# An Energy Inequality and Its Applications of Nonlocal Boundary Conditions of Mixed Problem for Singular Parabolic Equations in Nonclassical Function Spaces

Authors: Moussa Zakari Djibibe; Kokou Tcharie; N. Iossifovich Yurchuk
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation:∂v/∂t-(a(t)/x2)(∂/∂x)(x2∂v/∂x)+b(x,t)v=g(x,t),
v(x,0)=ψ(x),
0≤x≤ℓ, v(ℓ,t)=E(t), 0≤t≤T, ∫0ℓx3v(x,t)dx=G(t), 0≤t≤ℓ. It is important to know that a priori estimates established in nonclassical function spaces is a necessary tool to prove the uniqueness of a strong solution of the studied problems.


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## Body

## 1. Introduction

In this paper, we deal with a class of parabolic equations with time-  and space-variable characteristics, with a nonlocal boundary condition. The precise statement of the problem is a follows: letℓ>0,T>0, and Ω={(x,t)∈ℝ2:0<x<ℓ,0<t<T}. We will determine a solution v, in Ω of the differential equation
(1.1)∂v∂t-a(t)x2∂∂x(x2∂v∂x)+b(x,t)v=g(x,t),(x,t)∈Ω,
satisfying the initial condition
(1.2)v(x,0)=ψ(x),0≤x≤ℓ,
the classical condition
(1.3)v(ℓ,t)=E(t),0≤t≤T,
and the integral condition
(1.4)∫0ℓ‍x3v(x,t)dx=G(t),0≤t≤T.
For consistency, we have
(1.5)∫0ℓ‍x3ψ(x)=G(0),ψ(ℓ)=E(0),
where ℓ and T are fixed but arbitrary positive numbers, a(t) and  b(x,t) are the known fuctions satisfying the following condition.Condition 1.
Fort∈[0,T] and x∈[0,ℓ], we assume that  (i)
d0≤a(t)≤d1,(ii)
b(x,t)≤d2,(iii)
da(t)/dt≤d3.  The notion of nonlocal condition has been introduced to extend the study of the classical initial value problems and it is more precise for describing natural phenomena than the classical condition since more information is taken into account, thereby decreasing the negative effects incurred by a possibly erroneous single measurement taken at the initial value. The importance of nonlocal conditions in many applications is discussed in [1, 2].It can be a part in the contribution of the development of a priori estimates method for solving such problems. The questions related to these problems are so miscellaneous that the elaboration of a general theory is still premature. Therefore, the investigation of these problems requires at every time a separate study.This work can be considered as a continuation of the results of  Yurchuk [3],  Benouar and  Yurchuk[4],  Bouziani [5–7],  Bouziani and  Benouar [8], Djibibe et al. [9], and  Djibibe and  Tcharie [10]. Our results generalize and deepen ones from corresponding work in [11, 12].We should mention here that the presence of an integral term in the boundary condition can greatly complicate the application of standard functional and numerical techniques.This paper is organized as follows. After this introduction, in Section2, we present the preliminaries. Finally, in Section 3, we establish an energy inaquality and give its several applications.


