Electronic Journal of Differential Equations, Vol. 2006(2006), No. 88, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
TO PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL
EQUATIONS WITH INFINITE DELAY
HASSANE BOUZAHIR
Abstract. In this paper, we establish results concerning, existence, uniqueness, global continuation, and regularity of integral solutions to some partial
neutral functional differential equations with infinite delay. These equations
find their origin in the description of heat flow models, viscoelastic and thermoviscoelastic materials, and lossless transmission lines models; see for example
[15] and [38].
1. Introduction
In this article, we consider the following nonlinear partial neutral functional
differential equations with infinite delay
∂
Dut = ADut + F (t, ut ),
∂t
u0 = φ ∈ B,
t ≥ 0,
(1.1)
where A : D(A) ⊆ E → E is a linear operator on a Banach space (E, | · |), B is the
phase space of functions mapping (−∞, 0] into E, which will be specified later, D
is a bounded linear operator from B to E defined by
Dϕ = ϕ(0) − D0 ϕ
for ϕ ∈ B.
The operator D0 is a bounded and linear from B to E and for each u : (−∞, b] → E,
b > 0, and t ∈ [0, b], ut represents, as usual, the mapping defined from (−∞, 0] to
E by
ut (θ) = u(t + θ) for θ ∈ (−∞, 0].
The operator F is an E-valued nonlinear continuous mapping on R+ × B.
Throughout this paper, we suppose that (B, k · kB ) is a (semi)normed abstract
linear space of functions mapping (−∞, 0] to E, and satisfies the following fundamental axioms which were introduced in [20] and widely discussed in [25].
(A1) There exist a positive constant H and functions K(.), M (.) form R+ → R+ ,
with K continuous and M locally bounded, such that for any σ ∈ R and
2000 Mathematics Subject Classification. 34K30, 34K40, 35R10, 45K05.
Key words and phrases. Infinite delay; integrated semigroup; neutral type;
phase space; regularity.
c
2006
Texas State University - San Marcos.
Submitted Janaury 6, 2005. Published August 9, 2006.
1
2
H. BOUZAHIR
EJDE-2006/88
a > 0, if x : (−∞, σ + a] → E, xσ ∈ B and x(.) is continuous on [σ, σ + a],
then for every t in [σ, σ + a] the following conditions hold:
(i) xt ∈ B,
(ii) |x(t)| ≤ H kxt kB , which is equivalent to
(ii’) |ϕ(0)| ≤ HkϕkB , for every ϕ ∈ B
(iii) kxt kB ≤ K(t − σ) supσ≤s≤t |x(s)| + M (t − σ)kxσ kB .
(A2) For the function x(.) in (A1), t 7→ xt is a B-valued continuous function for
t in [σ, σ + a].
(B1) The space B is complete.
Example. Define for a constant γ the following standard space
Cγ := {φ : (−∞, 0] → E continuous such that lim eγθ φ(θ) exists in E}.
θ→−∞
It is known from [25] that Cγ with the norm kφkγ = supθ≤0 eγθ |φ(θ)|, φ ∈ Cγ ,
satisfies the axioms (A1), (A2) and (B) with H = 1, K(t) = max(1, e−γt ) and
M (t) = e−γt for all t ≥ 0.
Throughout, we also assume that the operator A satisfies the Hille-Yosida condition:
(H1) There exist M̄ ≥ 0 and ω ∈ N such that ]ω, +∞[⊂ ρ(A) and
sup{(λ − ω)n k(λI − A)−n k : n ∈ N, λ > ω} ≤ M̄ .
(1.2)
Let A0 be the part of the operator A in D(A), which is defined by
D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)},
A0 x = Ax, for x ∈ D(A0 ).
It is well known that D(A0 ) = D(A) and the operator A0 generates a strongly
continuous semigroup (T0 (t))t≥0 on D(A).
Rt
From [33], we recall that for all x ∈ D(A) and t ≥ 0, one has 0 T0 (s)x ∈ D(A0 )
and
Z
t
(A
T0 (s)xds) + x = T0 (t)x.
(1.3)
0
We also recall that (T0 (t))t≥0 coincides on D(A0 ) with the derivative of the locally
Lipschitz integrated semigroup (S(t))t≥0 generated by A on E. Which is, according
to [8] and [27], a family of bounded linear operators on E, that satisfies
(i) S(0) = 0,
(ii) for any y ∈ RE, t → S(t)y is strongly continuous with values in E,
s
(iii) S(s)S(t) = 0 (S(t + r) − S(r))dr for all t, s ≥ 0, and for any τ > 0 there
exists a constant l(τ ) > 0 such that
kS(t) − S(s)k ≤ l(τ )|t − s| for all t, s ∈ [0, τ ].
This integrated semigroup is exponentially bounded, that is, there exist two constants M̄ and ω such that kS(t)k ≤ M̄ eωt for all t ≥ 0.
As stated in Hale [17], Hale and Lunel [21] and the references therein, very
much attention has been given to differential difference equations of neutral type.
