Bull. Korean Math. Soc. 46 (2009), No. 4, pp. 691–700
DOI 10.4134/BKMS.2009.46.4.691
EXISTENCE OF SOLUTIONS OF QUASILINEAR
INTEGRODIFFERENTIAL EVOLUTION EQUATIONS
IN BANACH SPACES
Krishnan Balachandran and Dong Gun Park
Abstract. We prove the local existence of classical solutions of quasilinear integrodifferential equations in Banach spaces. The results are
obtained by using fractional powers of operators and the Schauder fixedpoint theorem. An example is provided to illustrate the theory.
1. Introduction
The problem of existence of solutions of quasilinear evolution equations in
Banach spaces has been studied by many authors [1, 2, 5-7, 12, 15-24, 26].
Crandall and Souganidis [8] have proved the existence, uniqueness and continuous dependence of a continuously differentiable solution to the quasilinear
evolution equation
u0 (t) + A(u)u(t) = 0,
u(0) = u0
0 < t ≤ a,
under the assumptions similar to one considered by Kato [14]. Pazy [21] considered the following quasilinear equation
u0 (t) + A(t, u)u(t) = 0,
u(0) = u0
0 < t ≤ a,
and discussed the mild and classical solutions by using a fixed point argument.
The same problem has been studied to the nonhomogeneous quasilinear evolution equation
u0 (t) + A(t, u)u(t) = f (t, u),
u(0) = u0
0 < t ≤ a,
Received August 29, 2008.
2000 Mathematics Subject Classification. 34G20, 47H20.
Key words and phrases. existence of solution, quasilinear integrodifferential equation,
analytic semigroup, fixed point theorem.
This paper is supported by Dong-A University Research Fund 2006.
c
°2009
The Korean Mathematical Society
691
692
KRISHNAN BALACHANDRAN AND DONG GUN PARK
by Furuya [10], Kato [13], Sobolevski [25] and Yagi [27]. Bahuguna [3] proved
the existence, uniqueness and continuous dependance of a strong solution for
quasilinear integrodifferential equations of the form
Z t
0
u (t) + A(t, u)u(t) =
a(t − s)k(s, u(s))ds + f (t),
0 ≤ t ≤ a,
0
u(0) = u0
by using the method of lines. He also established a local classical solution
for the same equation in [4]. Oka [19] and Oka and Tanaka [20] investigated
the existence of classical solutions of abstract quasilinear integrodifferential
equations. An equation of this type occurs in a nonlinear conservation law
with memory
Z t
ut (t, x) + Ψ(u(t, x))x =
b(t − s)Ψ(u(t, x))x ds + f (t, x), t ∈ [0, a], x ∈ R,
0
u(0, x) = φ(x), x ∈ R.
It is interesting to investigate the existence problem for these type of equations in Banach spaces. The aim of this paper is to study the existence of
solutions of quasilinear integrodifferential equations in Banach spaces by using fractional powers of operators and the Schauder fixed-point theorem. The
results generalize the results of [4, 13, 21, 25].
2. Preliminaries
Consider the initial value problem
(1)
x0 (t) + A(t)x(t) = f (t)
0≤s<t≤a
x(s) = y
with the following assumptions:
(P1 ) The domain D(A(t)) = D of A(t), 0 ≤ t ≤ a is dense in X and
independent of t;
(P2 ) For t ∈ [0, a], the resolvent R(λ; A(t)) = (λI − A(t))−1 of A(t) exists
for all λ with Re λ ≤ 0 and there is a constant C such that
kR(λ; A(t))k ≤ C[|λ| + 1]−1 for Re λ ≤ 0, t ∈ [0, a];
(P3 ) There exist constants L and 0 < α ≤ 1 such that
k(A(t) − A(s))A(τ )k ≤ L|t − s|α for t, s, τ ∈ [0, a].
