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), 0𝑥, 𝑣(,𝑡)=𝐸(𝑡), 0𝑡𝑇, 0𝑥3𝑣(𝑥,𝑡)𝑑𝑥=𝐺(𝑡), 0𝑡. 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.

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: let >0,𝑇>0, and Ω={(𝑥,𝑡)20<𝑥<,0<𝑡<𝑇}. We will determine a solution 𝑣, in Ω of the differential equation 𝜕𝑣𝜕𝑡𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑣𝜕𝑥+𝑏(𝑥,𝑡)𝑣=𝑔(𝑥,𝑡),(𝑥,𝑡)Ω,(1.1) satisfying the initial condition 𝑣(𝑥,0)=𝜓(𝑥),0𝑥,(1.2) the classical condition 𝑣(,𝑡)=𝐸(𝑡),0𝑡𝑇,(1.3) and the integral condition 0𝑥3𝑣(𝑥,𝑡)𝑑𝑥=𝐺(𝑡),0𝑡𝑇.(1.4) For consistency, we have 0𝑥3𝜓(𝑥)=𝐺(0),𝜓()=𝐸(0),(1.5) where and 𝑇 are fixed but arbitrary positive numbers, 𝑎(𝑡) and 𝑏(𝑥,𝑡) are the known fuctions satisfying the following condition.

Condition 1. For 𝑡[0,𝑇] and 𝑥[0,], we assume that (i)𝑑0𝑎(𝑡)𝑑1, (ii)𝑏(𝑥,𝑡)𝑑2, (iii)𝑑𝑎(𝑡)/𝑑𝑡𝑑3. 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 [57], 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 Section 2, 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 function 𝑢 defined by 𝑣(𝑥,𝑡)=𝑢(𝑥,𝑡)+𝑤(𝑥,𝑡), where 𝑤(𝑥,𝑡)=5𝑥4𝐸(𝑡)205(𝑥)𝐺(𝑡).(2.1) Then, problem becomes 𝜕𝑢𝜕𝑡𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑢𝜕𝑥+𝑏(𝑥,𝑡)𝑢=𝑓(𝑥,𝑡),(2.2)𝑢(𝑥,0)=𝜑(𝑥),0𝑥,(2.3)𝑢(,𝑡)=0,0𝑡𝑇,(2.4)0𝑥3𝑢(𝑥,𝑡)𝑑𝑥=0,0𝑡,(2.5) where 𝜑(𝑥)=𝜓(𝑥)+201(𝑥)𝐺(0)(5𝑥4)𝐸(0),𝑓(𝑥,𝑡)=𝐹(𝑥,𝑡)+205(𝑥)𝑏(𝑥,𝑡)𝐺(𝑡)+𝐺1(𝑡)(5𝑥4)𝑏(𝑥,𝑡)𝐸(𝑡)+𝐸+(𝑡)105𝑥𝑎(𝑡)4.𝐸(𝑡)𝐺(𝑡)(2.6) 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 𝐷(𝐴)=𝜕𝑢,1𝜕𝑡𝑥2𝜕𝑢,𝜕𝜕𝑥2𝑢𝜕𝑥2𝐿2(Ω),(2.7) satisfying (2.4) and (2.5). The operator 𝐴 is considered from 𝐸 to 𝐹, where 𝐸 is the banach space consisting of 𝑢𝐿2(Ω) satisfying the boundary conditions (2.4) and (2.5) and having the finite norm: 𝑢2=Ω𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+sup0𝑡𝑇0𝑥2𝑢2(𝑥,𝑡)𝑑𝑥+0𝑥𝜕𝑢𝜕𝑡2𝑑𝑥+0𝑥𝜕𝑢𝜕𝑥2,𝑑𝑥(2.8) and 𝐹 is the Hilbert space of vector-value function =(𝑓,𝜑) having the norm (𝑓,𝜑)2=Ω𝑥2𝑓2(𝑥,𝑡)𝑑𝑥𝑑𝑡+0𝑥2𝜑2(𝑥)𝑑𝑥+0𝑥𝜕𝜑(𝑥)𝜕𝑥2𝑑𝑥,(2.9) where 𝐽𝑥=𝑥𝜃2(𝜃,𝑡)𝑑𝜃.

3. A Priori Estimate and Its Consequences

Theorem 3.1. Under Condition 1, for any function 𝑣𝐷(𝐴), one has the following a priori estimate 𝑣𝐸𝑐𝐴𝑣𝐹,(3.1) where 𝑐 is a positive constant independent of the solution 𝑣.

