SeMA Journal https://doi.org/10.1007/s40324-021-00243-4 Regularity results for some class of nonautonomous partial neutral functional differential equations with finite delay Bila Adolphe Kyelem1,2 Received: 30 October 2020 / Accepted: 12 February 2021 © The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada 2021 Abstract In this paper, we study the existence and regularity results for a general class of nonautonomous neutral functional partial differential equations with finite delay We assume the linear nonautonomous part generates C0 semigroup for each fixed positive parameter The nonlinear term is assumed to be continuous with respect to the uniform norm topology The existence and uniqueness of the strict solution is established via the existence and uniqueness of mild solution under some sufficient conditions An example is given to illustrate the theoretical developed results Keywords Nonautonomous operator · C0 semigroup · Neutral partial differential equations · Mild solution · Strict solution · Evolution system Mathematics Subject Classification Primary 34G20 · 34K30 · 34K40; Secondary 47N20 Introduction In the population dynamics theory and applications, the ordinary neutral differential equations understandably received most attention in the vast literature In this vast domain, one can cite the works done in [4,12,13] Also, it is well known that the partial differential neutral systems appear in transmission line theory For example, Wu and Xia in [17] proposed the following system of partial neutral functional differential difference equations on the unit circle S ∂2 ∂ [x(., t) − q x(., t − r )] = K [x(., t) − q x(., t − r )] + f (xt ) t ≥ ∂t ∂ξ where ξ ∈ S , K is a positive constant and ≤ q < which models a circular array of identical resistively coupled lossless transmission lines B Bila Adolphe Kyelem kyeleadoc@yahoo.fr Unité de Formation et de Recherche en Sciences et Technologies, Département de Mathématiques et Informatique, Université de Ouahigouya, 01 B.P 346, Ouahigouya 01, Burkina Faso LAboratoire de Mathématiques et d’Informatique (LAMI), Université Joseph-Ki-Zerbo, 03 B.P 7021, Ouagadougou 03, Burkina Faso 123 B A Kyelem In this work which focus on the non-autonomous partial neutral differential equations with finite delay, we denote by (X , ) the Banach space X endowed with the norm and L(X , Z ) the space of linear operators from X to Z Also, we refer to D(A(t)), the domain of the operator A(t) : X → X for every t ∈ [0, T ] For the convenience, we assume that there exists a Banach space Y densely and continuously embedded in X The space C = C([−r , 0], X ) endowed with the uniform norm topology φ C = sup −r ≤θ ≤0 φ(θ ) is a Banach space The main purpose of this work is to outline the existence and uniqueness of strict solution for the following non-autonomous class of partial neutral differential equations with finite delay ⎧ d ⎪ ⎨ D(u t ) = A(t)D(u t ) + f (t, u t ), t ∈ [0, T ] dt (1.1) ⎪ ⎩ u (θ ) = ϕ(θ ), θ ∈ [−r , 0], where f : R+ × C → X is a continuous function with value in the Banach space X and for all t ∈ [0, T ], A(t) : D(A(t)) ⊆ X → X is a closed linear operator and generates an C0 -semigroup with T be some fixed positive real number We denote by u t for t ∈ [0, T ], the historic function defined on [−r , 0] by u t (θ ) = u(t + θ ) for θ ∈ [−r , 0], where u is a function from [0, T ] into X D is a bounded linear operator from C = C([−r , 0], X ) into X defined by D(φ) = φ(0) − D0 (φ) for φ ∈ C , where the operator D0 is given by D0 (φ) = −r η(θ )φ(θ ) for φ ∈ C , and η : [−r , 0] → L(X ) is of bounded variation and non atomic at zero; that is, there exists a continuous nondecreasing function γ : [0, r ] → [0, +∞) such that γ (0) = and −s dη(θ )φ(θ ) ≤ γ (s) φ C for φ ∈ C , s ∈ [0, r ] For the study of the similar non-autonomous equation of the form (1.1), the notion of evolution system associated to the family of linear operators {A(t)}t∈[0,T ] appeared The evolution system introduced for the first time in 1974 by Howland in [11], remains an important tool in the study of the quantitative and qualitative results in non-autonomous evolution equations In the same track, Friendman in [5] imposed optimal conditions to the family {A(t)}t∈[0,T ] and obtained the regularity results It is also important to note that the study of non-autonomous evolution equations stays actively the subject of many theoretical and applied branches of mathematics Among the classical relative works in the subject, we refer explicitly to [2,3,6,8,10,14] Note also that in the autonomous case where A(t) = A, the problem (1.1) has been the subject of various quantitative and qualitative studies (see [9,16]) 123 Regularity results for some class of nonautonomous partial Our paper is organized as follows: in Sect 2, we made some preliminary results and assumptions which play an important role in this paper The Sect did essentially the study of existence and uniqueness of mild solution of Eq (1.1) via some fixed point theory We dealed in Sect with the existence and uniqueness of strict solution of (1.1) The last section focused on an application to our studied theoretical results Preliminary The following definitions given