Theorem 2.1. Under the assumptions (P1 ) − (P3 ) there is a unique evolution
system U (t, s) on 0 ≤ s ≤ t ≤ a, satisfying
(i) kU (t, s)k ≤ M0 for 0 ≤ s ≤ t ≤ a
QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS
693
(ii) For 0 ≤ s ≤ t ≤ a, U (t, s) : X → D and t → U (t, s) is strongly
∂
differentiable in X. The derivative ∂t
U (t, s) ∈ B(X) and it is strongly
continuous on 0 ≤ s < t ≤ a. More over,
∂
U (t, s) + A(t)U (t, s) = 0 for 0 ≤ s < t ≤ a,
∂t
∂
k U (t, s)k = kA(t)U (t, s)k ≤ M0 (t − s)−1
∂t
and
kA(t)U (t, s)A−1 (s)k < M0 for 0 < s < t < a.
(iii) For every v ∈ D and t ∈ [0, a], U (t, s)v is differentiable with respect to
s on 0 ≤ s ≤ t ≤ a and
∂
U (t, s)v = U (t, s)A(s)v.
∂t
Note that (P2 ) and the fact that D is dense in X imply that for every t ∈
[0, a], −A(t) is the infinitesimal generator of an analytic semigroup. We define
the classical solutions of (1) as functions x : [s, a] → X which are continuous
for s ≤ t ≤ a, continuously differentiable for s < t ≤ a, x(t) ∈ D for s <
t ≤ a, x(s) = y and x0 (t) + A(t)x(t) = f (t) holds for s < t ≤ a. We will
call a function x(t) a solution of the initial value problem (1) if it is a classical
solution of this problem.
Theorem 2.2. Let A(t), 0 ≤ t ≤ a satisfy the conditions (P1 ) − (P3 ) and let
U (t, s) be the evolution system in Theorem 2.1. If f is Holder continuous on
[0, a], then the initial value problem (1) has, for every y ∈ X, a unique solution
x(t) given by
Z t
(2)
x(t) = U (t, s)y +
U (t, τ )f (τ )dτ.
s
The proofs of the above theorems can be found in [9, 21].
Now consider the quasilinear integrodifferential evolution equations of the
form
Z t
x0 (t) + A(t, x(t))x(t) = f (t, x(t)) +
k(t, s)g(s, x(s))ds,
(3)
0
x(0) = x0 ,
where −A(t, x) is the infinitesimal generator of an analytic semigroup in a
Banach space X. The nonlinear operators f, g : J × X → X are uniformly
bounded and continuous in all of its arguments and k : ∆ → J is continuous.
Here J = [0, a] and ∆ = {(t, s) : 0 ≤ s ≤ t ≤ a}. Throughout the paper Ci ’s
are positive constants.
Let r > 0 and take Br = {y ∈ X : kyk < r}, and assume the following
conditions:
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KRISHNAN BALACHANDRAN AND DONG GUN PARK
(i) The operator A0 = A(0, x0 ) is a closed operator with domain D dense
in X and
k(λI − A0 )−1 k ≤ C[|λ| + 1]−1
for all λ with Re λ ≤ 0 and C > 0.
(ii) The operator A−1
0 is a completely continuous operator in X.
(iii) For some α ∈ [0, 1) and for any y ∈ Br the operator A(t, A−α
0 y) is
well defined on D for all t ∈ J. Further more for any t, τ ∈ J and for
y, z ∈ Br
−α
−1
²
ρ
kA(t, A−α
(τ, A−α
0 y) − A(τ, A0 z)]A
0 z)k ≤ C1 [|t − τ | + ky − zk ],
where 0 < ² ≤ 1, 0 < ρ ≤ 1.
(iv) For every t, τ ∈ J and y, z, ∈ Br
−α
²
ρ
kf (t, A−α
0 y) − f (τ, A0 z)k ≤ C2 [|t − τ | + ky − zk ].
(v) For every t ∈ J and y, z ∈ Br
−α
ρ
kg(s, A−α
0 y) − g(s, A0 z)k ≤ C3 ky − zk .
(vi) For every t, s, τ ∈ J
|k(t, s) − k(τ, s)| ≤ C4 |t − τ |² .
(vii) x0 ∈ D(Aβ0 ) for some β > α and
kAα
0 x0 k < r.