Proof. Firstly, applying operator 𝐽𝑥 to (2.1), multiplying the obtained result with 𝐽𝑥(𝜕𝑢/𝜕𝑡), and integrating over Ω𝜏=(0,)×(0,𝜏), oberve that Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡Ω𝜏𝐽𝑥𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢+𝜕𝑡𝑑𝑥𝑑𝑡Ω𝜏𝐽𝑥(𝑏(𝑥,𝑡)𝑢)𝐽𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡=Ω𝜏𝐽𝑥(𝑓(𝑥,𝑡))𝐽𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.2) Integrating by parts of the second integral on the left-hand side of (3.2), we get Ω𝜏𝐽𝑥𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡=Ω𝜏𝑥2𝑎(𝑡)𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.3) Substituting (3.3) into (3.2), we get Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+Ω𝜏𝑥2𝑎(𝑡)𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢+𝜕𝑡𝑑𝑥𝑑𝑡.Ω𝜏𝐽𝑥(𝑏(𝑥,𝑡)𝑢)𝐽𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡=Ω𝜏𝐽𝑥(𝑓(𝑥,𝑡))𝐽𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.4) In the second time, multiplying the equality (2.1) with 𝑥2𝜕𝑢/𝜕𝑡, and integrating the obtained equality over Ω𝜏, we get Ω𝜏𝑥𝜕𝑢𝜕𝑡2𝑑𝑥𝑑𝑡Ω𝜏𝜕𝑎(𝑡)𝑥𝜕𝑥2𝜕𝑢𝜕𝑥𝜕𝑢+𝜕𝑡𝑑𝑥𝑑𝑡Ω𝜏𝑥2𝑏(𝑥,𝑡)𝑢𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡=Ω𝜏𝑥2𝑓(𝑥,𝑡)𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.5) The standard integration by parts of the second term on the left-hand side of (3.5), leads to Ω𝜏𝜕𝑎(𝑡)𝑥𝜕𝑥2𝜕𝑢𝜕𝑥𝜕𝑢1𝜕𝑡𝑑𝑥𝑑𝑡=20𝑎(𝜏)𝑥2𝜕𝑢𝜕𝑥(𝑥,𝜏)21𝑑𝑥20𝑎(0)𝑥2𝜕𝜑𝜕𝑥212Ω𝜏𝑎(𝑡)𝑥2𝜕𝑢𝜕𝑥2𝑑𝑥𝑑𝑡.(3.6) Substituting (3.6) into (3.5), we get Ω𝜏𝑥𝜕𝑢𝜕𝑡21𝑑𝑥d𝑡+20𝑎(𝜏)𝑥2𝜕𝑢𝜕𝑥(𝑥,𝜏)21𝑑𝑥20𝑎(0)𝑥2𝜕𝜑𝜕𝑥212Ω𝜏𝑎(𝑡)𝑥2𝜕𝑢𝜕𝑥2𝑑𝑥𝑑𝑡+Ω𝜏𝑥2𝑏(𝑥,𝑡)𝑢𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡=Ω𝜏𝑥2𝑓(𝑥,𝑡)𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.7) Finally, adding (3.4) to (3.7), we have Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+Ω𝜏𝑥𝜕𝑢𝜕𝑡21𝑑𝑥𝑑𝑡+20𝑎(𝜏)𝑥2𝜕𝑢(𝜕𝑥𝑥,𝜏)2=𝑑𝑥Ω𝜏𝐽𝑥(𝑓(𝑥,𝑡))𝐽𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+Ω𝜏𝑥2𝑓(𝑥,𝑡)𝜕𝑢1𝜕𝑡𝑑𝑥𝑑𝑡+20𝑎(0)𝑥2𝜕𝜑𝜕𝑥2Ω𝜏𝑥2𝑏(𝑥,𝑡)𝑢𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡Ω𝜏𝐽𝑥(𝑏(𝑥,𝑡)𝑢)𝐽𝑥𝜕𝑢+1𝜕𝑡𝑑𝑥𝑑𝑡2Ω𝜏𝑎(𝑡)𝑥2𝜕𝑢𝜕𝑥2𝑑𝑥𝑑𝑡Ω𝜏𝑥2𝑎(𝑡)𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.8) In the light of Cauchy inequality, certain terms of (3.8) are then majorized as follows: Ω𝜏𝐽𝑥(𝑓(𝑥,𝑡))𝐽𝑥𝜕𝑢𝛼𝜕𝑡𝑑𝑥𝑑𝑡12Ω𝜏𝐽2𝑥(1𝑓(𝑥,𝑡))𝑑𝑥𝑑𝑡+2𝛼1Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡,(3.9)Ω𝜏𝑥2𝑓(𝑥,𝑡)𝜕𝑢𝛼𝜕𝑡𝑑𝑥𝑑𝑡22Ω𝜏𝑥2𝑓2(1𝑥,𝑡)𝑑𝑥𝑑𝑡+2𝛼2Ω𝜏𝑥𝜕𝑢𝜕𝑡2𝑑𝑥𝑑𝑡,(3.10)Ω𝜏𝑥2𝑎(𝑡)𝜕𝑢𝐽𝜕𝑥𝑥𝜕𝑢𝛼𝜕𝑡𝑑𝑥𝑑𝑡32Ω𝜏𝑎2𝑥(𝑡)𝜕𝑢𝜕𝑥21𝑑𝑥𝑑𝑡+2𝛼3Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡,(3.11)Ω𝜏𝑥2𝑏(𝑥,𝑡)𝑢𝜕𝑢𝛼𝜕𝑡𝑑𝑥𝑑𝑡42Ω𝜏𝑥2𝑏2(𝑥,𝑡)𝑢21(𝑥,𝑡)𝑑𝑥𝑑𝑡+2𝛼4Ω𝜏𝑥𝜕𝑢𝜕𝑡2𝑑𝑥𝑑𝑡,(3.12)Ω𝜏𝐽𝑥(𝑏(𝑥,𝑡)𝑢)𝐽𝑥𝜕𝑢𝛼𝜕𝑡𝑑𝑥𝑑𝑡52Ω𝜏𝐽2𝑥1(𝑏(𝑥,𝑡)𝑢)𝑑𝑥𝑑𝑡+2𝛼5Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡.(3.13) 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 𝜆1Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+𝜆2Ω𝜏𝑥𝜕𝑢𝜕𝑡21𝑑𝑥𝑑𝑡+20𝑥𝑎(𝜏)𝜕𝑢(𝜕𝑥𝑥,𝜏)21𝑑𝑥2Ω𝜏𝛼3𝑎2(𝑡)+𝑎𝑥(𝑡)𝜕𝑢𝜕𝑥2𝛼𝑑𝑥𝑑𝑡+42Ω𝜏𝑥2𝑏2(𝑥,𝑡)𝑢2+𝛼(𝑥,𝑡)𝑑𝑥𝑑𝑡52Ω𝜏𝐽2𝑥𝛼(𝑏(𝑥,𝑡)𝑢)𝑑𝑥𝑑𝑡+12Ω𝜏𝐽2𝑥+𝛼(𝑓(𝑥,𝑡))𝑑𝑥𝑑𝑡22Ω𝜏𝑥2𝑓21(𝑥,𝑡)𝑑𝑥𝑑𝑡+20𝑎𝑥(0)𝜕𝜑𝜕𝑥2,(3.14) where 𝜆11=121𝛼1+1𝛼3+1𝛼5𝜆21=121𝛼2+1𝛼4.(3.15)