The reason was applications on lossless transmission lines. The development has
concerned the general theory of partial neutral functional differential equations.
The origin of the special form (1.1) is the description of heat flow models and of the
viscoelastic and thermoviscoelastic materials dynamics; see [15] and the references
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EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
3
therein. The recent study of (1.1) has been initiated in the case of finite delay by
Hale in [18] and [19]. The motivation was a model for a continuous circular array
of resistively coupled transmission lines with mixed initial boundary conditions
introduced by Wu and Xia ([39], [40]). In addition, Magal and Ruan have stated
in [29] that (1.1) is also a special case of age structured populations model.
Chen [12] proved some results concerning the existence, uniqueness, and asymptotic behavior of (local and global) solutions of (1.1) in the case where the delay is
finite and A generates a compact C0 -semigroup on E. Based mainly on a detailed
discussion in the book by Wu [38], Adimy and Ezzinbi have published some other
interesting results about (1.1) but also with finite delay (cf. [4]-[7]).
This work (such as [1], [2] and [30]) contributes to the construction of a complete
theory about the infinite delay case. It can be seen as an extension to the case of
neutral type of some earlier results about functional differential equations with
infinite delay in [3]. We do not suppose a global Lipschitz condition as in [1] or [30]
nor a compact condition as in [2]. Under a local Lipschitz condition on F , we state
the local existence, uniqueness, continuation and regularity.
We recall that in general, neutral functional differential equations with infinite
delay are functional differential equations depending on all past and present values, which involve derivatives with infinite delay as well as the unknown function
itself. In [23] and [24], existence and regularity of solutions were established to the
following neutral functional differential equations with infinite delay
d
[x(t) − G(t, xt )] = Ax(t) + F (t, xt ),
dt
x0 = ϕ ∈ B,
t ≥ 0,
(1.4)
where A generates a strongly continuous semigroup on E. G and F are appropriate
continuous functions from [0, +∞) × B to E. The authors have essentially used
the analytic semigroup theory. More recently, in [22] the same theory was used to
prove existence of mild solutions for the so-called partial neutral functional integrodifferential equations with infinite delay using the Leray-Schauder alternative.
Finally, more discussion about the comparison between the study of (1.1) and of
(1.4) can be found in [1, 4, 9].
2. Preliminaries
Consider the system
∂
Dut = ADut if t ≥ 0,
∂t
u(θ) = ϕ(θ) if θ ∈ (−∞, 0] with ϕ ∈ B.
(2.1)
Using (1.3), we can see that a necessary condition for u : (−∞, b) → E, b > 0, to
be a solution of (2.1) is that it verifies the following integrated equation on (−∞, b)
Dut = T0 (t)Dϕ,
u0 = ϕ,
t ≥ 0,
(2.2)
where ϕ ∈ Y := {ϕ ∈ B : Dϕ ∈ D(A)}.
The following result is only the combination of [2, Lemma 3] and [1, Proposition
11] which are proved in a general framework. Precisely, here it suffices to take
h(t) := T0 (t)Dϕ.
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H. BOUZAHIR
EJDE-2006/88
Proposition 2.1. Assume that Condition (H1) is satisfied and kD0 kK(0) < 1.
Then, for given ϕ ∈ Y there exists a unique function u which is continuous on
[0, T ) and solves (2.2) on (−∞, T ). Moreover, the family of operators (T (t))t≥0
defined on Y by T (t)ϕ = ut (., ϕ) is a C0 -semigroup on Y.
We now define a fundamental integral solution Z(t) associated to (1.1). Consider
for given c ∈ E the following equation
Dzt = S(t)c if t ≥ 0,
z(t) = 0 if t ∈ (−∞, 0].
(2.3)
To our purpose, we make the following condition
(H2) There exists a continuous nondecreasing function δ : [0, +∞) → [0, +∞[,
δ(0) = 0 and a family of continuous linear operators Wε : B → E, ε ∈
[0, +∞), such that
|D0 ϕ − Dε ϕ| ≤ δ(ε)kϕkB
for ε ∈ [0, +∞) and ϕ ∈ B,
where the linear operator Dε : B → E is defined, for ε ∈ [0, +∞), by
D ε = W ε ◦ τε ,
τε (ϕ)(θ) = ϕ(θ − ε)
for ϕ ∈ B and θ ∈ (−∞, 0].
Note that Assumption (H2) implies that the operator D0 does not depend very
strongly upon ϕ(0). It is the infinite delay version of the one introduced in [6, 7].
Proposition 2.2. Assume that Conditions (H1) and (H2) are satisfied such that
K(0)kD0 k < 1. Then, for given c ∈ E, (2.3) has a unique integral solution z :=
z(.)c : (−∞, +∞) → E. Moreover, the operator Z(t) : E → B defined by
Z(t)c = zt (.)c
satisfies, for any continuous function f : [0, +∞) → E, the following properties
(i) For each T > 0, there exists a function α(.) ∈ L∞ ([0, T ], R+ ) and β ∈ R,
such that kZ(t)k ≤ α(t)etβ for all t ∈ [0, T ];
(ii) Z(t)(E) ⊆ Y, for all t ≥ 0;
(iii) For all τ > 0 there exists a constant k(τ ) > 0 such that
kZ(t)c − Z(s)ckB ≤ k(τ )|t − skc|
for all t, s ∈ [0, τ ] and c ∈ E.
