Volume 2008, Article ID 814947, 8 pagesdoi:10.1155/2008/814947 Research Article On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation Said Mesloub Dep
Trang 1Volume 2008, Article ID 814947, 8 pages
doi:10.1155/2008/814947
Research Article
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
Said Mesloub
Department of Mathematics, College of Sciences, King Saud University, P.O Box 2455,
Riyadh 11451, Saudi Arabia
Received 31 August 2007; Accepted 5 February 2008
Recommended by Martin Schechter
This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem We prove the existence and uniqueness of solutions in a weighted Sobolev space Proofs are based on some a priori estimates and on the Schauder fixed point theorem we also give a result which helps to establish the regularity of a solution.
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In this paper, we are concerned with a one-dimensional nonlinear parabolic integrodifferential equation with Bessel operator, having the form
u t − u xx− 1
x
0
ξu ξ, tdξ, 0
wherex, t ∈ Q T 0, 1 × 0, T.
Well posedness of the problem is proved in a weighted Sobolev space when the problem data is a related weighted space In 1, a model of a one-dimensional quasistatic contact problem in thermoelasticity with appropriate boundary conditions is given and this work is motivated by the work of Xie 1, where the author discussed the solvability of a class of nonlinear integrodifferential equations which arise from a one-dimensional quasistatic contact problem in thermoelasticity The author studied the existence, uniqueness, and regularity of solutions We refer the reader to1,2, and references therein for additional information In the present paper, following the method used in1, we will prove the existence and uniqueness
of W σ,2 2,1 Q T see below for definition solutions of a nonlinear parabolic integrodifferential
Trang 2equation with Bessel operator supplemented with a one point boundary condition and an initial condition The proof is established by exploiting some a priori estimates and using a fixed point argument
2 The problem
We consider the following problem:
u t − u xx− 1
x
0
ξu ξ, tdξ, 0
f, x, t ∈ Q T 0, 1 × 0, T, 2.1
where gx and fx, t are given functions with assumptions that will be given later.
In this paper, · 2
L2
μ Q Tdenotes the usual norm of the weighted space L2
μ Q T , where we use the weights μ σ, ρ and σ x2while ρ x The respective inner products on L2
ρ Q T and
L2
σ Q T are given by
u, v L2Q T
Q T
σ Q T
Q T
Let W σ,2 1,0 Q T be the subspace of L2Q T with finite norm
u2
W σ,2 1,0 Q T u2
L2
σ Q T u x2
L2
and V σ W 2,1
σ,2 Q T be the subspace of W 1,0
σ,2 Q T whose elements satisfy u t , u xx ∈ L2
σ Q T In
general, a function in the space W σ,p i,j Q T , with i, j nonnegative integers possesses x-derivatives
up to ith order in the L p σ Q T , and tth derivatives up to jth order in L p
σ Q T We also use
weighted spaces in the interval0, 1 such as L2
σ 0, 1 and H1
σ 0, 1, whose definitions are analogous to the spaces on Q T We set
W σ,20
0, 1 L2
σ
0, 1, W σ,21
0, 1 H1
σ
0, 1, W σ,2 0,0
Q T
L2
σ
Q T
. 2.6 For general references and proprieties of these spaces, the reader may consult3
Throughout this paper, the following tools will be used
1 Cauchy inequality with ε see, e.g., 4,
|ab| ≤ ε
2|a|2 1
which holds for all ε > 0 and for arbitrary a and b.
