On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation

Volume 2008, Article ID 814947,8pages 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

Correspondence should be addressed to Said Mesloub,[email protected] 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.

Copyrightq2008 Said Mesloub. This is an open access article distributed under the Creative 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 xu_x d

dtmax

x 0

ξuξ, tdξ,0

f, 1.1 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 ofW_σ,2^2,1QT see below for definition solutions of a nonlinear parabolic integrodifferential

equation 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:

ut−uxx−1 xux d

dtmax

x 0

ξuξ, tdξ,0

f, x, t∈QT 0,1×0, T, 2.1 u_x1, t 0, t∈0, T, 2.2

ux,0 gx, x∈0,1, 2.3

wheregxandfx, tare given functions with assumptions that will be given later.

In this paper, · ²_L2

μQTdenotes the usual norm of the weighted spaceL²_μQT,where we use the weightsμσ, ρandσ x²whileρx. The respective inner products onL²_ρQTand L²_σQTare given by

u, v_L²_ρQT

QT

xuv dx dt, u, v_L²_σQT

QT

x²uv dx dt, 2.4 LetW_σ,2^1,0QTbe the subspace ofL²QTwith finite norm

u²_W1,0

σ,2QTu²_L2

σQTux²_L2

σQT, 2.5

andV_σ W_σ,2^2,1QTbe the subspace ofW_σ,2^1,0QTwhose elements satisfy u_t, u_xx ∈ L²_σQT. In general, a function in the spaceW_σ,p^i,jQT, withi,jnonnegative integers possessesx-derivatives up toith order in the L^p_σQT, and tth derivatives up to jth order inL^p_σQT.We also use weighted spaces in the interval0,1such asL²_σ0,1andH_σ¹0,1,whose definitions are analogous to the spaces onQ_T.We set

W_σ,2⁰ 0,1

L²_σ 0,1

, W_σ,2¹ 0,1

H_σ¹ 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.

1Cauchy inequality withεsee, e.g.,4,

|ab| ≤ε

2|a|² 1

2ε|b|², 2.7

which holds for allε >0 and for arbitraryaandb.

2An inequality of Poincar´e type, Ixu²

L²QT _x

0

uξ, tdξ ²

L²QT≤ 1

2u²_L2QT, 2.8

whereIxu_x

0uξ, tdξsee 5, Lemma 1.

3The well-known Gronwall lemmasee, e.g.,6, Lemma 7.1.

Remark 2.1. The need of weighted spaces here is because of the singular term appearing in the left-hand side of2.1and 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 ofVσsolutions of problem2.1–

2.3. We first start with a uniqueness result.

Theorem 3.1. Letf ∈L²_σQTandgx∈W_σ,2¹ 0,1.Then problem2.1–2.3, has at most one solution inV_σ.

Proof. Let u₁ andu₂ be two solutions of the problem 2.1–2.3and letθx, t w₁x, t− w₂x, t, where

wix, t _t

0

uix, τdτ, i1,2, 3.1 then the functionθx, tsatisfies

Lθθt−1 x

xθx

xmax

x 0

ξu1ξ, tdξ,0

−max

x 0

ξu2ξ, tdξ,0

, 3.2

θx1, t 0, 3.3

θx,0 0. 3.4

If we denote by

β_ix, t max

x 0

ξu_iξ, tdξ,0

, i1,2, 3.5

then calculating the two integrals

QT2x²θLθ dx dt,

QT2x²θtLθ dx dt,using conditions3.3, 3.4, and a combining with−

QT2xθxLθ dx dt,we obtain 2θ_t²

L²σQT2θ_x²

L²σQTθ_x²

L²QTθ·, T²

L²σ0,1θ_x·, T²

L²σ0,1

−2 θ, θx

L²ρQT2

θt, β1−β2

L²σQT2

θ, β1−β2

L²σQT−2

θx, β1−β2

L²ρQT.

3.6

In light of inequalities2.7and2.8, each term of the right-hand side of3.6is estimated as follows:

−2 θ, θ_x

L²ρQT≤ θ²_L2

σQTθ_x²

L²QT, 2

θ, β1−β2

L²σQT≤4θ²_L2 σQT1

8θt²_L2

σQT, 2

θ_t, β₁−β₂

L²σQT≤θ_t²

L²σQT1 2θ_t²

L²σQT,

−2

θx, β1−β2

L²ρQT≤4θx²_L2

σQT1 8θt²_L2

σQT.