(iv) For any continuous function f : [0, +∞) → E, the functions
Z t
Z t
t 7→
Z(t − s)f (s) ds and t 7→
S(t − s)f (s) ds
0
0
are continuously differentiable for all t ≥ 0 and satisfy
Z t
Z
d
1 t
(
Z(t − s)f (s) ds) = lim
T (t − s)Z(h)f (s)ds for all t ≥ 0.
dt 0
h→0+ h 0
d
D(
dt
Z
0
t
Z
1 t 0
Z(t − s)f (s) ds) = lim+
S (t − s)S(h)f (s)ds
h→0 h 0
Z t
d
=
S(t − s)f (s)ds.
dt 0
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EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
5
Sketch of Proof. Recalling that kS(t)k ≤ M̄ eω̄t for all t ≥ 0, the proof of existence,
uniqueness and (i) is only a particular case of [2, Lemma 3] where h(t) = S(t)c and
v0 = 0. To prove (ii), it suffices to remark that for any c ∈ E, S(t)c ∈ D(A) for all
t ≥ 0 and D(Z(t)c) = S(t)c. Then Z(t)c ∈ Y for all t ≥ 0. We infer (iii) from the
fact that S(.) is locally Lipschitz continuous. Finally, the proof of (iv) is exactly the
Rt
same as in [7]. Note that (iv) also ensures that 0 Z(t − s)f (s)ds is differentiable
with respect to t.
For convenience of the reader about the main equation (1.1), we recall the following definitions.
Definition 2.3. Let T > 0 and ϕ ∈ B. We consider the following definitions.
We say that a function u := u(., ϕ) : (−∞, T ) → E, 0 < T ≤ +∞, is an integral
solution of (1.1) if:
(1) u is continuous on [0, T ),
Rt
(ii) 0 Dus ds ∈ D(A) for t ∈ [0, T ),
Rt
Rt
(iii) Dut = Dϕ + A 0 Dus ds + 0 F (s, us ) ds for t ∈ [0, T ), (iv) u(t) = ϕ(t),
for all t ∈ (−∞, 0].
We deduce from [1] and [36] that integral solutions of (1.1) are given for ϕ ∈ B
such that Dϕ ∈ D(A) by the system
Z
d t
Dut = S 0 (t)Dϕ +
S(t − s)F (s, us )ds, t ∈ [0, T ),
dt 0
(2.4)
u(t) = ϕ(t), t ∈ (−∞, 0],
Definition 2.4. Let ϕ ∈ B. We say that a function u := u(., ϕ) : (−∞, T ) →
E, 0 < T ≤ +∞, is a strict solution of Eq. (1.1) if the following conditions hold:
(i) t → Dut ∈ C 1 ([0, T ), E) ∩ C([0, T ), D(A)),
(ii) u satisfies (1.1) on (−∞, T ).
Remark 2.5. It was proved in [1] that if u := u(., ϕ) : (−∞, T ) → E, 0 < T ≤
+∞, is an integral solution of (1.1) such that t → Dut belongs to C 1 ([0, T ), E),
then t → Dut belongs to C([0, T ), D(A)).
Since our method of proof needs computing integrals in B from integrals in E,
we suppose that B is normed and satisfies one of the following two extra axioms.
(C1) If (φn )n≥0 is a Cauchy sequence in B and if (φn )n≥0 converges compactly
to φ on (−∞, 0], then φ is in B and kφn − φkB → 0, as n → ∞.
(D1) For a sequence (ϕn )n≥0 in B, if kϕn kB → 0, as n → ∞, then |ϕn (θ)| → 0,
as n → ∞, for each θ ∈ (−∞, 0].
We remark that Axiom (D1) implies that the space B is normed.
Lemma 2.6 ([31]). Let B be a normed space which satisfies Axiom (C1) and f :
[0, a] → B, a > 0, be a continuous function such that f (t)(θ) is continuous for
(t, θ) ∈ [0, a] × (−∞, 0]. Then,
Z
Z a
a
f (t)(θ)dt, θ ∈ (−∞, 0].
f (t)dt (θ) =
0
0
In [1], we have also obtained the following similar result using (D1).
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H. BOUZAHIR
EJDE-2006/88
Lemma 2.7 ([1, 9]). Assume that B satisfies Axiom (D1) and f : [0, a] → B is a
continuous function. Then, for all θ ∈ (−∞, 0], the function f (.)(θ) is continuous
on [0, a] and satisfies
Z
Z a
a
f (t)dt (θ) =
f (t)(θ)dt, θ ∈ (−∞, 0].