in [14] are essential to the study of the problem developed in this paper Definition Let {T (t)}t≥0 be a C0 semigroup and let A be its infinitesimal generator A subspace Y of X is said A-admissible if it is an invariant subspace of {T (t)}t≥0 and the restriction of {T (t)}t≥0 to Y is a C0 semigroup in Y (i.e., it is strongly continuous in the norm Y ) Definition Let X be a Banach space A family {A(t)}t∈[0,T ] of infinitesimal generators of C0 semigroups on X is said stable if there are constants M ≥ and ω (called the stability constants) such that ]ω, +∞[⊂ ρ(A(t)) for t ∈ [0, T ] (2.1) and k R(λ; A(t j )) j=1 L(X ) ≤ M(λ − ω)−k for λ > ω (2.2) and any every sequence ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ T , k = 1, 2, Here, ρ(A(t)) is the resolvent set of the operator A(t) and R(λ; A(t)) defines the resolvent operator associated to A(t) at the point λ Remark The stability of a family {A(t)}t∈[0,T ] of infinitesimal generators of C0 semigroups on X is preserved when we replace the norm in X by an equivalent norm For t ∈ [0, T ], let A(t) be the infinitesimal generator of a C0 semigroup {Tt (s)}s≥0 on X Pazy did the following assumptions to obtain some useful results for the study of classical solutions in the non-autonomous hyperbolic problem: (H1 ) {A(t)}t∈[0,T ] is a stable family with the stability constants M, ω ˜ ˜ of (H2 ) Y ⊂ X is A(t)-admissible for t ∈ [0, T ] and the family { A(t)} t∈[0,T ] of parts A(t) ˜ ω A(t) in Y is a stable family in Y with the stability constants M, ˜ (H3 ) for t ∈ [0, T ], Y ⊂ D(A(t)), A(t) is a bounded operator from Y into X and t → A(t) is continuous in the space L(Y , X ) equipped with the uniform norm topology L(Y ,X ) Proposition [14] Let {A(t)}t∈[0,T ] be the infinitesimal generator of a C0 semigroup {Tt (s)}s≥0 on X If the family {A(t)}t∈[0,T ] satisfies the conditions (H1 ), (H2 ) and (H3 ), then there exists a unique evolution system {U (t, s) : ≤ s ≤ t ≤ T } in X verifying U (t, s) L(X ) ≤ Meω(t−s) for ≤ s ≤ t ≤ T ∂ U (t, s)v = A(s)v for v ∈ Y , ≤ s ≤ t ≤ T ∂t 123 B A Kyelem ∂ U (t, s)v = −U (t, s)A(s)v for v ∈ Y , ≤ s ≤ t ≤ T ∂s In order to obtain an evolution system that satisfies the two last conditions of the Proposition 1, Pazy formulated this additional assumption: (H+ ) There exists a family {Q(t)}t∈[0,T ] of isomorphisms of Y onto X such that, for every v ∈ Y , Q(t)v is continuously differentiable in X on [0, T ] and Q(t)A(t)Q −1 (t) = A(t) + B(t) (2.3) where {B(t)}t∈[0,T ] is a strongly continuous family of bounded operators on X Using the above additional condition, the following two results were obtained by Pazy Lemma [14, Lemma 4.4] The conditions (H1 ) and (H+ ) imply the condition (H2 ) Theorem [14] Let {A(t)}t∈[0,T ] be the infinitesimal generator of a C0 semigroup {Tt (s)}s≥0 on X If the family A(t) t ∈ [0, T ] satisfies the conditions (H1 ), (H+ ) and (H3 ) then there exists a unique evolution system {U (t, s) : ≤ s ≤ t ≤ T } in X satisfying the following: (a1 ) U (t, s) L(X ) ≤ Meω(t−s) for ≤ s ≤ t ≤ T (a2 ) ∂t∂ U (t, s)v = A(s)v for v ∈ Y , ≤ s ≤ t ≤ T ∂ U (t, s)v = −U (t, s)A(s)v for v ∈ Y , ≤ s ≤ t ≤ T (a3 ) ∂s (a4 ) U (t, s)Y ⊂ Y for ≤ s ≤ t ≤ T (a5 ) For every v ∈ Y , U (t, s)v is continuous in Y for ≤ s ≤ t ≤ T Let us give the notion of solutions which will be studied in this paper Definition Let φ ∈ C A function u : [−r , T ] → X is called a mild solution of Eq (1.1) associated to φ if: ⎧ t ⎪ ⎪ U (t, s) f (s, u s )ds for t ∈ [0, T ] ⎨ D(u t ) = U (t, s)D(u ) + ⎪ ⎪ ⎩ u0 = φ on [−r , 0] Definition Let φ ∈ C A continuous function u : [−r , T ] → X is called a strict solution of Eq (1.1) associated to φ if: ⎧ t −→ D(u t ) is continuously differentiable on [0, T ] ⎪ ⎪ ⎪ ⎪ ⎨ D(u t ) ∈ D(A(t)) for t ∈ [0, T ] ⎪ ⎪ ⎪ ⎪ ⎩ u(t) satisfies the system (1.1) for t ∈ [0, T ] Now, we will make the following assumptions which give us some sufficient conditions to obtain the regularity results: (C1 ) The domain D(A(t)) = D is independent of t ∈ [0, T ] In this case, we define on D a norm by y Y = y + A(0)y for all y ∈ D = Y Using the closedness of A(0), then Y = (D, Y ) is a Banach space (C2 ) The application t → A(t)x for all x ∈ D is continuously differentiable on R+ 123 (2.4) Regularity results for some class of nonautonomous partial (C3 ): D0 L(C ,X ) < Remark It is well known that the operator A(0) ∈ L(Y , X ) Using the fact that the family {A(t)}t∈[0,T ] has the common closed domain, the closed graph theorem gives that A(t) ∈ L(Y , X ) If the application t → A(t)x is continuously differentiable on [0, T ] then, this condition lead to supt∈[0,T ] A(t) L(Y ,X ) < +∞ via the principle of uniform boundedness The similar argument gives supt∈[0,T ] A (t) L(Y ,X ) < +∞ Remark [14] Let λ0 ∈ ρ(A(t)) Using (C1 ), {Q(t)}t∈[0,T ] given by Q(t) = λ0 I − A(t) is a family of isomorphism operators from Y to X and satisfies Q(t)A(t)Q −1 (t) = A(t) + B(t) (2.5) where B(t) = for all t ∈ [0, T ] is a strongly continuous and bounded operator on X Remark [14] When (C1 ) is satisfied and the application t → A(t)x is continuously differentiable on [0, T ] then, the conditions (H+ ) and (H3 ) are verified Now, we are able to make our first result which is the existence and uniqueness