## 2. Preliminares

We transform the problem with nonhomogeneous boundary conditions into a problem with homogeneous boundary conditions. For this, we introduce a new unknown functionu defined by v(x,t)=u(x,t)+w(x,t), where
(2.1)w(x,t)=(5xℓ-4)E(t)-20ℓ5(x-ℓ)G(t).
Then, problem becomes
(2.2)∂u∂t-a(t)x2∂∂x(x2∂u∂x)+b(x,t)u=f(x,t),(2.3)u(x,0)=φ(x),0≤x≤ℓ,(2.4)u(ℓ,t)=0,0≤t≤T,(2.5)∫0ℓ‍x3u(x,t)dx=0,0≤t≤ℓ,
where
(2.6)φ(x)=ψ(x)+20ℓ(x-ℓ)G(0)-1ℓ(5x-4ℓ)E(0),f(x,t)=F(x,t)+20ℓ5(x-ℓ)(b(x,t)G(t)+G′(t))-1ℓ(5x-4ℓ)(b(x,t)E(t)+E′(t))=+10ℓ5xa(t)(ℓ4E(t)-G(t)).
We introduce appropriate function spaces. Let L2(Ω) be the Hilbert space of square integrable functions. To problem (2.1), (2.2), (2.3), (2.5), we associate the operator A with the domain of definition
(2.7)D(A)={∂u∂t,1x2∂u∂x,∂2u∂x2∈L2(Ω)},
satisfying (2.4) and (2.5). The operator A is considered from E to F, where E is the banach space consisting of u∈L2(Ω) satisfying the boundary conditions (2.4) and (2.5) and having the finite norm:
(2.8)∥u∥2=∫Ω‍Jx2(∂u∂t)dxdt+sup0≤t≤T{∫0ℓ‍x2u2(x,t)dx+∫0ℓ‍(x∂u∂t)2dx+∫0ℓ‍(x∂u∂x)2dx},
and F is the Hilbert space of vector-value function ℱ=(f,φ) having the norm
(2.9)∥(f,φ)∥2=∫Ω‍x2f2(x,t)dxdt+∫0ℓ‍x2φ2(x)dx+∫0ℓ‍(x∂φ(x)∂x)2dx,
where Jxh=∫xℓθ2h(θ,t)dθ.