0
0
Proposition 2.8. Let B be a normed space which satisfies Axiom (C1) or Axiom (D1) with K(0)kD0 k < 1. If there exists an integral solution u := u(., ϕ) :
(−∞, T ) → E, 0 < T ≤ +∞, of (1.1), then the function [0, T ) 3 t 7→ ut ∈ B
satisfies
Z
d t
Z(t − s)F (s, us )ds
ut = T (t)ϕ +
dt 0
(2.5)
Z
1 t
= T (t)ϕ + lim
T (t − s)Z(h)F (s, us )ds.
h→0+ h 0
Conversely, if there exists a function v ∈ C([0, T ), B) such that
v(t) = T (t)ϕ +
d
dt
Z
t
Z(t − s)F (s, v(s))ds,
t ∈ [0, T )
(2.6)
0
then v(t) = ut for all t ∈ [0, T ), where
(
v(t)(0)
u(t) =
ϕ(t)
t ∈ [0, T )
t ∈ (−∞, 0]
and u(.) is an integral solution of (1.1).
Proof. First, by Proposition 2.2, it is immediate that for any continuous function
f : [0, T ) → E,
Z t
W (t) :=
Z(t − s)f (s)ds
0
0
is continuously differentiable and W (0) = 0. Set
(
W (t)(0) if t ≥ 0
w(t) =
0
if t ∈ (−∞, 0].
By Axiom (A1)(ii’), w(t) is continuously differentiable. Lemma 2.6 or Lemma 2.7
implies that for all t ∈ [0, T ),
Z t
w(t) = (
Z(t − s)f (s)ds)(0)
0
Z t
=
(Z(t − s)f (s))(0)ds
0
Z t
=
z(t − s)f (s)ds.
0
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EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
7
In general, for all t ∈ [0, T ) and θ ∈ (−∞, 0],
Z t
(W (t))(θ) = (
Z(t − s)f (s)ds)(θ)
0
Z t
(Z(t − s)f (s))(θ)ds
=
0
Z t
z(t + θ − s)f (s)ds.
=
0
Moreover, since z(s) = 0 for all s ∈ (−∞, 0],
Z t
Z t+θ
z(t + θ − s)f (s)ds =
z(t + θ − s)f (s)ds
0
0
and (W (t))(θ) = w(t + θ). Which is equivalent to W (t) = wt . On the other hand,
we can see that for all t ∈ [0, T ) and θ ∈ (−∞, 0],
(W 0 (t))(θ) = w0 (t + θ).
Hence W 0 (t) = (w0 )t for all t ∈ [0, T ).
Now, suppose that v(., ϕ) is a solution of (2.6). The function T (t)ϕ = xt with
x : (−∞, T ) → E is the integral solution of Dxt = S 0 (t)Dϕ such that x0 = ϕ. Set
Z t
w(t) =
z(t − s)F (s, v(s))ds.
0
Then
v(t) = xt + (w0 )t = (x + w0 )t .
If we set u(t) = x(t) + w0 (t), we obtain v(t) = ut and
Z
d t
ut = T (t)ϕ +
Z(t − s)F (s, v(s))ds
dt 0
Z
d t
= T (t)ϕ +
Z(t − s)F (s, us )ds.
dt 0
Since D(T (t)ϕ) = S 0 (t)Dϕ and by Proposition 2.2,
Z
d Z t
d t
Z(t − s)F (s, us )ds =
S(t − s)F (s, us )ds,
D
dt 0
dt 0
so that u(t) is an integral solution of (1.1). Conversely, let u(., ϕ) be an integral
solution of (1.1) on (−∞, T ). Then
Z
d t
Dut = S 0 (t)Dϕ +
S(t − s)F (s, us )ds.
dt 0
By the definition of T (t),
Z
d t
Dut = D(T (t)ϕ + D(
Z(t − s)F (s, us )ds
dt 0
Z
d t
= D T (t)ϕ +
Z(t − s)F (s, us )ds
dt 0
0
= D(xt + (w )t ),
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H. BOUZAHIR
EJDE-2006/88
where x : (−∞, T ) → E is the integral solution of Dxt = S 0 (t)Dϕ, and w(t) is
defined by
Z t
z(t − s)F (s, v(s))ds.
w(t) =
0
We deduce that, D[(u − (x + w0 ))t ] = 0, and hence u − (x + w0 ) = 0. Consequently,
Z
d t
ut = xt + (w0 )t = T (t)ϕ +
Z(t − s)F (s, us )ds.
dt 0
Which completes the proof.
3. Existence and regularity of local solutions
To obtain our results on existence, uniqueness and regularity of solutions to (1.1),
we add an extra condition
(H3) F : [0, +∞[×B is Lipschitz continuous with respect to ϕ on the balls of B;
i.e., for each r > 0 there exists a constant c0 (r) > 0 such that if t ≥ 0,
ϕ1 , ϕ2 ∈ B and kϕ1 kB , kϕ2 kB ≤ r then
|F (t, ϕ1 ) − F (t, ϕ2 )| ≤ c0 (r)kϕ1 − ϕ2 kB .