Lemma 3.2. For 𝑥(0,), the following inequalities hold: Ω𝜏𝐽2𝑥(𝑢)𝑑𝑥𝑑𝑡220𝑥2𝑢2𝜆𝑑𝑥,20𝑥2𝑢2𝑑𝑥𝜆20𝑥2𝜑2(𝑥)𝑑𝑥+𝜆2Ω𝜏𝑥2𝑢2(𝑥,𝑡)𝑑𝑥𝑑𝑡+𝜆2Ω𝜏𝑥𝜕𝑢𝜕𝑡2𝑑𝑥𝑑𝑡.(3.16) It follows by using Lemma 3.2 and (3.18) that 𝜆1Ω𝜏𝐽2𝑥𝜕𝑢1𝜕𝑡𝑑𝑥𝑑𝑡+20𝑥𝑎(𝜏)𝜕𝑢(𝜕𝑥𝑥,𝜏)2𝜆𝑑𝑥+220𝑥2𝑢21𝑑𝑥2Ω𝜏𝛼3𝑎2(𝑡)+𝑎𝑥(𝑡)𝜕𝑢𝜕𝑥2𝑑𝑥𝑑𝑡+2𝛼4+𝛼524Ω𝜏𝑥2𝑏2(𝑥,𝑡)𝑢2+𝛼(𝑥,𝑡)𝑑𝑥𝑑𝑡12+2𝛼24Ω𝜏𝑥2𝑓2𝜆(𝑥,𝑡)𝑑𝑥𝑑𝑡+220𝑥2𝜑21(𝑥)𝑑𝑥+20𝑎𝑥(0)𝜕𝜑𝜕𝑥2.(3.17) Therefore, by formula (3.17) and Condition 1, we obtain Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+0𝑥𝜕𝑢(𝜕𝑥𝑥,𝜏)2𝑑𝑥+0𝑥2𝑢2𝑑𝑥𝜆3Ω𝜏𝑥2𝑓2(𝑥,𝑡)𝑑𝑥𝑑𝑡+0𝑥2𝜑2(𝑥)𝑑𝑥+0𝑥𝜕𝜑𝜕𝑥2𝑑𝑥+𝜆4Ω𝜏𝑥𝜕𝑢𝜕𝑥2𝑑𝑥𝑑𝑡+Ω𝜏𝑥2𝑢2,(𝑥,𝑡)𝑑𝑥𝑑𝑡(3.18) where 𝜆3=𝛼max12+2𝛼2/4,𝜆2/2,𝑑1/2𝜆min1,𝜆2/2,𝑑0/2,𝜆4=𝛼max52+2𝛼4𝑑22/4,𝑑3+𝛼3𝑑21/2𝜆min1,𝜆2/2,𝑑0./2(3.19) Eliminating the last term on the right-hand side of inequality (3.18). To this end, using Gronwall's lemma, it follows that Ω𝜏𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+0𝑥𝜕𝑢(𝜕𝑥𝑥,𝜏)2𝑑𝑥+0𝑥2𝑢2𝑑𝑥𝜆5Ω𝑥2𝑓2(𝑥,𝑡)𝑑𝑥𝑑𝑡+0𝑥2𝜑2(𝑥)𝑑𝑥+0𝑥𝜕𝜑𝜕𝑥2,𝑑𝑥(3.20) 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 Ω𝐽2𝑥𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑡+sup0𝑡𝑇0𝑥2𝑢2(𝑥,𝑡)𝑑𝑥+0𝑥𝜕𝑢𝜕𝑥2𝑑𝑥𝑐Ω𝑥2𝑓2(𝑥,𝑡)𝑑𝑥𝑑𝑡+0𝑥2𝜑2(𝑥)𝑑𝑥+0𝑥𝜕𝜑(𝑥)𝜕𝑥2,𝑑𝑥(3.21) where 𝑐=𝜆5=𝜆3𝑒𝜆4𝑇/2. This completes the proof of Theorem 3.1.