3. Main result
Theorem 3.1. If the hypotheses (i)-(vii) are satisfied, then there exists at least
one continuously differentiable solution of the equation (3) on (0, T ] for some
T ≤ a.
Proof. In order to study the existence problem, we must introduce a set S of
functions x(t), t ∈ [0, T ] and a transformation zx = Φx defined by zx = Aα
0 z,
where z is the unique solution of
Z t
dz
−α
+ Ax (t)z = f (t, A0 x(t)) +
k(t, s)g(s, A−α
0 x(s))ds,
dt
0
z(0) = x0 .
We then show that Φ has a fixed point, that is, there is a function y ∈ S such
that Φy = y, and so x = A−α
0 y is the required solution of our problem (3).
Define the set
S = {x ∈ Y : kx(t) − x(τ )k ≤ K|t − τ |η for t, τ ∈ [0, T ], x(0) = Aα
0 x0 },
where K is a positive constant and η is any number satisfying 0 < η < β − α
and Y is a Banach space C(J, X) with usual supnorm. From hypothesis (vii),
and the definition of S it follows that if T is sufficiently small (depending on
K, η, kAα
0 x0 k), then
kx(t)k < r for t ∈ [0, T ].
QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS
695
Hence the operator Ax (t) = A(t, Aα
0 x(t)) is well defined and satisfies the conditions
²
ρ
k(Ax (t) − Ax (τ ))A−1
0 k ≤ C5 [|t − τ | + kx(t) − x(τ )k ]
≤ C6 |t − τ |µ ,
where µ = min{², ρη}. Further, if x(0) = Aα
0 x0 ,
−α α
Ax (0) = A(0, A−α
0 x(0)) = A(0, A0 A0 x0 ) = A(0, x0 ) = A0 ,
and it follows that for every t ∈ [0, T ] and λ with Reλ ≤ 0
k[λI − Ax (t)]−1 k ≤ C7 [|λ| + 1]−1 ,
µ
k[Ax (t) − Ax (τ )]A−1
x (s)k ≤ C8 |t − τ | for any t, τ, s ∈ [0, T ].
By the hypotheses (i)-(iii) there exists a fundamental solution Ux (t, s) corresponding to Ax (t), and all estimates for fundamental solutions derived in
Theorem 2.1 hold uniformly with respect to x ∈ S. From our assumptions, we
have
−β
β−α
.
kAα
0 [Ux (t1 , 0) − Ux (t2 , 0)]A0 k ≤ C9 |t1 − t2 |
From (v) and (vi), we can see that there exist constants M1 > 0, M2 > 0 such
that
kg(t, A−α
0 x(t))k ≤ M1 and |k(t, s)| ≤ M2 .
Let us take
Z
fx (t) =
f (t, A−α
0 x(t)),
t
gx (t) =
0
k(t, s)g(s, A−α
0 x(s))ds.
Then, it follows that the function fx (t) is Holder continuous such that
kfx (t) − fx (τ )k ≤ C10 |t − τ |µ , kgx (t) − gx (τ )k ≤ C11 |t − τ |µ .
Since fx (0) = f (0, A−α
0 x(0)) and gx (0) = 0 are independent of x, we have from
the above inequalities
kfx (t)k ≤ M3 , kgx (t)k ≤ M4 ,
and
° ·Z
° α
°A0
°
M3 > 0, M4 > 0
Z
t1
t2
Ux (t1 , s)(fx (s) + gx (s))ds −
0
0
¸°
°
Ux (t2 , s)(fx (s) + gx (s))ds °
°
≤ C12 |t1 − t2 |1−α .
We shall show that the operator Φ : S → Y defined by
Z t
α
(4)
Φx(t) = Aα
U
(t,
0)x
+
A
Ux (t, s)[fx (s) + gx (s)]ds
0
0 x
0
0
has a fixed point. This fixed point is the solution of equation (3). Clearly S
is closed convex and bounded subset of Y . First we show that Φ maps S into
itself. Obviously Φx(0) = Aα
0 x0 .