Theorem 3.1. Let B be a normed space which satisfies Axiom (C1) or Axiom (D1)
with K(0)kD0 k < 1. Assume that (H1) (H2) and (H3) hold. Let ϕ ∈ B such that
Dϕ ∈ D(A). Then, there exists a maximal interval of existence (−∞, bϕ ), bϕ > 0,
and a unique mild solution u(., ϕ) of (1.1), defined on (−∞, bϕ ) and either bϕ =
+∞ or
lim sup |u(t, ϕ)| = +∞.
t→b−
ϕ
Moreover, u(t, ϕ) is a continuous function of ϕ, in the sense that if ϕ ∈ B, Dϕ ∈
D(A) and t ∈ [0, bϕ ), then there exist positive constants β and α such that, for
ψ ∈ B, Dψ ∈ D(A) and kϕ − ψkB < α, we have t ∈ [0, bψ ) and
|u(s, ϕ) − u(s, ψ)| ≤ βkϕ − ψkB for all s ∈ [0, t].
Proof. The first part of the proof is contained in [10]. We prove that the solution
depends continuously on the initial data. Let ϕ ∈ B such that Dϕ ∈ D(A) and
t ∈ [0, bϕ ) be fixed. Set
r = 1 + sup kus (., ϕ)kB ,
0≤s≤t
ωt
c(t) = M e
exp(M eωt c0 (r)kt).
Let α ∈ (0, 1) be such that c(t)α < 1 and ψ ∈ B, Dψ ∈ D(A) such that kϕ − ψkB <
α. We have
kψkB ≤ kϕkB + α < r.
Let
b0 = sup{s ∈ (0, bψ ) : kuσ (., ψ)kB ≤ r for all σ ∈ [0, s]}.
Suppose that b0 < t. We can see similarly as in [10] that for s ∈ [0, b0 ],
Z s
ωt
kus (., ϕ) − us (., ψ)kB ≤ M e (kϕ − ψkB + c0 (r)k
kuσ (., ϕ) − uσ (., ψ)kB dσ).
0
By Gronwall’s lemma, we deduce that
kus (., ϕ) − us (., ψ)kB ≤ c(t)kϕ − ψkB .
(3.1)
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9
This implies
kus (., ψ)kB ≤ c(t)α + r − 1 < r
for all s ∈ [0, b0 ].
By continuity, there exists δ > 0 such that
kus (., ψ)kB ≤ c(t)α + r − 1 < r for all s ∈ [0, b0 + δ].
It follows that b0 cannot be the largest number s > 0 such that kuσ (., ψ)kB ≤ r, for
all σ ∈ [0, s]. Thus, b0 ≥ t and t < bψ . Furthermore, kus (., ψ)kB ≤ r, for s ∈ [0, t].
Then, using inequality (3.1), we deduce the continuous dependence on the initial
data. This completes the proof of Theorem 3.1.
As in [3], we can obtain the strictness of the integral solution to (1.1) under
similar restrictive conditions on ϕ and F ; namely, (3.4) below, (H3) and
(H4) F : [0, +∞) × B → E is continuously differentiable and the derivatives Dt F
and Dϕ F satisfy the locally Lipschitz condition (H3), i.e., for each r > 0
there exist constants C1 (r), C2 (r) > 0 such that if t ≥ 0, ϕ, ψ ∈ B and
kϕkB , kψkB ≤ r then
|Dt F (t, ϕ) − Dt F (t, ψ)| ≤ C1 (r)kϕ − ψkB ,
(3.2)
kDϕ F (t, ϕ) − Dϕ F (t, ψ)k ≤ C2 (r)kϕ − ψkB .
(3.3)
Theorem 3.2. Suppose that (H4) and the assumptions of Theorem 3.1 are satisfied.
In addition, let an element ϕ of B be continuously differentiable such that
ϕ0 ∈ B,
Dϕ ∈ D(A),
Dϕ0 ∈ D(A),
Dϕ0 = ADϕ + F (0, ϕ).
(3.4)
Then, the integral solution asserted by Theorem 3.1 is a strict solution of (1.1).
Proof. Let ϕ ∈ B such that ϕ0 ∈ B, Dϕ ∈ D(A), Dϕ0 ∈ D(A) and Dϕ0 = ADϕ +
F (0, ϕ). Let u := u(., ϕ) be the unique integral solution of (1.1) on (−∞, bϕ ).
To prove that u is also a strict solution, by Remark 2.5, it suffices to show that
t 7→ Dut is continuously differentiable on [0, bϕ ). For that purpose, consider the
linear equation
∂
Dvt = ADvt + Dt F (t, ut ) + Dϕ F (t, ut ) vt , t ≥ 0,
∂t
(3.5)
v0 = ϕ0 .
