WITHDRAWN: Existence and stability for a heat equation with a non local boundary condition

Accepted Manuscript Existence and stability for a heat equation with a non local boundary condition Salim A. Messaoudi, Farida Belhannache PII: DOI: Reference:

S0893-9659(16)30014-3 http://dx.doi.org/10.1016/j.aml.2016.01.002 AML 4921

To appear in:

Applied Mathematics Letters

Received date: 6 December 2015 Revised date: 8 January 2016 Accepted date: 8 January 2016 Please cite this article as: S.A. Messaoudi, F. Belhannache, Existence and stability for a heat equation with a non local boundary condition, Appl. Math. Lett. (2016), http://dx.doi.org/10.1016/j.aml.2016.01.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Existence and Stability for a heat equation with a non local boundary condition Salim A. Messaoudi1 & Farida Belhannache2 1) Department of Math. & Stat. KFUPM, Dhahran 31261, Saudi Arabia E-mail: [email protected] 2) Department of Mathematics, University of Jijel, Jijel, Algeria E-mail : [email protected] Abstract We consider a heat equation associated with a non local condition and show how the semigroup theory can be used to establish the well posedness. We also prove the exponential decay of the solution, using the multiplier method.

1

Introduction

Physical phenomena and Engineering problems can be modeled and described in terms of nonlocal problems, such as mixed problems with boundary integral conditions. These nonlocal boundary conditions arise mainly when the data on the boundary are not accessible and cannot be measured directly, but their average values are obtained. As a consequence, standard boundary conditions, such as Dirichlet, Neumann and Robin type, which are usually given pointwise, are not always adequate as it depends on the physical context which data can be measured at the boundary of the physical domain. In some situations, it is impossible to prescribe the solution u (pressure, temperature, ) pointwise at the boundary, because only the average value of the solution can be measured along the boundary or along some part of it. Parabolic problems with classical and nonlocal (integral) boundary conditions have been extensively investigated and several existence and uniqueness results have been established. Different tools, such the Fourier analysis, potential method, and the Galerkin method, have been used to establish the well posedness. We quote in this context the work of Bouziani [1,2] Cannon [3,4], Kamynin [5], and Ionkin [6]. But, the ”so called” energy method was mostly used by people treating problems with nonlocal boundary conditions. We cite, among others, Benouar et al. [7], Deneche and Marhoun [8], Mesloub [9], Mesloub and Bouziani [10], Mesloub et al. [11], Pulkina [12], and Yurchuk [13]. 1

In this work, we intend to discuss the following heat equation         

ut (x, t) − uxx (x, t) = 0, u(x, 0) = u0 (x), ux (L, t) = 0 and

Z

L

0

0 < x < L, t > 0 0≤x≤L

(1.1)

u(x, t)dx = 0, ∀t ≥ 0

associated with a Neumann and nonlocal conditions and L < +∞. Our aim is to employ the semi-group theory, based on the maximal monotone operators [14]. To the best of our knowledge, this approach has never been used to tackle such problems. We will prove the existence and uniqueness of solutions and establish the exponential decay of these solutions. Our approach will open the road to treat many other problems in a simpler way and avoid all the cumbersome calculations encountered in the other methods.

2

Preliminaries

In this section, we present some material needed in the proof of our result. Definition 1.1. For every function v ∈ L1 (0, L), we set Λv(x) =

Z

0

x

v(s)ds and Λ2 v(x) =

Z

0

x

(

Z

s

0

v(ξ) dξ)ds

We also introduce L2∗ (I)

Z

2

= {v ∈ L (I) /

L

0

v(x)dx = 0}, I = (0, L).

Remark 1.1. It is well known that L2∗ (I) is a Banach under the usual L2 -norm. Remark 1.2. It is easy to verify that (v, w) = −

Z

L

0

v(x)Λ2 w(x)dx

(2.1)

is an inner product on L2∗ (I) and (v, w) =

Z

L

0

Λv(x)Λw(x)dx = (w, v).

The associated norm is given by ||v||∗ =

Z

0

L

2

!1/2

|Λv(x)| dx

.

(2.2)

L2∗ (I) equipped with (2.1) is an inner product space. Let H be its completion with respect to the norm (2.2). Remark 1.3. L2∗ (I), equipped with the L2 -norm, is continuously embedded in H and ||v||∗ ≤ L||v||L2(I) . (2.3) 2

Definition 1.2. We define the space V = {v ∈ H such that there exists w ∈ H :

Z

L 0

′′

vφ dx =

Z

L 0

wφdx, ∀φ ∈ C0∞ (I)}.

We then call w = v ′′ . Note that H 2 (I) ⊂ V.

3

Well Posedness

In this section, we sate and prove an existence result. As we mentioned in the introduction, the main tool here is the semigroup theory. To this end, we rewrite (1.1) in the form ( ut + Au = 0, t > 0 (3.1) u(0) = u0 , where A : D(A) ⊂ H → H, with A = −∂xx and D(A) = {v ∈ V ∩ L2∗ (I) such that vx (L) = 0}. Theorem 2.1. Let u0 ∈ H, then Problem (3.1) has a unique solution u ∈ C ([0, +∞), H) ∩ C ((0, +∞), D(A)) ∩ C 1 ((0, +∞), H) . Proof. We verify that Hille-Yosida theorem requirements are satisfied. We show that A is monotone maximal and symmetric, hence self-adjoint. See [14, Ch. 7]. Let v ∈ D(A), we have Z

(Av, v) =

L

0

vxx Λ2 vdx = vx Λ2 v|L0 −

= −vΛu|L0 +

Z

L

0

v 2 dx =

Z

0

L

Z

L

0

vx Λ vdx

v 2 dx ≥ 0.

