)), t [, τ]. Proof. Let w C([, τ], D(A)) be a solution of the equation w(t) = Ax + F (, x) + A w(s)ds + t F (s, u(s)) + D xf (s, u(s))w(s)ds, t [, τ]. From the results in Section 4 concerning globally Lipschitz perturbations, it is clear that the solution w(t) exists and is uniquely determined. Since u(t) exists on [, τ], let t [, τ) be fixed. We have for each h (, τ t) that u(t + h) u(t) = 1 [+h ] h h A u(s)ds u(s)ds + 1 [+h ] F (s, u(s))ds F (s, u(s))ds h Therefore, [ = A + u(t + h) u(t) h Denote and u(s + h) u(s) ds h ] + 1h h A u(s)ds F (s + h, u(s + h)) F (s, u(s)) ds + 1 h h w(t) = A h [ u(s + h) u(s) h h ] w(s) ds + 1 h h A u(s)ds + 1 F (s, u(s))ds Ax F (, x) h [ F (s + h, u(s + h)) F (s + h, u(s)) + h [ F (s + h, u(s)) F (s, u(s)) + ] h t F (s, u(s)) ds. v h (t) := x h := 1 h h A u(s)ds + 1 h u(t + h) u(t) h h F (s, u(s))ds. ] D x F (s, u(s))w(s) ds w(t) F (s, u(s))ds Ax F (, x).

28 168 Pierre Magal and Shigui Ruan We have Set 1 v h (t) = x h + A v h (s)ds + D x F (l (u(s + h) u(s)) + u(s)) ( ) u(s + h) u(s) w(s) dlds h K = [D x F (l (u(s + h) u(s)) + u(s)) D x F (u(s))] w(s)dlds [ F (s + h, u(s)) F (s, u(s)) ]ds. h t F (s, u(s)) sup D x F (l (u(s + h) u(s)) + u(s)) L(X,X) < +. l [,1],s [,τ],h [,τ s] Let τ > such that MV (S A,, t) 1, t [, τ]. 8(K + 1) 1 Choose γ > max(, ω A ) so that 4 1 e < 1 (ω A γ)bτ 2. Then by Proposition 2.14, we have for all γ > max(, ω A ) that e γt v h (t) M x h sup e γs v h (s) s [,τ] 1 + sup e γs s [,τ] 1 + sup e γs s [,τ] which implies that [D x F (l(u(s + h) u(s)) + u(s)) D x F (u(s))] w(s)dl [ F (s + h, u(s)) F (s, u(s)) ]dl h t F (s, u(s)), e γt v h (t) 2M x h sup e γs [D x F (l(u(s + h) u(s)) + u(s)) D x F (u(s))] w(s)dl s [,τ] 1 [ F (s + h, u(s)) F (s, u(s)) + 2 sup e γs ]dl s [,τ] h t F (s, u(s)). We now claim that lim x h =. h

29 Indeed, we have On semilinear Cauchy problems with non-dense domain 169 u(h) u() h and by Lemma 6.4, we have so = 1 h h A u(s)ds + 1 h h u(h) u() lim = Ax + F (, x), h + h lim x h =. h We conclude that for each t [, τ), we will have u(t + h) u(t) lim = w(t). h + h F (s, u(s))ds Since w C ([, τ], X), we deduce that u C 1,+ ([, τ], X). By using Lemma 6.1, we obtain the result. To extend the differentiability result to the case where F (, x ) / D(A), we notice that, since u(t) D(A) for all t [, T ], a necessary condition for the differentiability is Ax + F (, x) D(A). In fact, this condition is also sufficient. Indeed, taking any bounded linear operator B L(X), if u satisfies then we have u(t) = x + A u(t) = x + (A + B) u(s)ds + u(s)ds + F (s, u(s))ds, t [, T ], (F (s, u(s)) Bu(s)) ds, t [, T ]. So to prove the differentiability of u(t) it is sufficient to find B such that (A + B)x D(A). Take B(ϕ) = x (ϕ)ax, where x X is a continuous linear form with x (x) = 1 if x, which is possible by the Hahn-Banach theorem. We then have x D(A) = D(A + B) and (A + B)x D(A) = D(A + B). Moreover, assuming that Ax + F (, x) D(A), we obtain F (, x ) Bx D(A). By using Theorem 2.12, we deduce that A + B satisfies Assumptions 2.1 and 2.8 and we have the following theorem.