## 3. A Priori Estimate and Its Consequences

Theorem 3.1.
Under Condition1, for any function v∈D(A), one has the following a priori estimate
(3.1)∥v∥E≤c∥Av∥F,
where c is a positive constant independent of the solution v.Proof.
Firstly, applying operatorJx to (2.1), multiplying the obtained result with Jx(∂u/∂t), and integrating over Ωτ=(0,ℓ)×(0,τ), oberve that
(3.2)∫Ωτ‍Jx2(∂u∂t)dxdt-∫Ωτ‍Jx(a(t)x2∂∂x(x2∂u∂x))Jx(∂u∂t)dxdt+∫Ωτ‍Jx(b(x,t)u)Jx(∂u∂t)dxdt=∫Ωτ‍Jx(f(x,t))Jx(∂u∂t)dxdt.
Integrating by parts of the second integral on the left-hand side of (3.2), we get
(3.3)-∫Ωτ‍Jx(a(t)x2∂∂x(x2∂u∂x))Jx(∂u∂t)dxdt=∫Ωτ‍x2a(t)∂u∂xJx(∂u∂t)dxdt.
Substituting (3.3) into (3.2), we get
(3.4)∫Ωτ‍Jx2(∂u∂t)dxdt+∫Ωτ‍x2a(t)∂u∂xJx(∂u∂t)dxdt.+∫Ωτ‍Jx(b(x,t)u)Jx(∂u∂t)dxdt=∫Ωτ‍Jx(f(x,t))Jx(∂u∂t)dxdt.
In the second time, multiplying the equality (2.1) with x2∂u/∂t, and integrating the obtained equality over Ωτ, we get
(3.5)∫Ωτ‍(x∂u∂t)2dxdt-∫Ωτ‍a(t)∂∂x(x2∂u∂x)∂u∂tdxdt+∫Ωτ‍x2b(x,t)u∂u∂tdxdt=∫Ωτ‍x2f(x,t)∂u∂tdxdt.
The standard integration by parts of the second term on the left-hand side of (3.5), leads to
(3.6)-∫Ωτ‍a(t)∂∂x(x2∂u∂x)∂u∂tdxdt=12∫0ℓ‍a(τ)x2(∂u∂x(x,τ))2dx-12∫0ℓ‍a(0)x2(∂φ∂x)2-12∫Ωτ‍a′(t)x2(∂u∂x)2dxdt.
Substituting (3.6) into (3.5), we get
(3.7)∫Ωτ‍(x∂u∂t)2dxdt+12∫0ℓ‍a(τ)x2(∂u∂x(x,τ))2dx-12∫0ℓ‍a(0)x2(∂φ∂x)2-12∫Ωτ‍a′(t)x2(∂u∂x)2dxdt+∫Ωτ‍x2b(x,t)u∂u∂tdxdt=∫Ωτ‍x2f(x,t)∂u∂tdxdt.
Finally, adding (3.4) to (3.7), we have
(3.8)∫Ωτ‍Jx2(∂u∂t)dxdt+∫Ωτ‍(x∂u∂t)2dxdt+12∫0ℓ‍a(τ)x2(∂u∂x(x,τ))2dx=∫Ωτ‍Jx(f(x,t))Jx(∂u∂t)dxdt+∫Ωτ‍x2f(x,t)∂u∂tdxdt+12∫0ℓ‍a(0)x2(∂φ∂x)2=-∫Ωτ‍x2b(x,t)u∂u∂tdxdt-∫Ωτ‍Jx(b(x,t)u)Jx(∂u∂t)dxdt=+12∫Ωτ‍a′(t)x2(∂u∂x)2dxdt-∫Ωτ‍x2a(t)∂u∂xJx(∂u∂t)dxdt.
In the light of Cauchy inequality, certain terms of (3.8) are then majorized as follows:
(3.9)∫Ωτ‍Jx(f(x,t))Jx(∂u∂t)dxdt≤α12∫Ωτ‍Jx2(f(x,t))dxdt+12α1∫Ωτ‍Jx2(∂u∂t)dxdt,(3.10)∫Ωτ‍x2f(x,t)∂u∂tdxdt≤α22∫Ωτ‍x2f2(x,t)dxdt+12α2∫Ωτ‍(x∂u∂t)2dxdt,(3.11)-∫Ωτ‍x2a(t)∂u∂xJx(∂u∂t)dxdt≤α32∫Ωτ‍a2(t)(x∂u∂x)2dxdt+12α3∫Ωτ‍Jx2(∂u∂t)dxdt,(3.12)-∫Ωτ‍x2b(x,t)u∂u∂tdxdt≤α42∫Ωτ‍x2b2(x,t)u2(x,t)dxdt+12α4∫Ωτ‍(x∂u∂t)2dxdt,(3.13)-∫Ωτ‍Jx(b(x,t)u)Jx(∂u∂t)dxdt≤α52∫Ωτ‍Jx2(b(x,t)u)dxdt+12α5∫Ωτ‍Jx2(∂u∂t)dxdt.
Combining the inequalities (3.9), (3.10), (3.11) with (3.8), choosing α1,α2,α3,α4,α5 which that α1+α3+α5<2α1α3α5 and α2+α4<2α2α4, we get
(3.14)λ1∫Ωτ‍Jx2(∂u∂t)dxdt+λ2∫Ωτ‍(x∂u∂t)2dxdt+12∫0ℓ‍a(τ)(x∂u∂x(x,τ))2dx≤12∫Ωτ‍(α3a2(t)+a′(t))(x∂u∂x)2dxdt+α42∫Ωτ‍x2b2(x,t)u2(x,t)dxdt++α52∫Ωτ‍Jx2(b(x,t)u)dxdt+α12∫Ωτ‍Jx2(f(x,t))dxdt++α22∫Ωτ‍x2f2(x,t)dxdt+12∫0ℓ‍a(0)(x∂φ∂x)2,
where
(3.15)λ1=1-12(1α1+1α3+1α5)λ2=1-12(1α2+1α4).Lemma 3.2.
Forx∈(0,ℓ), the following inequalities hold:
(3.16)∫Ωτ‍Jx2(u)dxdt≤ℓ22∫0ℓ‍x2u2dx,λ2∫0ℓ‍x2u2dx≤λ2∫0ℓ‍x2φ2(x)dx+λ2∫Ωτ‍x2u2(x,t)dxdt+λ2∫Ωτ‍(x∂u∂t)2dxdt.