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KRISHNAN BALACHANDRAN AND DONG GUN PARK
For any 0 ≤ α < β ≤ 1 and 0≤ t1 ≤ t2 ≤ T , we have
kΦx(t1 ) − Φx(t2 )k
−β
β
≤ kAα
0 [Ux (t1 , 0) − Ux (t2 , 0)]A0 kkA0 x0 k
° Z t1
Z
° α
α
+°
A
U
(t
,
s)[f
(s)
+
g
(s)]ds
−
A
x
1
x
x
0
° 0
0
t2
0
°
°
Ux (t2 , s)[fx (s) + gx (s)]ds°
°.
Thus, for T sufficiently small,
kΦx(t1 ) − Φx(t2 )k ≤ rC9 |t1 − t2 |β−α + C12 |t1 − t2 |1−α
≤ K|t1 − t2 |η for some K > 0, η < β − α.
Hence Φ maps S into itself.
Next we show that this operator is continuous on the space Y . Let x1 , x2 ∈ S
−α
and set z1 = A−α
0 Φx1 , z2 = A0 Φx2 . Then,
dzi
+ Axi (t)zi
dt
zi (0)
=
fxi (t) + gxi (s)
=
x0 , i = 1, 2.
Therefore,
d
(z1 − z2 ) + Ax1 (t)(z1 − z2 )
dt
= [Ax2 (t) − Ax1 (t)]z2 + fx1 (t) − fx2 (t) + gx1 (t) − gx2 (t).
(5)
It is easy to see that the functions Ax2 (t)z2 (t) and A0 A−1
x2 (t) are uniformly
−1
Holder continuous, and so A0 z2 (t) = [A0 Ax2 (t)]Ax2 (t)z2 (t) is uniformly Holder
continuous. Similarly the functions
fx1 (t) − fx2 (t), gx1 (t) − gx2 (t)
are also uniformly Holder continuous in [τ, T ], τ > 0. Hence, we have
[z1 (t) − z2 (t)]
Z
³
Ux1 (t, s) [Ax2 (s) − Ax1 (s)]z2 (s)
0
´
+ [fx1 (s) − fx2 (s)] + [gx1 (s) − gx2 (s)] ds.
t
= Ux1 (t, τ )[z1 (τ ) − z2 (τ )] +
Since A0
Rt
0
Ux2 (t, s)[fx2 (s) + gx2 (s)]ds is a bounded function, it follows that
kA0 z2 (t)k ≤ C13 tβ−1 .
Hence we can take τ → 0 in the above equation and we get
Z t
³
[z1 (t) − z2 (t)] =
Ux1 (t, s) [Ax2 (s) − Ax1 (s)]z2 (s)
0
´
+[fx1 (s) − fx2 (s)] + [gx1 (s) − gx2 (s)] ds.
QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS
697
−α
Since z1 = A−α
0 Φx1 and z2 = A0 Φx2 and from (iii), (iv), (v) and (vi) it
follows that
Z
kΦx1 (t) − Φx2 (t)k
t
≤
0
≤
kAα
0 Ux1 (t, s)k[k[Ax2 (s) − Ax1 (s)]z2 (s)k
+ kfx1 (s) − fx2 (s)k + kgx1 (s) − gx2 (s)k]ds
Z t
C14 |t − s|−α [C15 kx1 (s) − x2 (s)kρ sβ−1
0
+ C16 kx1 (s) − x2 (s)kρ ]ds.
Hence
(6)
kΦx1 − Φx2 kY ≤ K ∗ T β−α kx1 − x2 kρY for some K ∗ > 0.
This shows that Φ : S → Y is continuous. We shall show that this operator
is completely continuous. We now claim that the set ΦS is contained in a
compact subset of Y . Indeed, the functions x(t) of S are uniformly bounded
and equicontinuous. By Arzela-Ascoli’s theorem it is sufficient to show that
for each t the set {Φx(t); x ∈ S} is contained in a compact subset of X. For
γ
each t ∈ [0, T ], we can write Φx(t) = A−γ
0 A0 Φx(t), (0 < γ < β − α). Since
γ
{A0 Φx(t) : x ∈ S} is a bounded subset of X, and since A−γ
is completely
0
continuous, it follows that the set {Φx(t) : x ∈ S} is contained in a compact
subset of X. Therefore by the Schauder fixed point theorem, Φ has a fixed
point z ∈ S such that Φz(t) = z(t) which satisfies
Z
z(t) =
Aα
0 Uz (t, 0)x0
+
Aα
0
t
Uz (t, s)[fz (s) + gz (s)]ds.