30 17 Pierre Magal and Shigui Ruan Theorem 6.6. Let Assumptions 2.1 and 2.8 be satisfied. Let τ > be fixed and F : [, τ] D(A) X be continuously differentiable. Assume that there exists an integrated solution u C ([, τ], X) of the Cauchy problem du(t) = Au(t) + F (t, u(t)), t [, τ], u() = x X. dt Assume in addition that x D(A) and Ax + F (, x) D(A). Then u C 1 ([, τ], X) C([, τ], D(A)) and u (t) = Au(t) + F (t, u(t)), t [, τ]. We now consider the nonlinear generator A N ϕ = Aϕ + F (, ϕ), ϕ D(A N ) = D(A). As in the linear case, one may define A N, (the part A N in D(A)) as follows: { } A N, = A N on D(A N, ) = y D(A) : A N y D(A). Of course, one may ask about the density of the domain D(A N, ) in D(A). Lemma 6.7. Under Assumptions and 3.2, the domain D(A N, ) is dense in X = D(A). Assume in addition that X has a positive cone X + and that Assumption 3.4 is satisfied. Then D(A N, ) X + is dense in X +. Proof. Let y D(A) be fixed. Consider the following fixed-point problem: x λ D(A) satisfies Denote (λi A F )x λ = λy x λ = λ(λi A) 1 y + (λi A) 1 F (, x λ ). Φ λ (x) = λ(λi A) 1 y + (λi A) 1 F (, x λ ), x X. Fix r >. Since y D(A), by Lemma 2.13, lim λ + (λi A) 1 L(X) =, thus we deduce that there exists λ > ω A such that Φ λ (B X (y, r)) B X (y, r), λ λ, where B(y, r) denotes the ball centered at y with radius r in X. Moreover, there exists λ 1 λ, such that for each λ λ 1, Φ λ is a strict contraction on B(y, r). Hence, for all λ λ 1, there exists x λ B(y, r) such that Φ λ (x λ ) =

31 On semilinear Cauchy problems with non-dense domain 171 x λ. Finally, using the fact that y D(A), we have lim λ(λi λ + A) 1 y = y, so lim x λ = y. λ + The proof of the positive case is similar. 7. Stability of Equilibria In this section we first investigate the local stability of an equilibrium. Proposition 7.1. Let Assumptions 2.1 and 2.8 be satisfied. Let F : D(A) X be a continuous map. Assume that (a) there exists x D(A) such that Ax + F (x) = ; (b) there exist M 1, ω <, and L L (X, X) such that T (A+L) (t) Me bωt, t ; L(X ) (c) F L Lip(BX (x,r),x) as r. Then for each γ ( ω, ) there exists ε >, such that for each x B X (x, ε), there exists a unique solution U(.)x C ([, + ), X ) of which satisfies U(t)x = x + A U(s)xds + F (U(s)x) ds, t U(t)x x e γt 2 M x x, t, x X. Proof. Without loss of generality we can assume that x =, L =, ω A <, and F Lip(BX (xη),x) as η. Choose η > such that F Lip(BX (xη ),X) < +. Let φ : (, + ) [, + ) be a Lipschitz continuous map such that =, if 2 α φ(α) [, 1], if 1 α 2 = 1, if α 1. Set F r (x) = φ(r x )F (x), x X, r >.

32 172 Pierre Magal and Shigui Ruan Then F r (x) = {, if 2 r x, F (x), if x 1 r. Choose η (, η ] and fix r = 2 η. Let x, y X. Define ϕ : [, 1] R by ϕ(t) = F r (t (x y) + y) F r (y), t [, 1]. Since F Lip(BX (x,η),x) < +, the map ϕ is Lipschitz continuous, thus we have for each pair t, s [, 1] that ( ) ϕ(t) ϕ(s) F Lip(BX (x,η),x) 2 φ Lip + 1 x y t s. In particular, for t = 1 and s =, we deduce that ( ) F r (x) F r (y) F Lip(BX (x, 2 r ),X) 2 φ Lip + 1 (x y). Thus for all r 2 η, F r Lip (X, X) and F r Lip(X,X) F Lip(B X (x, 2 r ),X) ( ) 2 φ Lip + 1 as r +. (3.5) For each r 2 η, we consider the nonlinear semigroup {U r (t)} which is a solution of U r (t)x = x + A U r (s)xds + F r (U r (s)x) ds, t. Let γ ( ω, ) be fixed. By Proposition 4.1 and (3.5), there exists r = r (γ) 2 η such that U r (t)x e γt 2M x, t, x X. Let ε (, 1 2r 1 2M ). Then, for each x BX (, ε), U r (t)x e γt 2M x 1 2r. 1 On the other hand, since F = F r on B X (, 2r ), we deduce that for each x B X (, ε), U r (.)x is a solution of U r (t)x = x + A U r (s)xds + F (U r (s)x) ds, t. The uniqueness of the solution with initial value x in B X (, ε) follows from the fact that F is locally Lipschitz continuous around and by using the arguments of Lemma 3.3 in Magal and Ruan [22].