of mild solution to the problem (1.1) Existence and uniqueness of mild solution The main result of this section is the following Theorem Let {A(t)}t∈[0,T ] be a stable family of infinitesimal generators of C0 -semigroups on X and assume (C1 ), (C2 ) and (C3 ) hold Furthermore, suppose that the continuous f : R+ × C → X is lipschitzian with respect to its second argument i.e., there exists positive constant L > such that for ϕ, ψ ∈ C f (t, ϕ) − f (t, ψ) ≤ L ϕ − ψ C for all t ≥ Then, for all φ ∈ C , there exists a unique mild solution associated to (1.1) on [−r , +∞) Proof The parameter set {A(t)}t∈[0,T ] is assumed to be a stable family of infinitesimal generators of C0 -semigroups on X Consequently, the hypothesis (H1 ) hods Also, the conditions (C1 ) and (C2 ) imply the hypothesis (H+ ) and (H3 ) via the Remarks and Hence, using theorem 1, one obtains the existence of the unique evolution system {U (t, s) : ≤ s ≤ t ≤ T } associated to the family of linear operators {A(t)}t∈[0,T ] and satisfying (a1 )–(a5 ) Now, let a > and Ma = C([0, a], X ) be the space of continuous functions from [0, a] to X provided with the uniform norm topology Let us set for φ ∈ C K (φ) = {z ∈ Ma : z(0) = φ(0)} For z ∈ K (φ), we introduce the extension z˜ of z on [−r , a] by ⎧ ⎨ z(t) for t ∈ [0, a] z˜ (t) = ⎩ φ(t) for t ∈ [−r , 0] 123 B A Kyelem Moreover, consider the operator T defined on K (φ) by t T (z)(t) = U (t, 0)D(φ) + D0 (˜z t ) + U (t, s) f (s, z˜ s )ds for t ∈ [0, a] We begin to prove that T (K (φ)) ⊂ K (φ) For that, let z ∈ K (φ), t1 , t ∈ [0, a] with t1 < t Then, t T (z)(t) − T (z)(t1 ) = U (t, 0)D(φ) + D0 (˜z t ) + U (t, s) f (s, z˜ s )ds − U (t1 , 0)D(φ) + D0 (˜z t1 ) + t1 U (t1 , s) f (s, z˜ s )ds = (U (t, 0)D(φ) − U (t1 , 0)D(φ)) + D0 (˜z t ) − D0 (˜z t1 ) t1 + U (t, s) − U (t1 , s) f (s, z˜ s )ds + t U (t, s) f (s, z˜ s )ds t1 Since (t, 0) → U (t, 0)D(φ) is continuous then, lim U (t, 0)D(φ) − U (t1 , 0)D(φ) = t→t1 Note also that the linear operator D0 is continuous from C to X Therefore, using the condition (C3 ) one has D0 (˜z t ) − D0 (˜z t1 ) ≤ D0 ≤ z˜ t − z˜ t1 sup θ ∈[−r ,0] C z˜ (t + θ ) − z˜ (t1 + θ ) Using the continuity of the function z˜ on [−r , a], it follows lim D0 (˜z t ) − D0 (˜z t1 ) = in X -norm t→t1 Recalling that the function f is continuous and lipschitzian with respect to its second argument Then, f (s, z˜ s ) ≤ L sup sup s∈[0,t] where R > is some real constant Let t1 z˜ s − φ + sup s∈[0,t] f (s, φ) < R s∈[0,t] > with ≤ t1 We can write (U (t, s) − U (t1 , s)) f (s, z˜ s )ds t1 − ≤ (U (t, s) − U (t1 , s)) f (s, z˜ s )ds + t1 t1 − U (t, s) − U (t1 , s) f (s, z˜ s )ds t1 − ≤ (U (t, t1 − ) − U (t1 , t1 − )) U (t1 − , s) f (s, z˜ s )ds + 123 t1 t1 − U (t, s) − U (t1 , s) f (s, z˜ s )ds (3.1) Regularity results for some class of nonautonomous partial t1 − ≤ (U (t, t1 − ) − U (t1 , t1 − )) U (t1 − , s) f (s, z˜ s )ds eωs ds + +M R t−t1 + eωs ds t−t1 Thus, t1 U (t, s) − U (t1 , s) f (s, z˜ s )ds → 0, as t → t1 Also, we have t t U (t, s) f (s, z˜ s )ds ≤ t1 U (t, s) f (s, z˜ s ) ds t1 t ≤ RM eω(t−s) ds t1 t−t1 = RM eωs ds Then, t U (t, s) f (s, z˜ s )ds → 0, as t → t1 t1 Finally, T (z)(t) − T (z)(t1 ) → as t → t1 and a ≥ t > t1 Using a similar argument, one obtains that for t1 , t ∈ [0, a] with t1 > t T (z)(t1 ) − T (z)(t) → as t → t1 Hence T (z) ∈ K (φ) for all z ∈ K (φ) Now, let us show that T (z) is a strict contraction on K (φ) For that, let z, u ∈ K (φ) and t ∈ [0, a] t (T (z)(t) − T (u)(t)) = D0 (˜z t ) − D0 (u˜ t ) + U (t, s) f (s, z˜ s ) − f (s, u˜ s ) ds Therefore, it follows t (T (z)(t) − T (u)(t)) ≤ D0 (˜z t ) − D0 (u˜ t ) + U (t, s)[ f (s, z˜ s ) − f (s, u˜ s )] ds (T (z)(t) − T (u)(t)) ≤ D0 L(C ,X ) t z˜ t − u˜ t + M L eω(t−s) z˜ s − u˜ s C ds Since z˜ (θ ) − u(θ ˜ ) = for all θ ∈ [−r , 0], then z˜ s − u˜ s C ≤ sup 0≤τ ≤s z(τ ) − u(τ ) So, T (z)(t) − T (u)(t) ≤ D0 L(C ,X ) t + ML eωs ds z−u C , 123 B A Kyelem where z − u denotes the supremum norm in C([0, a], X ) Using (C3 ), one can choose a C small enough such that D0 L(C ,X ) a + ML eωs ds < Then, T is a strict contraction on K (φ) Therefore, T has a unique fixed point u which is the unique mild solution of Eq (1.1) on [0, a] Moreover, one can extend the solution u to [a, 2a] To prove this extension of the obtained solution, we consider the following equation ⎧ d ⎪ ⎨ D(z t ) = A(t)D(z t ) + f (t, z t ) for t ∈ [a, 2a] dt (3.2) ⎪ ⎩ z(t) = u(t) for t ∈ [−r , a] To show that Eq (3.2) has a unique mild solution, we consider the operator Ta defined on functional space K a (φ) = {z ∈ C([a, 2a], X ) : z(a) = u(a)} by t Ta (z)(t) = U (t, a)D(u a ) + D0 (˜z t ) + U (t, s) f (s, z˜ s )ds for t ∈ [a, 2a], a where the function z˜ is defined by z˜ (t) = ⎧ ⎨ z(t) for t ∈ [a, 2a] ⎩ u(t) for t ≤ a Using the similar argument, one obtains that Ta is a strict contraction on [a, 2a] that gives a unique mild solution of (3.2) on [a, 2a] which is an extension of u Proceeding inductively, the solution u is uniquely and continuously extended to [na, (n + 1)a] for all n ≥ Finally, we obtain that Eq (1.1) has a unique mild solution on [−r , +∞) Now, we make the following lemma which will play an important role for the study of existence of strict solution Lemma Let {A(t)}t∈[0,T ] be