So A is monotone. Next, let f ∈ H and consider the following weak formulation Z

0

L

uvdx +

Z

0

L

Λu Λvdx = −

One can easily check that the bilinear form a(u, v) = R

Z

0

L

Z

L

f Λ2 vdx, ∀v ∈ L2∗ (I).

0

uvdx +

Z

0

L

(3.2)

Λu Λvdx

and the linear form F (v) = − 0L f Λ2vdx, defined on L2∗ (I), are bounded and a is coercive , with a(u, u) ≥ ||u||2L2(I) . Therefore, the Lax-Milgram theorem guarantees 3

the existence of a unique u ∈ L2∗ (I) such that (3.2) holds. To show that u ∈ D(A), it suffices to show that u ∈ V and ux (L) = 0. Let φ ∈ C0∞ (I). Then, v = φ′′ ∈ C0∞ (I) and φ′′ ∈ L2∗ (I). Thus, replacing in (3.2) and performing some routine integration by parts, we obtain Z

L

0

uφ′′ dx −

Z

0

L

u φdx = −

Z

0

L

f φdx.

(3.3)

Consequently, we have Z

0

L

′′

u φ dx =

Z

L

0

(−f + u)φdx, ∀φ ∈ C0∞ (I);

which implies that u ∈ V and u′′ = −f + u ∈ H. It remains to show that ux (L) = 0. For this purpose, we perform, again integration by parts in (3.2) to get −ux (L)

Z

0

L

Λvdx +

Z

0

L

(uxx − u + f ) Λ2 vdx = 0,

∀v ∈ L2∗ (I).

This yields ux (L) = 0 since v is arbitrary. So u ∈ D(A) and we conclude that A is maximal. Also, direct computations show that A is symmetric and, hence, self -adjoint. Therefore, applying Hille-Yosida theorem [14], we conclude our existence result. Remark 2.1. The same result holds for the nonhomogeneous problem         

utt (x, t) − uxx (x, t) = f (x, t), u(x, 0) = u0 (x), ux (L, t) = 0 and

Z

0

L

0 < x < L, t > 0 0≤x≤L

u(x, t)dx = 0, ∀t ≥ 0

provided that f ∈ C 1 ([0, +∞), H) . See [14, Theorem 7.10].

4

Exponential Decay

In this section, we show that the solution of (1.1) decays exponentially in the norm ||.||∗. We have the following theorem Theorem 3.1. The solution of (1.1) satisfies t

||u(., t)||∗ ≤ ||u0||∗ e− L , ∀t ≥ 0. Proof. We multiply equation in (1.1) by −Λ2 u and integrate over (0, L), to obtain 4

Z

L

0

This leads to

d1 dt 2 By using (2.3), we arrive at

Λut Λudx + Z

0

L

2

Z

0

L

u2 dx = 0.

|Λu| dx = −

Z

0

L

u2 dx.

d 2 ||u(., t)||2∗ ≤ − ||u(., t)||2∗. dt L Direct integration gives the desired result. This completes the proof. Acknowledgment The first author thanks KFUPM for its support.

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References 1. A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation. J. Appl. Math. Stochastic Anal. 9 No. 3 (1996), 323-330. 2. A. Bouziani, N. Merazga and S. Benamira, Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlinear Analysis 69 (2008), 1515-1524 3. J.R. Cannon, The solution of the heat equation subject to the specification of energy. Quart. Appl. Math. 21 (1963), 155-160. 4. J.R. Cannon, S. P. Esteve, and J. Van der Hoeck, A Galerkin procedure for the diffusion equation subject to the specification of mass. SIAM J. Numer. Anal. 24 (1987), 499-515. 5. A.V. Kartynnik, A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic, Differential Equations 26 No. 9 (1991), 1160-1166. 6. N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition. (Russian) Differencial’nye Uravneniya 13 No. 2 (1977), 294-304. 7. N.E. Benouar and N.I. Yurchuk, A mixed problem with an integral conditions for parabolic equations with a Bessel operator, Differential Equations 27 No. 12 (1991), 1482-1487 8. M. Denche and A. L. Marhoune, A Three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator, Appl. Math. Letters 13 (2000), 85 - 89. 5

9. S. Mesloub, On a nonlocal problem for a pluriparabolic equation. Acta Sci. Math. (Szeged) 67 No. 1-2 (2001), 203-219. 10. S. Mesloub and A. Bouziani, Mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. J. Appl. Math. Stochastic Anal. 15 No. 3 (2002), 291-300. 11. S. Mesloub, A. Bouziani and N. Kechkar, A strong solution of an evolution problem with integral conditions, Georgian Math. J. Vol. 9 No. 1 (2002), 149-159 12. L.S. Pulkina, A non-local problem with integral conditions for hyperbolic equations, Electron. J. Differential Equations Vol. 1999 No. 45 (1999), 6 pp. (electronic) 13. N.I. Yurchuk, A mixed problem with an integral condition for some parabolic equations (Russian) Differentsial’nye Uravneniya 22 No.12 (1986), 2117-2126. 14. H. Brezis, Functional Analysis, Sobolev spaces, and PDE’s, Springer, Universitex 2011

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