Using Axiom (A2), we can set r := sup0≤s≤T kus kB for each 0 ≤ T < bϕ . Then the
fact that F is continuously differentiable and (3.3) imply that there exists β0 > 0
such that kDϕ F (t, ut )k ≤ β0 for all t ∈ [0, T ] where 0 ≤ T < bϕ . Hence for all
0 ≤ T < bϕ , the function G : [0, T ] × B → E defined by G(t, ψ) := Dt F (t, ut ) +
Dϕ F (t, ut )ψ is uniformly Lipschitzian with respect to ψ. Then, using the same
reasoning as in the proof in [1, Theorem 7], one can show that (3.5) has a unique
integral solution v on (−∞, bϕ ) given by
Z
d t
0
0
S(t − s)(Dt F (s, us ) + Dϕ F (s, us )vs )ds, t ∈ [0, bϕ )
Dvt = S (t)Dϕ +
dt 0
v0 = ϕ0 .
Let w : (−∞, bϕ ) → E be the function defined by
(
ϕ(t)
if t ∈ (−∞, 0],
Rt
w(t) =
ϕ(0) + 0 v(s) ds if t ∈ [0, bϕ ).
10
H. BOUZAHIR
EJDE-2006/88
Then, using Lemma 2.6 or Lemma 2.7,
Z t
vs ds for t ∈ [0, bϕ ).
wt = ϕ +
0
Integrating the equation of vt , we get
Z t
Z t
0
Dvs ds = S(t)Dϕ +
S(t − s)(Dt F (s, us ) + Dϕ F (s, us )vs )ds.
0
(3.6)
0
Since
t
Z
Z t
Dvs ds = D(
vs ds) = Dwt − Dϕ,
0
0
equality (3.6) becomes
Dwt = Dϕ + S(t)Dϕ0 +
t
Z
S(t − s)(Dt F (s, us ) + Dϕ F (s, us )vs )ds.
0
On the other hand, from the assumption, Dϕ0 = ADϕ + F (0, ϕ). Then
S(t)Dϕ0 = S(t)ADϕ + S(t)F (0, ϕ).
Since Dϕ ∈ D(A), we have S(t)ADϕ = S 0 (t)Dϕ − Dϕ. Hence
S(t)Dϕ0 = S 0 (t)Dϕ − Dϕ + S(t)F (0, ϕ).
Thus wt satisfies
Z
0
Dwt = S (t)Dϕ + S(t)F (0, ϕ) +
t
S(t − s)(Dt F (s, us ) + Dϕ F (s, us )vs )ds. (3.7)
0
Note that
Z
t
Z
S(t − s)F (s, ws )ds =
0
t
S(s)F (t − s, wt−s ) ds.
0
Since t 7→ wt is continuously differentiable and F (t − s, ϕ) is also continuously
differentiable, it follows that F (t−s, wt−s ) is continuously differentiable with respect
to t. Thus
Z
d t
S(t − s)F (s, ws )ds
dt 0
Z t
d
= S(t)F (0, ϕ) +
S(s)(Dt F (t − s, wt−s ) + Dϕ F (t − s, wt−s ) wt−s )ds
dt
0
Z t
= S(t)F (0, ϕ) +
S(t − s) Dt F (s, ws ) + Dϕ F (s, ws )vs ds.
0
We deduce that
Z
Z t
d t
S(t)F (0, ϕ) =
S(t−s)F (s, ws )ds−
S(t−s)(Dt F (s, ws )+Dϕ F (s, ws )vs )ds.
dt 0
0
Therefore, (3.7) becomes
Z
d t
Dwt = S 0 (t)Dϕ +
S(t − s)F (s, ws )ds
dt 0
Z t
S(t − s)(Dt F (s, ws ) + Dϕ F (s, ws )vs )ds
−
0
Z t
+
S(t − s)(Dt F (s, us ) + Dϕ F (s, us )vs )ds.
0
EJDE-2006/88
EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
11
Since the integral solution u satisfies
Dut = S 0 (t)Dϕ +
d
dt
Z
t
S(t − s)F (s, us )ds,
0
we get
Z
d t
D(ut − wt ) =
S(t − s)(F (s, us ) − F (s, ws ))ds
dt 0
Z t
S(t − s)(Dt F (s, us ) − Dt F (s, ws ))ds
−
0
Z t
S(t − s)(Dϕ F (s, us ) − Dϕ F (s, ws ))vs ds.
−
0
Let 0 ≤ T < bϕ and choose T1 := min{ε, T − T /2} with ε ∈ (0, T ], we obtain for
t ∈ [0, T1 ] and θ ∈ (−∞, 0]
−∞ < t + θ − ε ≤ t − ε ≤ 0.
Since u(θ) = w(θ) = ϕ(θ) for all θ ≤ 0, it follows that
τε (ut )(θ) = ut (θ − ε) = u(t + θ − ε) = ϕ(t + θ − ε),
τε (wt )(θ) = wt (θ − ε) = w(t + θ − ε) = ϕ(t + θ − ε).
Since Wε is linear,
Dε (ut − wt ) = Wε ◦ τε (ut − wt ) = 0,
and
D(ut − wt ) = u(t) − w(t) − D0 (ut − wt )
= u(t) − w(t) − (D0 (ut − wt ) − Dε (ut − wt )).