a stable family of infinitesimal generators of C0 -semigroups on X and assume (C1 ), (C2 ) and (C3 ) hold Consider φ ∈ C and h ∈ C(R+ , X ) such that D(φ) = h(0) Then, there exists a unique continuous function x on R+ which solves the following problem ⎧ ⎨ D(xt ) = h(t) for t ≥ 0, (3.3) ⎩ x(t) = φ(t) for t ∈ [−r , 0] ∞ (R+ , R+ ) such that Moreover, there exist two functions α and β in L loc xt C ≤ α(t) φ C + β(t) sup h(s) for t ≥ 0≤s≤t Proof We define for p > the space W = {x ∈ C([0, p], X ) : x(0) = φ(0)} 123 (3.4) Regularity results for some class of nonautonomous partial endowed with the uniform norm topology For x ∈ W , we define its extension x˜ on [−r , 0] by ⎧ ⎨ x(t) for t ∈ [0, p] x(t) ˜ = ⎩ φ(t) for t ∈ [−r , 0] Since φ ∈ C then, one can see that t → x˜t is continuous from [0, p] to C Let us define the function K on W by (K(x))(t) = D0 (x˜t ) + h(t) for t ≥ (3.5) We have prove that the application K has a unique fixed point on W Firstly, let us show K(W ) ⊂ W It is known that h ∈ C(R+ , X ) Consequently, h ∈ C([0, p], X ) Also, it is clear to note that h(0) = D(φ) = φ(0) − D0 (φ) It follows that (K(x))(0) = φ(0) Thus, K(W ) ⊂ W Moreover, we have to prove that the application K is a strict contraction To this, let x, y ∈ W with their respective extensions x˜ and y˜ associated to φ Then, (K(x) − K(y))(t) ≤ D0 x˜t − y˜t L(C ,X ) ≤ D0 sup L(C ,X ) 0≤s≤t ≤ D0 x(s) − y(s) x−y L(C ,X ) C W where W is the supremum norm defined on the functional space W Taking account to the assumption (C3 ), one obtains that K is a strict contraction Consequently, the problem (3.3) has a unique solution x defined on [−r , p] ∞ ([−r , p], R+ ) verifying Now, let us show the existence of the two functions α and β in L loc (3.4) For that, let x ∈ W solution of (3.3) Then, x(t) ≤ D0 (xt ) + h(t) ≤ D0 ≤ D0 ≤ D0 xt L(C ,X ) C + h(t) φ sup L(C ,X ) s∈[0,t] φ L(C ,X ) + D0 C + x(s) + h(t) C sup L(C ,X ) 0≤s≤t x(s) + h(t) Consequently, sup 0≤s≤t x(s) ≤ sup D0 0≤s≤t ≤ D0 L(C ,X ) L(C ,X ) φ C φ C + D0 + D0 sup L(C ,X ) 0≤τ ≤s sup L(C ,X ) 0≤s≤t x(τ ) + h(s) x(s) + sup h(s) 0≤s≤t 123 B A Kyelem Using (C3 ), on obtains that − D0 sup x(s) ≤ 0≤s≤t D0 L(C ,X ) − D0 > So, L(C ,X ) φ + C L(C ,X ) h(s) sup − D0 L(C ,X ) 0≤s≤t and φ C + sup x(s) ≤ 0≤s≤t φ − D0 + C L(C ,X ) 1 − D0 sup L(C ,X ) h(s) 0≤s≤t Since xt C ≤ φ C + sup x(s) , 0≤s≤t ∞ ([0, p], R+ ) such that then, we claim the existence of two functions α and β in L loc xt C ≤ α(t) φ C + β(t) sup for t ∈ [0, p] h(s) 0≤s≤t To extend the solution x on [ p, p], we consider the following functional space W1 = {u ∈ C ([ p, p], X ) : u( p) = x( p)} endowed with the uniform norm topology and the following extension function ⎧ ⎨ u(t) for t ∈ [ p, p], u(t) ˜ = ⎩ x(t) for t ∈ [−r , p] We define the application K1 on W1 by (K1 (u))(t) = D0 (u˜ t ) + h(t), for t ∈ [ p, p] Using the same arguments as above, we show that K1 is a strict contraction on W1 That leads to the existence of a unique solution u of problem (3.3) defined on [−r , p] and u is the extension of x on [−r , p] Moreover, for t ∈ [ p, p] and x ∈ W1 one has, x(t) ≤ D0 (xt ) + h(t) ≤ D0 ≤ D0 ≤ D0 L(C ,X ) xt C + h(t) xp sup L(C ,X ) s∈[ p,t] L(C ,X ) xp C + D0 C + h(t) + x(s) sup L(C ,X ) p≤s≤t x(s) + h(t) Therefore, sup p≤s≤t x(s) ≤ sup p≤s≤t ≤ D0 123 D0 L(C ,X ) L(C ,X ) xp C xp + D0 C + sup p≤τ ≤s x(τ ) sup L(C ,X ) p≤s≤t + h(s) x(τ ) + sup p≤s≤t h(s) B A Kyelem D1 f and D2 f both locally Lipschitz continuous with respect to the second variable It is worth to some necessary techniques to exhibit the strict solution of (1.1) So, let us give the main result of this work Theorem Let {A(t)}t∈[0,T ] be a stable family of infinitesimal generators of C0 -semigroups on X and assume (C1 ), (C2 ) and (C3 ) hold Furthermore, suppose that f : [0, T ] × C → X is continuously differentiable with partial derivatives D1 f and D2 f locally Lipschitz continuous with respect to the second variable i.e., there exist some positive constants L, L and L such that for all ϕ, ψ ∈ C , (i) f (t, ϕ) − f (t, ψ) ≤ L ϕ − ψ (ii) D1 f (t, ϕ) − D1 f (t, ψ) ≤ L ϕ − ψ (iii) D2 f (t, ϕ) − D2 f (t, ψ) ≤ L ϕ − ψ C for all t ∈ [0, T ]; C C for all t ∈ [0, T ]; for all t ∈ [0, T ] Let φ ∈ C([−r , 0], X ) be continuously differentiable such that D(φ) ∈ Y , D(φ ) ∈ Y and D(φ ) = A(0)D(φ) + f (0, φ) Then, the mild solution u of the problem (1.1) is a strict solution of the problem (1.1) Proof Let λ0 ∈ ρ(A(t)) Then, using remark 3, we have the existence of the family {Q(t)}t∈[0,T ] of isomorphism operators from Y to X defined as follows Q(t) = λ0 I − A(t) Moreover, let p ∈ (0, T ] and u be the mild solution of the problem (1.1) associated to φ Furthermore, we consider the following Cauchy problem of unknown function y given by ⎧ d −1 ⎪ ⎪ ⎪ dt D(yt ) = A(t)D(yt ) + Q (t)[D1 f (t, u t ) + D2 f (t, u t )Φu (y)t ] ⎪ ⎪ ⎨ (4.1) +Q −1 (t)[A(t)Q −1 (t) f (t, u t ) − λ0 f (t, u t )] for t ∈ [0, T ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y0 (θ ) = Q −1 (0)(φ (θ ) − λ0 φ(θ )) for θ ∈ [−r , 0] where Φu : C([−r , T ], X ) → C([−r , T ], X ) is some continuous function defined as follows: for all w ∈ C([−r , T ], X ) ⎧ t ⎪ ⎪ D(Φu (w)s )ds = Q −1 (t) f (t, u t ) − D(wt ) on [0, T ] ⎨ D(φ) + (4.2) ⎪ ⎪ ⎩ Φu (w)(θ ) = φ (θ ) on [−r , 0] Using the fact that f , D1 f , D2 f are lipschitz functions with respect to their second argument and also the uniform boundedness of the operators A(t), Q −1 (t) for t ∈ [0, T ] then, the Eq (4.1) has a unique mild and continuous solution y on [−r , T ] given by ⎧ t ⎪ ⎪ D (y ) = U (t, 0) D (y ) + U (t, s)Q −1 (s) A(s)Q −1 (s) f (s, u s ) − λ0 f (s, u s ) ds ⎪ t ⎪ ⎪ ⎪ ⎨ t + U (t, s)Q −1 (s) D1 f (s, u s ) + D2 f (s, u s )Φu (y)s ds, for t ∈ [0, T ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y0 (θ ) = Q −1 (0)(φ (θ ) − λ0 φ(θ )) for θ ∈ [−r , 0] (4.3) 123 Regularity results for some class of nonautonomous partial Now, we consider the function z ∈ C([−r , T ], X ) associated to the above mild solution y and defined by ⎧ t ⎪ ⎪ Φu (y)(s)ds, for t ∈ [0, p] ⎨ φ(0) + z(t) = (4.4) ⎪ ⎪ ⎩ φ(t) for t ∈ [−r , 0] Some simple computations give t D(z t ) = D(φ) + D(Φu (y)s )ds (4.5) Our objective is to show u = z on [−r , T ] Using (4.2), (4.3) and (4.5) one can write for t ∈ [0, p] t D(z t ) = D(φ) + D(Φu (y)s )ds = Q −1 (t) f (t, u t ) − D(yt ) = Q −1 (t) f (t, u t ) − U (t, 0)D(y0 ) t − U (t, s)Q −1 (s) A(s)Q −1 (s) f (s, u s ) − λ0 f (s, u s ) ds t − U (t, s)Q −1 (s) D1 f (s, u s ) + D2 f (s, u s )Φu (y)s ds Note also that D(y0 ) = Q −1 (0)(D(φ ) − λ0 D(φ)) (4.6) and the compatibility condition is D(φ ) = A(0)D(φ) + f (0, φ) (4.7) Using (4.7) in (4.6), it follows, D(y0 ) = Q −1 (0) f (0, φ) − D(φ) (4.8) As a consequence, we deduce that for t ∈ [0, p], D(z t ) = Q −1 (t) f (t, u t ) − U (t, 0)Q −1 (0) f (0, φ) + U (t, 0)D(φ) t − U (t, s)Q −1 (s) A(s)Q −1 (s) f (s, u s ) − λ0 f (s, u s ) ds t − U (t, s)Q −1 (s) D1 f (s, u s ) + D2 f (s, u s )Φu (y)s ds Coming back to (4.4), it is clear that z is differentiable function on [0, p] with z (t) = Φu (y)(t) Also, using the fact that Q −1 is differentiable on [0, p] with (Q −1 ) (t) = −Q −1 (t)Q (t)Q −1 (t) and Q (t) = −A(t) then, for all s ∈ [0, p] ∂ (Q −1 (s) f (s, z s )) ∂s ∂ ( f (s, z s )) ∂s = Q −1 (s)A(s)Q −1 (s) f (s, z s ) + Q −1 (s)[D1 f (s, z s ) + D2 f (s, z s )Φu (y)s ] = −Q −1 (s)Q (s)Q −1 (s) f (s, z s ) + Q −1 (s) 123 B A Kyelem = Q −1 (s)gz (s) with (4.9) gz (s) = D1 f (s, z s ) + D2 f (s, z s )Φu (y)s + A(s)Q −1 (s) f (s, z s ) Moreover, for all s, t ∈ [0, p] with s ≤ t we have ∂ (U (t, s)Q −1 (s) f (s, z s )) = U (t, s)Q −1 (s)gz (s) − U (t, s)A(s)Q −1 (s) f (s, z s ) ∂s = −U (t, s) λ0 I − Q(s) Q −1 (s) f (s, z s ) + U (t, s)Q −1 (s)gz (s) = U (t, s) f (s, z s ) + U (t, s)Q −1 (s) gz (s) − λ0 f (s, z s ) Consequently, integrating over to t, one has t U (t, s) f (s, z s )ds = Q −1 (t) f (t, z t ) − U (t, 0)Q −1 (0) f (0, φ) t − U (t, s)Q −1 (s)[gz (s) − λ0 f (s, z s )]ds (4.10) t which yields U (t, s) f (s, z s )ds ∈ Y Also, t U (t, 0)Q −1 (0) f (0, φ) = Q −1 (t) f (t, z t ) − U (t, s) f (s, z s )ds t − U (t, s)Q −1 (s)[gz (s) − λ0 f (s, u s )]ds We can summarize as follows, D(z t ) = Q −1 (t) f (t, u t ) − U (t, 0)Q −1 (0) f (0, φ) + U (t, 0)D(φ) t − t − U (t, s)Q −1 (s) D1 f (s, u s ) + D2 f (s, u s )Φu (y)s ds U (t, s)Q −1 (s) A(s)Q −1 (s) f (s, u s ) − λ0 f (s, u s ) ds t = Q −1 (t) f (t, u t ) − f (t, z t ) + U (t, 0)D(φ) + U (t, s) f (s, z s )ds t + U (t, s)Q −1 (s) A(s)Q −1 (s)[ f (s, z s ) − f (s, u s )] ds t + U (t, s)Q −1 (s) D1 [ f (s, z s ) − f (s, u s )] ds t + t + U (t, s)Q −1 (s) D2 [ f (s, z s ) − f (s, u s )]Φu (y)s ds U (t, s)Q −1 (s) λ0 [ f (s, u s ) − f (s, z s )] ds Consequently, t D(z t ) − D(u t ) = Q −1 (t) f (t, u t ) − f (t, z t ) + 123 U (t, s)[ f (s, z s ) − f (s, u s )]ds Regularity results for some class of nonautonomous partial t + U (t, s)Q −1 (s) A(s)Q −1 (s)[ f (s, z s ) − f (s, u s )] ds t + U (t, s)Q −1 (s) D1 [ f (s, z s ) − f (s, u s )] ds t + t + U (t, s)Q −1 (s) D2 [ f (s, z s ) − f (s, u s )]Φu (y)s ds U (t, s)Q −1 (s) λ0 [ f (s, u s ) − f (s, z s )] ds Now, let us set for all t ∈ [0, p] h (t) = Q −1 (t) f (t, u t ) − f (t, z t ) ; t h (t) = U (t, s)[ f (s, z s ) − f (s, u s )]ds; t h (t) = U (t, s)Q −1 (s) A(s)Q −1 (s)[ f (s, z s ) − f (s, u s )] ds; t h (t) = U (t, s)Q −1 (s) D1 [ f (s, z s ) − f (s, u s )] ds; t h5 = t h (t) = U (t, s)Q −1 (s) D2 [ f (s, z s ) − f (s, u s )]Φu (y)s ds; U (t, s)Q −1 (s) λ0 [ f (s, u s ) − f (s, z s )] ds The Remark implies the existence of some positive real number R0 such that for all t ∈ [0, T ], ≤ R0 (4.11) A(t) L(Y ,X ) Since t → A(t) is uniformly bounded with respect to the L(Y , X )-norm when t runs on [0, T ] then, the mapping t → Q(t) = λ0 I − A(t) is uniformly bounded with respect to the L(Y , X )-norm when t runs on [0, T ] Note also that for all t ∈ [0, T ], when λ0 > ω, one has Q −1 (t) L(X ,Y ) = R(λ0 , A(t)) L(X ,Y ) ≤ M λ0 − ω (4.12) From (4.12), one can choose λ0 big enough to obtain the existence of some positive real number R1 < and L R1 < such that sup t∈[0,T ] Q −1 (t) L(X ,Y ) ≤ R1 < Consequently, we obtain for t ∈ [0, p] h (t) = Q −1 (t)[ f (t, z t ) − f (t, u t )] ≤ Q −1 (t) L(X ,Y ) ≤ R1 L u t − z t ≤ R1 L sup s∈[0,t] f (t, z t ) − f (t, u t ) C u s − zs C 123 B A Kyelem Also, one remembers that U (t, s) ≤ Meω(t−s) when ≤ s ≤ t ≤ T Then, t h (t) = U (t, s)[ f (s, z s ) − f (s, u s )]ds t ≤ L U (t, s) u s − zs C t ≤ LM eω(t−s) u s − z s C t