2 An inequality of Poincar´e type,
Ix u2
L2Q T
x
0
u ξ, tdξ
2
L2Q T≤ 1
2u2
whereIx ux
0 u ξ, tdξ see 5, Lemma 1
3 The well-known Gronwall lemma see, e.g., 6, Lemma 7.1.
Trang 3Remark 2.1 The need of weighted spaces here is because of the singular term appearing in the
left-hand side of2.1 and the annihilation of inconvenient terms during integration by parts
3 Existence and uniqueness of the solution
We are now ready to establish the existence and uniqueness of V σ solutions of problem2.1–
2.3 We first start with a uniqueness result
Theorem 3.1 Let f ∈ L2
σ Q T and gx ∈ W1
σ,2 0, 1 Then problem 2.1–2.3, has at most one
solution in V σ
w2x, t, where
w i x, t
t
0
then the function θx, t satisfies
Lθ θ t− 1
x
xθ x
x max
x
0
ξu1ξ, tdξ, 0
− max
x
0
ξu2ξ, tdξ, 0
If we denote by
β i x, t max
x
0
ξu i ξ, tdξ, 0
then calculating the two integrals
Q T 2x2θ Lθ dx dt,Q
T 2x2θ t Lθ dx dt, using conditions 3.3,
3.4, and a combining with −Q T 2xθ x Lθ dx dt, we obtain
2θ t2
L2
σ Q T 2θ x2
L2
σ Q Tθ x2
L2Q Tθ ·, T2
L2
σ 0,1θ x ·, T2
L2
σ 0,1
−2θ, θ x
L2Q T 2θ t , β1− β2
L2
σ Q T2
θ, β1− β2
L2
σ Q T− 2θ x , β1− β2
L2Q T.
3.6
In light of inequalities2.7 and 2.8, each term of the right-hand side of 3.6 is estimated as follows:
−2θ, θ x
L2Q T≤ θ2
L2
σ Q Tθ x2
L2Q T,
2
θ, β1− β2
L2
σ Q T≤ 4θ2
L2
σ Q T1
8θ t2
L2
σ Q T,
2
θ t , β1− β2
L2
σ Q T≤θ t2
L2
σ Q T 1
2θ t2
L2
σ Q T,
−2θ x , β1− β2
L2Q T≤ 4θ x2
L2
σ Q T1
8θ t2
L2
σ Q T.
3.7
Trang 4Therefore, using inequalities3.7, we infer from 3.6
θ t2
L2
σ Q Tθ ·, T2
L2
σ 0,1θ x ·, T2
L2
σ 0,1 ≤ 20θ2
L2
σ Q T 20θ x2
L2
σ Q T. 3.8
By applying Gronwall’s lemma to3.8, we conclude that
θ t2
L2
Hence u1 u2.
We now prove the existence theorem
Theorem 3.2 Let f ∈ L2
σ Q T and gx ∈ W1
σ,2 0, 1 be given and satisfying
f2
L2
σ Q T g2
W1
σ,2 0,1 ≤ c2
for c2> 0 small enough and that
σ,2 Q T of problem 2.1–2.3.
Proof We define, for positive constants C and D which will be specified later, a class of
functions W WC, D which consists of all functions v ∈ L2
σ Q T satisfying conditions 2.2,
2.3, and
v V σ ≤ C, v t
L2
Given v ∈ WC, D, the problem
u t− 1
x
xu x
x Jv f, x, t ∈ Q T ,
u x 1, t 0, t ∈ 0, T,
u x, 0 gx, x ∈ 0, 1,
3.13
where
x
0
ξv ξ, tdξ, 0
has a unique solution u ∈ V σ We define a mapping h such that u hv.
Once it is proved that the mapping h has a fixed point u in the closed bounded convex subset WC, D, then u is the desired solution.
Trang 5We, first, show that h maps WC, D into itself For this purpose we write u in the form
u w ζ, where w is a solution of the problem
and ζ is a solution of the problem
ζ t − ζ xx− 1
x ζ x fx, t, x, t ∈ Q T , 3.18
By multiplying3.15, 3.18, respectively, by the operators, O1w 2x2w 2x2w t − 6xw xand
O2ζ 2x2ζ 2x2ζ t − 6xζ x , then integrating over Q T , we obtain
2Lw, wL2
σ Q T 2Lw, w t
L2
σ Q T− 6Lw, w x
L2Q T
2Jv, w L2
σ Q T 2Jv, w t
L2
σ Q T− 6Jv, w x
L2Q T ,
3.21
2Lζ, ζL2
σ Q T 2Lζ, ζ t
L2
σ Q T− 6Lζ, ζ x
L2Q T
2f, ζ t
L2
σ Q T 2f, ζ L2
σ Q T− 6f, ζ x
L2Q T.