Consequently,
u(t) − w(t) =D0 (ut − wt ) − Dε (ut − wt )
Z
d t
+
S(t − s)(F (s, us ) − F (s, ws ))ds
dt 0
Z t
−
S(t − s)(Dt F (s, us ) − Dt F (s, ws ))ds
0
Z t
−
S(t − s)(Dϕ F (s, us ) − Dϕ F (s, ws ))vs ds.
0
Recall that by Proposition 2.2,
Z
d t
S(t − s)(F (s, us ) − F (s, ws )) ds
dt 0
Z
1 t 0
= lim
S (t − s)S(h)(F (s, us ) − F (s, ws )) ds.
h→0+ h 0
Since
1
lim sup kS(h)k < +∞.
h
+
h→0
(3.8)
12
H. BOUZAHIR
EJDE-2006/88
Hence, for suitable constants M , ω > 0 and for all t ∈ [0, T1 ],
Z
d t
S(t − s)(F (s, us ) − F (s, ws )) ds
dt 0
Z t
≤ M eωT1
|F (s, us ) − F (s, ws )| ds.
0
Since S(t) is assumed to be exponentially bounded, we have also for suitable positive
constants M̄ and ω,
Z
t
S(t − s)(Dt F (s, us ) − Dt F (s, ws )) ds
0
Z t
ωT1
|Dt F (s, us ) − Dt F (s, ws )| ds,
≤ M̄ e
0
and
Z
t
S(t − s)(Dϕ F (s, us ) − Dϕ F (s, ws ))vs ds
0
Z t
ωT1
≤ M̄ e
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
0
Set KT := max0≤t≤T K(t). Since u0 = w0 = ϕ, by Axiom (A1)(ii), for all 0 ≤ t ≤
T1 ,
kut − wt kB ≤ KT sup |u(s) − w(s)|.
0≤s≤t
From (H2) and inequality (3.8), we infer that
|u(t) − w(t)| ≤KT δ(ε) sup |u(s) − w(s)|
0≤s≤t
Z t
+ M eωT k
|F (s, us ) − F (s, ws )|ds
0
Z t
|Dt F (s, us ) − Dt F (s, ws )|ds
+ M̄ eωT
0
Z t
+ M̄ eωT
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
0
Choose ε small enough such that KT δ(ε) < 1. Thus for all t ∈ [0, T1 ],
kut − wt kB ≤ KT
sup |u(s) − w(s)|
0≤s≤T1
−1
≤ KT (1 − KT δ(ε))
ωT1
Me
t
Z
|F (s, us ) − F (s, ws )|ds
k
0
+ KT (1 − KT δ(ε))−1 M̄ eωT1
Z
+ KT (1 − KT δ(ε))−1 M̄ eωT1
Z
t
|Dt F (s, us ) − Dt F (s, ws )|ds
0
t
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
0
Set
r := max
sup kus kB ,
0≤s≤T1
sup kvs kB , sup kws kB .
0≤s≤T1
0≤s≤T1
EJDE-2006/88
EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
13
There exist C0 (r), C1 (r), C2 (r) > 0 such that, for s ∈ [0, T1 ],
|F (s, us ) − F (s, ws )| ≤ C0 (r)kus − ws kB ,
|Dt F (t, us ) − Dt F (t, ws )| ≤ C1 (r)kus − ws kB ,
kDϕ F (t, us ) − Dϕ F (t, ws )k ≤ C2 (r)kus − ws kB .
This implies that for suitable positive constants M and ω, for all t ∈ [0, T1 ],
Z t
KT M eωT1
kut − wt kB ≤
(kC0 (r) + C1 (r) + rC2 (r))
kus − ws kB ds.
1 − KT δ(ε)
0
By the Gronwall lemma, kut − wt kB for any t ∈ [0, T1 ]. Using Axiom (A1)(ii), we
deduce that u(t) = w(t) for all t ∈ [0, T1 ]. We can repeat the previous argument
on [T1 , T2 ], where T2 := min{2ε, T − T /22 } and ε ∈ (0, T ], KT δ(ε) < 1, with the
initial condition uT1 . We obtain for t ∈ [T1 , T2 ] and θ ∈ (−∞, 0],
−∞ < t + θ − ε ≤ t − ε ≤ T2 − ε ≤ ε ≤ T1 .
Since uT1 (θ) = wT1 (θ) for all θ ≤ 0, it follows that for t ∈ [T1 , T2 ],
τε (ut )(θ) = ut (θ − ε) = u(t + θ − ε) = w(t + θ − ε) = wt (θ − ε) = τε (wt )(θ)
Since Wε is linear, Dε (ut − wt ) = Wε ◦ τε (ut − wt ) = 0 and
D(ut − wt ) = u(t) − w(t) − D0 (ut − wt )
= u(t) − w(t) − (D0 (ut − wt ) − Dε (ut − wt )).