33 On semilinear Cauchy problems with non-dense domain 173 Remark 7.2. (1) If F is continuously differentiable in B X (x, r ), we set L = DF (x). Then by the formula F (x) F (y) = it is clear that 1 DF (s(x y) + y)(x y)ds, x, y B X (x, ε), F DF (x) Lip(BX (x,r),x) as r. Thus if x is an equilibrium (i.e., assertion (a) is satisfied) and T (A+DF (x)) (t) Me bωt, t L(X ) for some M 1 and ω <, the conclusion of the proposition holds. (2) In order to see an example where the condition (c) is more appropriate than the usual differentiability condition, we consider the following case. Assume that F is quasi-linear; that is, F (x) = L(x)x, where L : X L (X, X) is a Lipschitz continuous map (but not necessarily differentiable in a neighborhood of ). Then (F L()) x (F L()) y = (L(x) L()) x (L(y) L()) y (L(x) L())x (L(y) L())x + (L(y) L())x (L(y) L())y [ x + y ] L Lip x y. So (F L()) Lip(BX+ (,ε),x) 2ε L Lip as ε. Thus, in this case we can apply the condition (c), but F is not differentiable. We now investigate the global asymptotic stability of an equilibrium. Proposition 7.3. Let Assumptions 2.1 and 2.8 be satisfied. Let F : D(A) X be a Lipschitz continuous map. Assume that (a) there exists x D(A) such that Ax + F (x) = ; (b) there exist M >, ω <, and L L (X, X), such that T (A+L) (t) Me bωt, t. L(X ) Consider {U(t)} t the C -semigroup of nonlinear operators on X which is the solution of U(t)x = x + A U(s)xds + F (U(s)x) ds, t.

34 174 Pierre Magal and Shigui Ruan Then, for each γ ( ω, ), there exists δ = δ (γ) >, such that F L Lip(X,X) δ U(t)x x e γt 2 M x x, t, x X. Thus x is a globally exponentially stable equilibrium of {U(t)} t. Proof. Replacing U(t)x by V (t)x = U(t)(x + x) x and F (.) by G(.) = F (. + x) F (x), respectively, without loss of generality we can assume that x =. Moreover, using Theorem 2.12 and replacing M by M, ω A by ω, A by A + L, and F by F L, respectively, we can further assume that L = and ω A <. Fix τ > and set ε := Mδ(τ). Let γ (ω A, ) be fixed. Choose δ = δ (γ) > such that δ Then by Lemma 2.15 we have 2εe γτε ( 1 e (ω A γ)τ ε ) 1 2. L (ϕ) L(BC γ ([,+ ),X),BC γ ([,+ ),X )) 2εe ( γτε ) 1. 1 e (ω A γ)τ ε 2δ It is sufficient to consider the problem U(.)x BC γ ([, + ), X ), U(t)x = T A (t)x + L (Ψ (U(.)x)) (t), t [, + ), where Ψ : BC γ ([, + ), X ) BC γ ([, + ), X) is defined by Ψ (ϕ) (t) = F (ϕ (t)), t [, + ). If F Lip(X,X) δ, we have L Ψ Lip(BC γ ([,+ ),X ),BC γ ([,+ ),X )) 1/2, so for each t U(.)x BC γ ([,+ ),X ) M x U(.)x BC γ ([,+ ),X ) and the result follows. Let L : X X be a bounded linear operator. In order to apply the stability theorem, one needs to prove that there exist two constants M 1 and ω <, such that T(A+L) (t) L(X ) Me bωt, t, which is also equivalent to ω ((A + L) ) := ln ( T (A+L) (t) L(X ) lim t + t ) < ω.