It follows by using Lemma 3.2 and (3.18) that
(3.17)λ1∫Ωτ‍Jx2(∂u∂t)dxdt+12∫0ℓ‍a(τ)(x∂u∂x(x,τ))2dx+λ22∫0ℓ‍x2u2dx≤12∫Ωτ‍(α3a2(t)+a′(t))(x∂u∂x)2dxdt+(2α4+α5ℓ2)4∫Ωτ‍x2b2(x,t)u2(x,t)dxdt≤+α1ℓ2+2α24∫Ωτ‍x2f2(x,t)dxdt+λ22∫0ℓ‍x2φ2(x)dx+12∫0ℓ‍a(0)(x∂φ∂x)2.
Therefore, by formula (3.17) and Condition 1, we obtain
(3.18)∫Ωτ‍Jx2(∂u∂t)dxdt+∫0ℓ‍(x∂u∂x(x,τ))2dx+∫0ℓ‍x2u2dx≤λ3(∫Ωτ‍x2f2(x,t)dxdt+∫0ℓ‍x2φ2(x)dx+∫0ℓ‍(x∂φ∂x)2dx)≤+λ4(∫Ωτ‍(x∂u∂x)2dxdt+∫Ωτ‍x2u2(x,t)dxdt),
where
(3.19)λ3=max((α1ℓ2+2α2)/4,λ2/2,d1/2)min(λ1,λ2/2,d0/2),λ4=max((α5ℓ2+2α4)d22/4,d3+α3d12/2)min(λ1,λ2/2,d0/2).
Eliminating the last term on the right-hand side of inequality (3.18). To this end, using  Gronwall's lemma, it follows that
(3.20)∫Ωτ‍Jx2(∂u∂t)dxdt+∫0ℓ‍(x∂u∂x(x,τ))2dx+∫0ℓ‍x2u2dx≤λ5(∫Ω‍x2f2(x,t)dxdt+∫0ℓ‍x2φ2(x)dx+∫0ℓ‍(x∂φ∂x)2dx),
where λ5=λ3eλ4T.
The right-hand side of (3.20) is independent of τ, hence, replacing the left-hand side by the upper bound with respect to τ, We get
(3.21)∫Ω‍Jx2(∂u∂t)dxdt+sup0≤t≤T{∫0ℓ‍x2u2(x,t)dx+∫0ℓ‍(x∂u∂x)2dx}≤c(∫Ω‍x2f2(x,t)dxdt+∫0ℓ‍x2φ2(x)dx+∫0ℓ‍(x∂φ(x)∂x)2dx),
where c=λ5=λ3eλ4T/2. This completes the proof of Theorem 3.1.Lemma 3.3.
The operatorA:E→F with domain D(A) has a closure A-.Proof of Lemma3.2.
Suppose thatun∈D(A) is a sequence such that
(3.22)limn→+∞un=0,inE,(3.23)limn→+∞Aun=(f,φ),inF,
we must show that f≡0 and  φ≡0. Equality (3.22) implies that
(3.24)limn→+∞un=0,in𝒟′(Ω).
By virtue of the condition of derivation of 𝒟′(Ω) in 𝒟′(Ω), we get
(3.25)limn→+∞[∂un∂t-a(x,t)∂2un∂x2+b(x,t)∂un∂x+c(x,t)un]=0,in𝒟′(Ω).
Then from equality (3.23) it follows that
(3.26)limn→+∞[∂un∂t-a(t)x2∂∂x(x2∂un∂x)+b(x,t)un]=f,inL2(Ω).
therefore
(3.27)limn→+∞[∂un∂t-a(t)x2∂∂x(x2∂un∂x)+b(x,t)un]=f,in𝒟′(Ω).
By virtue of the uniqueness of the limit in 𝒟′(Ω), the identies (3.25) and (3.27) conduct to f≡0.
By analogy, from (3.23), we get
(3.28)limn→+∞un(x,0)=φ(x),inL2(0,ℓ).
We see via (3.22) and the obvious inequality
(3.29)∥un(x,0)∥L2(0,ℓ)≤∥un(x,t)∥E,∀n∈ℕ
that
(3.30)limn→+∞un(x,0)=0,inL2(0,ℓ).
By virtue of (3.28), (3.30) and the uniqueness of the limit in L2(0,ℓ) we conclude that  φ≡0.Definition 3.4.
A solution of the equation(3.31)A¯v=(f,φ),
is called a strong solution of problem (2.2), (2.3), (2.4), and (2.5).Consequence 3.5.
Under the conditions of Theorem3.1, there is a constant c>0 independent of v such that
(3.32)∥v∥E≤c∥A¯v∥F,∀v∈D(A¯).Consequence 3.6.
The rangeR(A¯) of the operator A¯ is closed and R(A)-=R(A)-.Consequence 3.7.
A strong solution of the problem (2.2), (2.3), (2.4), and (2.5) is unique and depends continuously on ℱ=(f,φ)∈F.



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