0
Then x(t) = A−α
0 z(t) satisfies
Z
x(t)
= UAα0 x (t, 0)x0 +
0
t
UAα0 x (t, s)[fAα0 x (s) + gAα0 x (s)]ds.
By Theorem 2.2, x(t) is a solution of (3).
¤
Theorem 3.2. Let the assumptions (i), (iii)-(v) hold with ρ = 1. Then the
assertion of Theorem 3.1 is valid and the solution is unique.
Proof. If ρ = 1, then from (6) shows that for T sufficiently small Φ is a contraction, that is kΦx1 − Φx2 k ≤ θkx1 − x2 k for some θ < 1. Hence by the Banach
fixed point theorem Φ has a unique fixed point.
¤
698
KRISHNAN BALACHANDRAN AND DONG GUN PARK
4. Example
Consider the following nonlinear parabolic integrodifferential equation
∂z
(7)
+ Σ|α|=2m aα (x, t, z, Dz, . . . , D2m−1 z)Dα z
∂t
Z t
2m−1
= f (x, t, z, Dz, . . . , D
z) +
k(x, t, s)g(x, s, z, Dz, . . . , D2m−1 z)ds,
0
∂j z
= 0 on ST = {(x, t) : x ∈ ∂Ω, 0 ≤ t ≤ T }, 0 ≤ j ≤ m − 1
∂ν j
u(x, 0) = 0 on Ω0 = {(x, 0) : x ∈ ∂Ω}
in a cylinder QT = Ω × (0, T ) with coefficients in QT , where Ω is a bounded
domain in Rn , ∂Ω the boundary of Ω, ν is the outward normal. Here the
parabolicity means that for any vector y 6= 0 and for arbitrary values of
z, Dz, . . . , D2m−1 z,
(−1)m Re{Σ|α|=2m aα (x, t, z, Dz, . . . , D2m−1 z)y α } ≥ C|y|2m , C > 0.
If z0 (x) ∈ C 2m−1 (Ω), then
A0 z = Σ|α|=2m aα (x, t, z, Dz, . . . , D2m−1 z)Dα z
is a strongly elliptic operator with continuous coefficients. So the condition
(i) holds. Let us take X to be Lp (Ω), 1 < p < ∞. Then A−1
maps bounded
0
subsets of Lp (Ω) in to bounded subsets of W 2m,p (Ω), so it is a completely
continuous operator in Lp (Ω). Further, if (2m − 1)/2m < α < 1, then [9]
Ω
|Dβ A0 −αz|Ω
0,p ≤ C|z|0,p , 0 ≤ |β| ≤ 2m − 1,
where C depends only on a bound on the coefficients A0 , on a module of strong
ellipticity and on a modulus of continuity of the leading coefficients. Here the
norm is defined as
 p1


X Z
α
p
|D
z(x)|
dx
=
|z|Ω
j,p


Ω
|α|≤j
for any nonnegative integer j and a real number p, 1 ≤ p < ∞. It follows that
if f and aα are continuously differentiable in all variables, then (iii) and (iv)
hold with σ = ρ = 1. Hence there exist fundamental operator solution Ux (t, s)
for the equation (7). The nonlinear functions f, g satisfy the conditions (iv),(v)
and k satisfies the condition (vi). Hence by the above theorem there exist a
local solution for the equation (7).
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Krishnan Balachandran
Department of Mathematics
Bharathiar University
Coimbatore 641 046, India
E-mail address: [email protected]
Dong Gun Park
Department of Mathematics
Dong-A University
Pusan 604-714, Korea
E-mail address: [email protected]