u s − zs ≤ L M sup C s∈[0,t] = L M sup C s∈[0,t] ds eω(t−s) ds t u s − zs ds eωs ds We can also write t h (t) = U (t, s)Q −1 (s) A(s)Q −1 (s)[ f (s, z s ) − f (s, u s )] ds t ≤ Q −1 (s) L U (t, s) ≤ R0 R12 M L t eω(t−s) u s − z s C t u s − zs ≤ R0 R12 M sup C s∈[0,t] C s∈[0,t] L(Y ,X ) Q −1 (s) L(X ,Y ) u s − zs C ds ds eω(t−s) ds t u s − zs = R0 R12 M L sup A(s) L(X ,Y ) eωs ds As above, one obtains t h (t) = U (t, s)Q −1 (s) D1 [ f (s, z s ) − f (s, u s )] ds t = L1 U (t, s) Q −1 (s) t = L M R1 L(X ,Y ) eω(t−s) u s − z s C u s − zs = L M R1 sup s∈[0,t] t C u s − zs C ds ds eωs ds We can emphasize that Φu (y) : [−r , T ] → X is bounded in the X -norm Then, there exists a positive real number R2 such that Φu (y)t C = ≤ sup Φu (y)(t + s) sup Φu (y)(s) ≤ R2 s∈[−r ,0] s∈[−r ,T ] It follows that t h (t) = 123 U (t, s)Q −1 (s) D2 [ f (s, z s ) − f (s, u s )]Φu (y)s ds Regularity results for some class of nonautonomous partial t ≤ Q −1 (s) L U (t, s) t ≤ L M R1 R2 eω(t−s) u s − z s C ≤ L M R1 R2 sup u s − zs s∈[0,t] u s − zs = L M R1 R2 sup s∈[0,t] u s − zs L(X ,Y ) t C Φu (y)s C ds ds eω(t−s) ds t C C eωs ds Otherwise, t h (t) = U (t, s)Q −1 (s) λ0 [ f (s, u s ) − f (s, z s )] ds t ≤ L|λ0 | U (t, s) Q −1 (s) t ≤ L|λ0 |M R1 L(X ,Y ) eω(t−s) u s − z s C ≤ L|λ0 |M R1 sup u s − zs s∈[0,t] = L|λ0 |M R1 sup u s − zs s∈[0,t] t C C ds ds eω(t−s) ds t C u s − zs eωs ds One obtains for t ∈ [0, p] D(z t − u t ) ≤ t L R1 + L M + R0 R12 M L + L M R1 + L M R1 R2 + L|λ0 |M R1 × sup s∈[0,t] u s − zs C eωs ds Moreover, since for all θ ∈ [−r , 0], u(θ ) = z(θ ), then one has for all s ∈ [0, t] u s − zs C ≤ max u(τ ) − z(τ ) 0≤τ ≤t Then, using lemma 2, we have for all t ∈ [0, p] zt − u t ≤ L R1 + L M + R0 R12 M L + L M R1 + L M R1 R2 + L|λ0 |M R1 × max 0≤τ ≤ p where p eωs ds u(τ ) − z(τ ) = sup0≤s≤ p β(s) One can choose p > small enough such that L R1 + L M + R0 R12 M L + L M R1 + L M R1 R2 + L|λ0 |M R1 p eωs ds < 123 B A Kyelem It follows that u = z in [−r , p] and that leads to u continuously differentiable on [0, p] with respect to the X -norm since z is continuously differentiable Furthermore, z ∈ C([0, p], Y ) Consequently, the mild solution u belongs to C([0, p], Y ) In order to extend the solution to [ p, p] in the case of p < T , we consider the following problems ⎧ ∀t ∈ [ p, p], D(yt ) = U (t, p)D(y p ) ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ + U (t, s)Q −1 (s)[A(s)Q −1 (s) f (s, u s ) − λ0 f (s, u s )]ds ⎪ ⎪ ⎪ p ⎪ ⎨ t U (t, s)Q −1 (s)[D1 f (s, u s ) + D2 (s, u s )Φu (y)s ]ds + ⎪ p ⎪ ⎪ ⎪ D(y ) = Q −1 (t) f (t, u ) − D(u ), t ∈ [0, p] ⎪ t t t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y0 = Q −1 (0)(φ − λ0 φ), on [−r , 0] with ⎧ ⎪ ⎪ ⎨ D(u p ) + ⎪ ⎪ ⎩ t p D(Φu (y)s )ds = Q −1 (t) f (t, u t ) − D(yt ) t ∈ [ p, p] Φu (y)(t) = u (t), t ∈ [−r , p] and z˜ ∈ C([−r , p], X ) defined by ⎧ t ⎪ ⎪ Φu (y)(s)ds for t ∈ [ p, p] ⎨ u( p) + p z˜ (t) = ⎪ ⎪ ⎩ z(t) for t ≤ p Using the same technique, one obtains that u = z˜ on [−r , p] Proceeding inductively, the solution u is uniquely and continuously extended to [−r , T ] Since t → D(z t ) is continuously differentiable on [0, T ], then t → D(u t ) is continuously differentiable Also, D(z t ) ∈ Y ; thus, D(u t ) ∈ D(A(t)) = Y for all t ∈ [0, T ] To end the proof, we need the following lemma: Lemma Assume that the conditions of Theorem hold Then, the mild solution u of (1.1) verifies (1.1) Proof We know that u = z on [0, T ] Then, using the mild solution form of u and taking into account the Eq (4.10), one can write for all t ∈ [0, T ], t D(u t ) = U (t, 0)D(φ) + U (t, s) f (s, u s )ds −1 = U (t, 0)D(φ) + Q (t) f (t, u t ) t − U (t, 0)Q −1 (0) f (0, φ) + U (t, s)Q −1 (s)[gu (s) − λ0 f (s, u s )]ds Using the fact that the function s → U (t, s)Q −1 (s)[gu (s) − λ0 f (s, u s )] is continuous on [0, t] for all t ∈ [0, T ], Q −1 (s)[gu (s) − λ0 f (s, u s )] ∈ Y and setting t v(t) = 123 U (t, s)Q −1 (s)[gu (s) − λ0 f (s, u s )]ds Regularity results for some class of nonautonomous partial then, we can write for all h = such that t + h ∈ [0, T ], t 1 (v(t + h) − v(t)) = h h + [U (t + h, s) − U (t, s)]Q −1 (s)[gu (s) − λ0 f (s, u s )]ds h t+h U (t + h, s)Q −1 (s)[gu (s) − λ0 f (s, u s )]ds t Also, the relation (4.9) gives ∂ Q −1 (t) f (t, u t ) = Q −1 (t)gu (t) ∂t Consequently, v (t) = lim = h→0 t (v(t + h) − v(t)) h A(t)U (t, s)Q −1 (s)[gu (s) − λ0 f (s, u s )]ds + Q −1 (t)[gu (t) − λ0 f (t, u t )] = A(t)v(t) + Q −1 (t)gu (t) − (A(t) + Q(t))Q −1 (t) f (t, u t ) ∂ Q −1 (t) f (t, u t ) − A(t)Q −1 (t) f (t, u t ) − f (t, u t ) = A(t)v(t) + ∂t Therefore, d D(u t ) = A(t)U (t, 0)D(φ) dt ∂ Q −1 (t) f (t, u t ) − A(t)U (t, 0)Q −1 (0) f (0, φ) − v (t) + ∂t = A(t) U (t, 0)D(φ) + Q −1 (t) f (t, u t ) − U (t, 0)Q −1 (0) f (0, φ) − v(t) + f (t, u t ) = A(t)D(u t ) + f (t, u t ) Using the condition D(φ ) = A(0)D(φ) + f (0, φ), we obtain finally that d D(u t ) = A(t)D(u t ) + f (t, u t ) for all t ∈ [0, T ] dt which implies that u satisfies (1.1) Coming back to the proof of Theorem 3, one has to use the Lemma to see that u verifies (1.1) Finally, we claim that u is strict solution defined on [−r , T ] of (1.1) Remark The proof of Theorem is quite long The methodology contains a lot of details and are often very