3.22
By using conditions 3.16, 3.17, 3.19, 3.20, an evaluation of the left-hand side of both equalities3.21 and 3.22 gives, respectively,
w x, T2
L2
σ 0.1 2w x2
L2
σ Q T 2w, w x
L2Q Tw x x, T2
L2
σ 0.1
2w t2
L2
σ Q T 2w t , w x
L2Q T 3w x2
L2Q T− 6w t , w x
L2Q T
2Jv, w L2
σ Q T 2Jv, w t
L2
σ Q T− 6Jv, w x
L2Q T,
3.23
and applying inequalities2.7, 2.8, and Gronwall’s lemma, we obtain the following estimat-es:
ζ2
V σ ≤ 7 exp7T f2
L2
σ Q T g2
W1
σ,2 0,1
≤ 7 exp7Tc2
2;
3.24
w2
V ≤ 7 exp7TJv2
Trang 6We also multiply by x and square both sides of 3.15, integrate over Q T, use the integral
−2Q T xw x Lw dx dt, then integrate by parts and using inequality 2.7, we obtain
w t2
L2
σ Q Tw xx2
L2
σ Q Tw x ·, T2
L2
σ Q T≤ 2Jv L2
σ Q T. 3.26 Direct computations yield
Jv2
L2
σ Q T≤ 1
4
2c21 7 exp7Tc2
2
By choosing c1and c2small enough in the previous inequality, we obtain
Jv L2
Inequalities3.21–3.25 then give
u2
V σ ≤ 2w2
V σ 2ζ2
V σ ≤ 14 exp7Tc22 c2
1
,
u t2
L2
σ Q T≤ 2w t2
L2
σ Q T 2ζ t2
L2
σ Q T4c21 14 exp7Tc2
2.
3.29
At this point we take C ≥ √14 exp7T/2 c2
1 c2
2 and D ≥ 4c21 14 exp7Tc2
2, so that it
follows from the last two inequalities thatu V σ ≤ C and u tL2
σ Q T≤ D from which we deduce that u ∈ W WC, D, hence h maps W into itself To show that h is a continuous mapping,
we consider v1, v2 ∈ W and their corresponding images u1and u2 It is straightforward to see
that U u1− u2satisfies
x
0
ξv1ξ, tdξ, 0
− d
x
0
ξv2ξ, tdξ, 0
,
U x 1, t 0, U x, 0 0.
3.30
Define the function px, t by the formula
p x, t
t
0
then it follows from3.26 and 3.28 that px, t satisfies
p t − p xx− 1
x p x F max
x
0
ξv1ξ, tdξ, 0
− max
x
0
ξv2ξ, tdξ, 0
,
p x 1, t 0, p x, 0 0.
3.32
Since
F2
L2
σ Q T≤v1− v22
L2
then
U2
L2Q ≤ 6v1− v22
Trang 7hv1− hv22
L2
σ Q T≤ 6v1− v22
L2
hence the continuity of the mapping h The compactness of the set W C, D is due to the
following
Theorem 3.3 Let E0⊂ E ⊂ E1with compact embedding (reflexive Banach spaces) (see [ 4 , 7 ]) Suppose
0, T; E0
, ω t ∈ L q
Note that L2
σ 0, T; L2
σ 0, 1 L2
σ Q T , hWC, D ⊂ WC, D ⊂ L2
σ Q T By the Schauder fixed point theorem the mapping h has a fixed point u in W C, D.
Remark 3.5 The following theorem gives an a priori estimate which may be used in establishing
a regularity result for the solution of2.1–2.3 More precisely, one should expect the solution
to be in W σ,p 2,1 Q T with p ≤ ∞.