3.7

Therefore, using inequalities3.7, we infer from3.6 θ_t²

L²σQTθ·, T²

L²σ0,1θ_x·, T²

L²σ0,1≤20θ²_L2

σQT20θ_x²

L²σQT. 3.8 By applying Gronwall’s lemma to3.8, we conclude that

θt²_L2

σQT0. 3.9

Henceu₁u₂.

We now prove the existence theorem.

Theorem 3.2. Letf∈L²_σQTandgx∈W_σ,2¹ 0,1be given and satisfying f²_L2

σQTg²_W1

σ,20,1≤c₂², 3.10

forc₂>0 small enough and that

g_x1 0. 3.11

Then there exists at least one solutionux, t∈W_σ,2^2,1QTof problem2.1–2.3.

Proof. We define, for positive constants C and D which will be specified later, a class of functionsWWC, Dwhich consists of all functionsv∈L²_σQTsatisfying conditions2.2, 2.3, and

v_Vσ ≤C, v_t

L²σQT≤D. 3.12

Givenv∈WC, D,the problem

u_t− 1 x

xu_x

xJvf, x, t∈Q_T, ux1, t 0, t∈0, T, ux,0 gx, x∈0,1,

3.13

where

Jv d dtmax

x 0

ξvξ, tdξ,0

, 3.14

has a unique solutionu∈V_σ. We define a mappinghsuch thatuhv.

Once it is proved that the mappinghhas a fixed pointuin the closed bounded convex subsetWC, D,thenuis the desired solution.

We, first, show thathmapsWC, Dinto itself. For this purpose we writeuin the form uwζ,wherewis a solution of the problem

wt−wxx−1

xwxJv, x, t∈QT, 3.15

w_x1, t 0, t∈0, T, 3.16

wx,0 0, x∈0,1, 3.17

andζis a solution of the problem ζt−ζxx−1

xζxfx, t, x, t∈QT, 3.18

ζx1, t 0, t∈0, T, 3.19

ζx,0 gx, x∈0,1. 3.20

By multiplying3.15,3.18, respectively, by the operators,O1w2x²w2x²wt−6xwxand O₂ζ2x²ζ2x²ζ_t−6xζ_x, then integrating overQ_T,we obtain

2Lw, w_L²_σQT2 Lw, wt

L²σQT−6 Lw, wx

L²ρQT

2Jv, w_L²_σQT2 Jv, w_t

L²σQT−6 Jv, w_x

L²ρQT,

3.21

2Lζ, ζ_L²_σQT2 Lζ, ζt

L²σQT−6 Lζ, ζx

L²ρQT

2 f, ζt

L²σQT2f, ζ_L²_σQT−6 f, ζx

L²ρQT.

3.22

By using conditions3.16,3.17,3.19,3.20, an evaluation of the left-hand side of both equalities3.21and3.22gives, respectively,

wx, T²

L²σ0.12w_x²

L²σQT2 w, w_x

L²ρQTw_xx, T²

L²σ0.1

2w_t²

L²σQT2 w_t, w_x

L²ρQT3w_x²

L²QT−6 w_t, w_x

L²ρQT

2Jv, w_L²_σQT2 Jv, wt

L²σQT−6 Jv, wx

L²ρQT,

3.23

and applying inequalities2.7,2.8, and Gronwall’s lemma, we obtain the following estimat- es:

ζ²_Vσ ≤7 exp7T f²_L2

σQTg²_W1 σ,20,1

≤7 exp7Tc²₂;

3.24 w²_Vσ ≤7 exp7TJv²_L2

σQT. 3.25

We also multiply by xand square both sides of 3.15, integrate overQT, use the integral

−2

QTxwxLw dx dt,then integrate by parts and using inequality2.7, we obtain wt²_L2

σQTwxx²_L2

σQTwx·, T²_L2

σQT≤2Jv_L²_σQT. 3.26

Direct computations yield

Jv²_L2 σQT≤1

4

2c₁²7 exp7Tc²₂

. 3.27

By choosingc1andc2small enough in the previous inequality, we obtain

Jv_L²_σQT≤c₁. 3.28 Inequalities3.21–3.25then give

u²_Vσ ≤2w²_Vσ2ζ²_Vσ ≤14 exp7T c²₂c²₁

, ut²

L²σQT≤2wt²

L²σQT2ζt²

L²σQT4c²₁14 exp7Tc²₂.