important to the objective of the section In fact, I avoid ignoring these details for fear of complicating the clear understanding Nevertheless, to bypass this long calculation done in this proof, future work in this sense could be subdivised into several lemmas 123 B A Kyelem 4.1 Application As an application, we consider the following partial neutral functional differential reaction diffusion equation: ⎧ ∂ ∂2 ⎪ ⎪ ⎪ u(t, x) − qu(t − r , x) = u(t, x) − qu(t − r , x) ⎪ ⎪ ∂t ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h(θ, u(t + θ, x))dθ, t ≥ 0, x ∈ [0, π], +a(t) u(t, x) − qu(t − r , x) + b(t) −r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u(t, 0) − qu(t − r , 0) = u(t, π) − qu(t − r , π) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(θ, x) = φ0 (θ, x), θ ∈ [−r , 0], (4.13) where a : R+ → R∗− is continuous function and is assumed to be non decreasing, b : R+ → R is a bounded function and is continuously differentiable bounded function on R+ q is a positive constant such that |q| < The initial data φ0 : [−r , 0] × [0, π] → R and the functional space C will be more precised in the next h : [−r , 0] × C → R is a continuously differentiable bounded function and satisfies (e1 ) There exists some positive constant C h such that for all φ1 , φ2 ∈ C , for all θ ∈ [−r , 0], h(θ, φ1 ) − h(θ, φ2 ) ≤ C h φ1 − φ2 C (e2 ) There exists some positive constante C h such that for all φ1 , φ2 ∈ C , for all θ ∈ [−r , 0], ∂h ∂h (θ, φ1 ) − (θ, φ2 ) ≤ C h φ1 − φ2 ∂x ∂x C ∂h denotes the partial differential of h with respect to its second argument ∂x In order to write the system (4.13) in an abstract form, we introduce the space X = L ([0, π], R) Let A be the operator defined on by ⎧ ⎨ D(A) = H ((0, π), R) ∩ H01 ((0, π), R), where ⎩ Ay = y , y ∈ D(A) Let us note for all t ≥ 0, the operator A(t) defined on X as follows A(t)y = (A + a(t)I )y = y + a(t)y Since a : R+ → R∗− is continuous function, the domain D(A(t)) of the operator A(t) is equals to D(A) Consequently, D(A(t)) = D(A) is independent of t ∈ [0, T ] and is given by D(A(t)) = y ∈ X : y, y are absolutely continuous y ∈ X y(0) = y(π) = Thus, the assumption (C1 ) is verified We equiped a subspace D = D(A(t)) with the graph norm x 123 Y = x + A(0)x for every x ∈ Y = D Regularity results for some class of nonautonomous partial which is a Banach space Using the fact that the function t → a(t) is continuously differentiable on R+ then, the application t → A(t)y = (A + a(t))y is continuously differentiable on R+ Hence, the condition (C2 ) is satisfied Note also that Ay = y for y ∈ D(A) In [15], it is well known that A has a discrete spectrum, the eigenvalues are {−n : n ∈ N∗ } with the corresponding normalized eigenvectors z n (ξ ) = π sin(nξ ) Thus for y ∈ D(A) = D there holds: +∞ −n (y, z n )z n , Ay = n=1 +∞ (−n + a(t))(y, z n )z n A(t)y = n=1 In the one hand, we have D = Y = D(A(t)) = X , A(t) is closed and R+ ⊂ ρ(A(t)) ∀t ∈ [0, T ] In the other hand, it is well known that for all λ > 0, R(λ, A) for all λ > 0, one obtains R(λ, A(t)) L(X ,Y ) = R(λ − a(t), A) L(X ,Y ) ≤ Consequently, λ L(X ,Y ) 1 ≤ ≤ λ − a(t) λ Hence, using Hill–Yosida theorem, one obtains that {A(t)}t∈[0,T ] is a family of generators of C0 -semigroups on X Moreover, since a is a non decreasing and negative function, one can see that ]ω, +∞[⊂ ρ(A(t)), (4.14) where ω is chosen in (−1 + a(T ), 0) Taking account to the chapter of [14] then, there exists a constant M j ≥ such that Tt j (s) L(X ,X ) ≤ M j eωs with s ≥ Consequently, k k Tt j (s j ) j=1 Thus, there exists M = k j=1 ≤ L(X ,X ) eωs j j=1 j=1 M j ≥ such that ⎛ k ≤ M exp ⎝ω Tt j (s j ) j=1 k Mj B(X ,X ) k ⎞ sj⎠ (4.15) j=1 Using (4.14) and (4.15), one claimed via the Theorem 2.2, p.131 of [14] that, {A(t)}t∈[0,T ] is a stable family 123 B A Kyelem Let us note C = C([−r , 0]; X ) the Banach space of all continuous functions from [−r , 0] to X equiped with the norm φ C = sup θ ∈[−r ,0] φ(θ ) for all φ ∈ C We define the functions f : R+ × C → R as follows f (t, φ)(x) = b(t) −r h(θ, φ(θ, x))dθ x ∈ [0, π] and φ ∈ C Also, let us define the operators D and D0 on C to X by D(φ)(x) = φ(0)(x) − qφ(−r )(x) for all x ∈ [0, π] and D0 (φ)(x) = qφ(−r )(x) for all x ∈ [0, π] Then, D(φ) = φ(0) − D0 (φ) Setting v(t)(x) = u(t, x) and φ(θ )(x) = φ0 (θ, x) for θ ∈ [−r , 0] and x ∈ [0, π] then, (4.13) can be written as the following abstract form: ⎧ d ⎪ ⎨ D(vt ) = A(t)D(vt ) + f (t, vt ) t ≥ 0, dt (4.16) ⎪ ⎩ v(θ ) = φ(θ ), θ ∈ [−r , 0], Proposition The operator D verifies D ∈ L(C , X ) Proof Let φ ∈ C Then, D0 (φ)(x) = qφ(−r )(x) for all x ∈ [0, π] So, D0 (φ) π = = π |D0 (φ)(s)|2 ds q |φ(−r )(s)|2 ds = q φ(−r ) ≤ q2 φ C < +∞ Hence, D0 ∈ L(C , X ) It is obvious that φ(0) ∈ L(C , X ) Therefore, it is clear that D ∈ L(C , X ) Since < q < 1, then D0 L(C ,X ) < Consequently, the condition (C3 ) holds Proposition There exists a positive constant L > such that for φ, ψ ∈ C and t ∈ [0, T ] f (t, φ) − f (t, ψ) ≤ L φ − ψ C Moreover, f : [0, T ] × C → X is continuously differentiable with D1 f and D2 f locally Lipschitz continuous with respect to the second variable 123 Regularity results for some class of nonautonomous partial Proof Consider φ, ψ ∈ C and t ∈ [0, T ] We have f (t, φ)(x) − f (t, ψ)(x) = b(t) ≤ b(t) h(θ, φ(θ )(x)) − h(θ, ψ(θ )(x)) dθ −r h(θ, φ(θ )(x)) − h(θ, ψ(θ )(x)) dθ −r ≤ C h b(t) −r = C h b(t) −r |φ(θ )(x) − ψ(θ )(x)|dθ |1| φ(θ )(x) − ψ(θ )(x) dθ = C h r b(t) 2 −r φ(θ )(x) − ψ(θ )(x) dθ Consequently, f (t, φ)(x) − f (t, ψ)(x) ≤ C h2 r b(t) −r φ(θ )(x) − ψ(θ )(x) dθ Thus, f (t, φ) − f (t, ψ) π = | f (t, φ)(s) − f (t, ψ)(s)|2 ds π ≤ C h2 r b(t) = C h2 r b(t) = C h2 r ≤ C h2 r b(t) b(t) = C h2 r b(t) −r 2 −r −r |φ(θ )(s) − ψ(θ )(s)|2 dθ ds π |φ(θ )(s) − ψ(θ )(s)|2 ds dθ φ(θ ) − ψ(θ ) dθ sup −r θ ∈[−r ,0] φ−ψ = C h2 r C12 φ − ψ φ(θ ) − ψ(θ ) dθ C −r dθ C where C12 = supt∈R+ b(t) Hence, there exists positive constant L = C h rC1 > such that for φ, ψ ∈ C and t ≥ f (t, φ) − f (t, ψ) ≤ L φ − ψ C Moreover, using the fact that b is bounded and continuously differentiable, one can say that f is continuously differentiable with respect to the first argument and D1 f (t, φ)(t0 )(x) = b (t) −r h(θ, φ(θ )(x))dθ × t0 for all t0 ∈ R+ 123 B A Kyelem One has for all φ, ψ ∈ C D1 f (t, φ)(x) − D1 f (t, ψ)(x) = b (t) ≤ |b (t)| −r (h(θ, φ(θ, x)) − h(θ, φ(θ, x))) dθ h(θ, φ(θ, x)) − h(θ, φ(θ, x)) dθ −r ≤ C h |b (t)| −r = C h |b (t)| −r |φ(θ, x) − ψ(θ, x)|dθ |1||φ(θ )(x) − ψ(θ )(x)|dθ = C h r |b (t)| −r |φ(θ )(x) − ψ(θ )(x)| dθ 2 So, D1 f (t, φ)(x) − D1 f (t, ψ)(x) ≤ C h2 r b (t) −r φ(θ )(x) − ψ(θ )(x) dθ Therefore, D1 f (t, φ) − D1 f (t, ψ) π = D1 f (t, φ)(s) − D1 f (t, ψ)(s) ds π ≤ C h2 r b (t) = C h2 r |b (t)|2 −r = C h2 r |b (t)|2 −r ≤ C h2 r |b (t)|2 ≤ Cb φ − ψ −r −r π φ(θ )(s) − ψ(θ )(s) dθ ds |φ(θ )(s) − ψ(θ )(s)|2 ds dθ φ(θ ) − ψ(θ ) dθ φ−ψ C dθ C where Cb = sup C h2 r |b (t)|2 Thus, the differential function D1 f is lipschizian with respect t≥0 to the second argument and is continuously differentiable In the other word, for all φ, ϕ ∈ C , y ∈ [0, π] one has D2 f (t, φ)(ϕ)(y) = b(t) −r ∂h θ, φ(θ )(y) ϕ(θ )(y)dθ ∂x One has for all φ, ψ, ϕ ∈ C and y ∈ [0, π] D2 f (t, φ)(ϕ)(y) − D2 f (t, ψ)(ϕ)(y) = b(t) −r ≤ C h |b(t)| 123 ∂h ∂h θ, φ(θ )(y) ϕ(θ )(y) − θ, ψ(θ )(y) ϕ(θ )(y) dθ ∂x ∂x −r |φ(θ )(y) − ψ(θ )(y)||ϕ(θ )(y)|dθ Regularity results for some class of nonautonomous partial = C h |b(t)| −r ≤ C h |b(t)| |1||φ(θ )(y) − ψ(θ )(y)||ϕ(θ )(y)|dθ −r = C h r |b(t)| |1|2 dθ −r −r |φ(θ )(y) − ψ(θ )(y)|2 |ϕ(θ )(y)|2 dθ |φ(θ )(y) − ψ(θ )(y)|2 |ϕ(θ )(y)|2 dθ 2 Therefore, |D2 f (t, φ)(ϕ)(y) − D2 f (t, ψ)(ϕ)(y)|2 ≤ (C h )2 r |b(t)|2 ) × −r |φ(θ )(y) − ψ(θ )(y)|2 |ϕ(θ )(y)|2 dθ Hence, there exists some constant C > such that D2 f (t, φ) − D2 f (t, ψ) ≤ C φ − ψ C Consequently, the function D2 f is lipschizian with respect to the second argument We make the following compatibility assumptions: (e3 ) φ0 ∈ C ([−r , 0] × [0, π]; R) and φ0 (0, ) − qφ0 (−r , ) ∈ D(A) such that (e4 ) ∂φ0 ∈ C ∂θ ∂φ0 ∂φ0 ∂2 (0, x) − q (−r , x) = − φ0 (0, x) − qφ0 (−r , x) + a(0) φ0 (0, x) − ∂θ ∂θ ∂x qφ0 (−r , x) + f (0, φ0 (−r , x)) for x ∈ [0, π] Theorem Under the assumptions (e1 ), (e2 ), (e3 ) and (e4 ), the Eq (4.13) has a unique strict solution defined on [0, T ] Proof The conditions (C1 ), (C2 ) and (C3 ) hold Therefore, the existence and uniqueness of mild solution of (4.13) associated to initial value φ ∈ C is proved Moreover, the nonlinear delayed term f : [0, T ] × C → X is continuously differentiable with D1 f and D2 f locally Lipschitz continuous with respect to the second variable Consequently, Eq (4.13) has a unique strict solution defined on [0, T ] Remark The present work is one of the avenues for the study of the stability by the linearization principle for some nonautonomous neutral partial differential equations with finite delay of the form (1.1) Using the phase space which was firstly introduced by Hale and Kato in [7], the theory discussed in this paper will also be extended to the wide class of nonautonomous neutral functional partial differential equations with infinite delay involving in Banach spaces Acknowledgements This work was done with the financial support of LAboratoire de Mathématiques et d’Informatique (LAMI) of Université Joseph Ki-Zerbo, Burkina Faso The author greets in advance, the editorial board and the referees for their careful read and suggestions References Adimy, M., Ezzinbi, K.: Existence and stability in the α-norm for partial functional differential equations of neutral type Ann Mat 185(3), 437–460 (2006) 123 B A Kyelem Akrid, T., Maniar, L., Ouhinou, A.: Periodic solutions of non-densely non-autonomous differential equations with delay Afr Diaspora J Math 15(1), 25–42 (2013) Engel, K., Nagel, R.: One-parameter Semigroups for Linear evolution equations, Graduate Texts in Mathematics, vol 194 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avenues for the study of the stability by the linearization principle for some nonautonomous neutral partial differential equations with finite delay of the form (1.1) Using the... wide class of nonautonomous neutral functional partial differential equations with infinite delay involving in Banach spaces Acknowledgements This work was done with the financial support of LAboratoire... φ(0)} 123 (3.4) Regularity results for some class of nonautonomous partial endowed with the uniform norm topology For x ∈ W , we define its extension x˜ on [−r , 0] by ⎧ ⎨ x(t) for t ∈ [0, p]