Consequently,
u(t) − w(t) =D0 (ut − wt ) − Dε (ut − wt )
Z
d t
S(t − s)(F (s, us ) − F (s, ws ))ds
+
dt T1
Z t
−
S(t − s)(Dt F (s, us ) − Dt F (s, ws ))ds
T1
t
Z
−
S(t − s)(Dϕ F (s, us ) − Dϕ F (s, ws ))vs ds.
T1
Recall that by Proposition 2.2,
Z
d t
S(t − s)(F (s, us ) − F (s, ws )) ds
dt T1
Z
1 t 0
= lim+
S (t − s)S(h)(F (s, us ) − F (s, ws )) ds,
h→0 h T1
since lim suph→0+ h1 kS(h)k < +∞. Hence, for suitable constants M , ω > 0 and for
all t ∈ [T1 , T2 ],
Z
Z t
d t
|
S(t − s)(F (s, us ) − F (s, ws )) ds| ≤ M eωT2
|F (s, us ) − F (s, ws )| ds.
dt T1
T1
14
H. BOUZAHIR
EJDE-2006/88
Since S(t) is assumed to be exponentially bounded, we have also for suitable positive
constants M̄ and ω,
Z
t
S(t − s)(Dt F (s, us ) − Dt F (s, ws )) ds
T1
t
Z
ωT2
|Dt F (s, us ) − Dt F (s, ws )| ds,
≤ M̄ e
T1
and
Z
t
S(t − s)(Dϕ F (s, us ) − Dϕ F (s, ws ))vs ds
T1
ωT2
Z
t
≤ M̄ e
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
T1
Note that maxT1 ≤t≤T K(t − T1 ) ≤ KT . Since uT1 = wT1 , by Axiom (A1)(iii), for
all T1 ≤ t ≤ T2 ,
kut − wt kB ≤ KT sup |u(s) − w(s)|,
T1 ≤s≤t
and
|u(t) − w(t)| ≤KT δ(ε) sup |u(s) − w(s)|
T1 ≤s≤t
ωT
+ Me
t
Z
|F (s, us ) − F (s, ws )|ds
k
+ M̄ eωT
+ M̄ eωT
Z
T1
t
|Dt F (s, us ) − Dt F (s, ws )|ds
T1
Z t
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
T1
Recall that KT δ(ε) < 1. Thus for all t ∈ [0, T1 ),
kut − wt kB ≤ KT
|u(s) − w(s)|
sup
T1 ≤s≤T2
−1
≤ KT (1 − KT δ(ε))
ωT2
Me
t
Z
|F (s, us ) − F (s, ws )|ds
k
0
+ KT (1 − KT δ(ε))−1 M̄ eωT2
Z
t
|Dt F (s, us ) − Dt F (s, ws )|ds
0
+ KT (1 − KT δ(ε))−1 M̄ eωT2
Z
t
kDϕ F (s, us ) − Dϕ F (s, ws )kkvs kB ds.
0
Set
r := max
sup
T1 ≤s≤T2
kus kB ,
sup
T1 ≤s≤T2
kvs kB ,
sup
kws kB ,
T1 ≤s≤T2
There exist C0 (r), C1 (r), C2 (r) > 0 such that, for s ∈ [T1 , T2 ],
|F (s, us ) − F (s, ws )| ≤ C0 (r)kus − ws kB ,
|Dt F (t, us ) − Dt F (t, ws )| ≤ C1 (r)kus − ws kB ,
kDϕ F (t, us ) − Dϕ F (t, ws )k ≤ C2 (r)kus − ws kB .
EJDE-2006/88
EXISTENCE AND REGULARITY OF LOCAL SOLUTIONS
15
This implies that for suitable positive constants M and ω, and all t ∈ [T1 , T2 ],
Z t
KT M eωT2
kut − wt kB ≤
(kC0 (r) + C1 (r) + rC2 (r))
kus − ws kB ds.
1 − KT δ(ε)
T1
By the Gronwall lemma, kut − wt kB = 0 for any t ∈ [T1 , T2 ]. Using Axiom (A1)(ii),
we deduce that u(t) = w(t) for all t ∈ [T1 , T2 ]. Proceeding inductively we obtain
u(t) = w(t) for all t ∈ [0, T ] for any T in [0, bϕ ). Finally, since
Z t
Z t
Dvs ds
t 7→ Dwt = Dϕ + D
vs ds = Dϕ +
0
0
is continuously differentiable, the function t 7→ Dut is continuously differentiable.
This completes the proof of Theorem 3.2.
Acknowledgments. The author would like to thank Professors M. Adimy and
K. Ezzinbi for helpful discussions; thanks also to Professor S. Ruan and the MSO
at the University of Miami, for the facilities offered. This research was supported
by TWAS under contract No. 04-150 RG/MATHS/AF/AC, and by the MoroccanAmerican Fulbright Visiting Scholar program.
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Hassane Bouzahir
LAMA, Université Ibn Zohr,
Ecole Nationale des Sciences Appliquées,
P. O. Box 1136 Agadir, 80 000 Morocco
E-mail address: hbouzahir@yahoo.fr
URL: www.geocities.com/hbouzahir