Theorem 3.6 Let u ∈ V σ be a solution of problem 2.1–2.3, then the following a priori estimate
holds:
sup
0≤t≤T
u ·, T2
W1
σ,2 0,1u t2
L2
σ Q Tu xx2
L2
σ Q Tu x2
L2
σ Q T
≤ 80 exp80T g2
W1
σ,2 0,1 f2
L2
σ Q T .
3.37
u t2
L2
σ Q Tu xx2
L2
σ Q Tu x ·, T2
L2
σ 0,1 − 2u t , u xL2Q T
g x2
L2
σ 0,1
Q T
x2
d
x
0
ξu ξ, tdξ, 0
f
2
dx dt.
3.38
Multiplying2.1 by 2x2u t , integrating over Q T , carrying out standard integrations by parts,
and using conditions2.2 and 2.3 yields
2u t2
L2
σ Q Tu x ·, T2
L2
σ 0,1 2u t , u x
L2Q T
g x2
L2
σ 0,1 2
Q
x2u t f dx dt 2
Q
x2u t d
x
0
ξu ξ, tdξ, 0
dx dt.
3.39
Trang 8Adding side to side equalities 3.38 and 3.39, then using inequalities 2.7 and 2.8 to estimate the involved integral terms to get
1
4u t2
L2
σ Q Tu xx2
L2
σ Q T 2u x ·, T2
L2
σ 0,1≤ 2g x2
L2
σ 0,1 6f2
L2
σ Q T. 3.40 Let be the elementary inequality
1
8u ·, T2
L2
σ 0,1≤ 1
8u t2
L2
σ Q T 1
8u2
L2
σ Q T 1
8g2
L2
σ 0,1 3.41 Adding the quantityu x2
L2
σ Q Tto both sides of3.38, then combining the resulted inequality with3.39, we obtain
u ·, T2
L2
σ 0,1u x ·, T2
L2
σ 0,1u t2
L2
σ Q Tu xx2
L2
σ Q Tu x2
L2
σ Q T
≤ 48 g2
W1
σ,2 0,1 f2
L2
σ Q T u2
L2
σ Q Tu x2
L2
σ Q T .
3.42
Applying Gronwall’s lemma to3.40 and then taking the supremum with respect to t over the
interval0, T, we obtain the desired a priori bound 3.37
Acknowledgments
The author is grateful to the anonymous referees for their helpful suggestions and comments which allowed to correct and improve the paper This work has been funded and supported
by the Research Center Project no Math/2008/19 at King Saud University
References
1 W Q Xie, “A class of nonlinear parabolic integro-differential equations,” Differential and Integral
Equations, vol 6, no 3, pp 627–642, 1993.
2 P Shi and M Shillor, “A quasistatic contact problem in thermoelasticity with a radiation condition for
the temperature,” Journal of Mathematical Analysis and Applications, vol 172, no 1, pp 147–165, 1993.
3 R A Adams, Sobolev Spaces, vol 65 of Pure and Applied Mathematics, Academic Press, New York, NY,
USA, 1975.
4 J.-L Lions, ´Equations Diff´erentielles Op´erationnelles et Probl`emes aux Limites, vol 111 of Die Grundlehren
der mathematischen Wissenschaften, Springer, Berlin, Germany, 1961.
5 O A Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, vol 49 of Applied
Mathematical Sciences, Springer, New York, NY, USA, 1985.
6 L Garding, Cauchy Problem for Hyperbolic Equations, Lecture Notes, University of Chicago, Chicago, Ill,
USA, 1957.
7 J Simon, “Compact sets in the space L p 0,T; B,” Annali di Matematica Pura ed Applicata, vol 146, no 1,
pp 65–96, 1987.
8 J.-P Aubin, “Un th´eor`eme de compacit´e,” Comptes Rendus de l’Acad´emie des Sciences, vol 256, pp 5042–
5044, 1963.
9 Ju A Dubinski˘ı, “Weak convergence for nonlinear elliptic and parabolic equations,” Matematicheskii
Sbornik, vol 67109, no 4, pp 609–642, 1965, Russian.