3.29

At this point we takeC ≥ √

14 exp7T/2

c²₁c²₂andD ≥

4c²₁14 exp7Tc₂²,so that it follows from the last two inequalities thatu_Vσ ≤Candut_L²_σQT≤Dfrom which we deduce thatu∈W WC, D,hencehmapsWinto itself. To show thathis a continuous mapping, we considerv₁, v₂∈Wand their corresponding imagesu₁andu₂.It is straightforward to see thatUu₁−u₂satisfies

Ut−Uxx−1

xUx d dtmax

x 0

ξv1ξ, tdξ,0

− d dtmax

x 0

ξv2ξ, tdξ,0

, U_x1, t 0, Ux,0 0.

3.30

Define the functionpx, tby the formula px, t

_t

0

Ux, τdτ, 3.31

then it follows from3.26and3.28thatpx, tsatisfies pt−pxx− 1

xpxFmax

x 0

ξv1ξ, tdξ,0

−max

x 0

ξv2ξ, tdξ,0

, p_x1, t 0, px,0 0.

3.32

Since

F²_L2

σQT≤v1−v2²

L²σQT, 3.33

then

U²_L2

σQT≤6v1−v2²

L²σQT, 3.34

or

hv1−hv2²

L²σQT≤6v1−v2²

L²σQT, 3.35

hence the continuity of the mapping h. The compactness of the set WC, D is due to the following.

Theorem 3.3. LetE₀⊂E⊂E₁with compact embedding (reflexive Banach spaces) (see [4,7]). Suppose thatp, q∈1,∞andT >0.Then

Σ

ω:ω∈L^p

0, T;E₀

, ω_t∈L^q

0, T;E₁

3.36

is compactly embedded inL^p0, T;E, that is, the bounded sets are relatively compact inL^p0, T;E.

Note thatL²_σ0, T;L²_σ0,1 L²_σQT,hWC, D⊂WC, D⊂L²_σQT.By the Schauder fixed point theorem the mappinghhas a fixed pointuinWC, D.

Remark 3.4. For compactness of the setWC, D, see also8,9.

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 inW_σ,p^2,1QTwithp≤ ∞.

Theorem 3.6. Letu ∈ V_σ be a solution of problem 2.1–2.3, then the following a priori estimate holds:

0≤t≤Tsup

u·, T²

W_σ,2¹ 0,1ut²

L²σQTuxx²

L²σQTux²

L²σQT

≤80 exp80T g²_W1

σ,20,1f²_L2 σQT

.

3.37

Proof. From2.1, we have u_t²

L²σQTu_xx²

L²σQTu_x·, T²

L²σ0,1−2ut, u_xL²_ρQT gx²_L2

σ0,1

QT

x² d

dtmax

x 0

ξuξ, tdξ,0

f 2

dx dt.

3.38

Multiplying2.1by 2x²u_t, integrating overQ_T,carrying out standard integrations by parts, and using conditions2.2and2.3yields

2u_t²

L²σQTu_x·, T²

L²σ0,12 u_t, u_x

L²ρQT

g_x²

L²σ0,12

QT

x²u_tf dx dt2

QT

x²u_td dtmax

x 0

ξuξ, tdξ,0

dx dt.

3.39

Adding 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

L²σQTu_xx²

L²σQT2u_x·, T²

L²σ0,1≤2g_x²

L²σ0,16f²_L2

σQT. 3.40

Let be the elementary inequality 1

8u·, T²

L²σ0,1≤1 8ut²

L²σQT1 8u²_L2

σQT1 8g²_L2

σ0,1. 3.41

Adding the quantityux²_L2

σQTto both sides of3.38, then combining the resulted inequality with3.39, we obtain

u·, T²_L2

σ0,1ux·, T²_L2

σ0,1ut²_L2

σQTuxx²_L2

σQTux²_L2

σQT

≤48 g²_W1

σ,20,1f²_L2

σQTu²_L2

σQTux²_L2

σQT

.

3.42

Applying Gronwall’s lemma to3.40and then taking the supremum with respect totover the interval0, T,we obtain the desired a priori bound3.37.

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/2008/34 at King Saud University.

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