Lemma 3.3. The operator 𝐴𝐸𝐹 with domain 𝐷(𝐴) has a closure 𝐴.

Proof of Lemma 3.2. Suppose that 𝑢𝑛𝐷(𝐴) is a sequence such that lim𝑛+𝑢𝑛=0,in𝐸,(3.22)lim𝑛+𝐴𝑢𝑛=(𝑓,𝜑),in𝐹,(3.23) we must show that 𝑓0 and 𝜑0. Equality (3.22) implies that lim𝑛+𝑢𝑛=0,in𝒟(Ω).(3.24) By virtue of the condition of derivation of 𝒟(Ω) in 𝒟(Ω), we get lim𝑛+𝜕𝑢𝑛𝜕𝜕𝑡𝑎(𝑥,𝑡)2𝑢𝑛𝜕𝑥2+𝑏(𝑥,𝑡)𝜕𝑢𝑛𝜕𝑥+𝑐(𝑥,𝑡)𝑢𝑛=0,in𝒟(Ω).(3.25) Then from equality (3.23) it follows that lim𝑛+𝜕𝑢𝑛𝜕𝑡𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑢𝑛𝜕𝑥+𝑏(𝑥,𝑡)𝑢𝑛=𝑓,in𝐿2(Ω).(3.26) therefore lim𝑛+𝜕𝑢𝑛𝜕𝑡𝑎(𝑡)𝑥2𝜕𝑥𝜕𝑥2𝜕𝑢𝑛𝜕𝑥+𝑏(𝑥,𝑡)𝑢𝑛=𝑓,in𝒟(Ω).(3.27) By virtue of the uniqueness of the limit in 𝒟(Ω), the identies (3.25) and (3.27) conduct to 𝑓0.
By analogy, from (3.23), we get lim𝑛+𝑢𝑛(𝑥,0)=𝜑(𝑥),in𝐿2(0,).(3.28) We see via (3.22) and the obvious inequality 𝑢𝑛(𝑥,0)𝐿2(0,)𝑢𝑛(𝑥,𝑡)𝐸,𝑛(3.29) that lim𝑛+𝑢𝑛(𝑥,0)=0,in𝐿2(0,).(3.30) By virtue of (3.28), (3.30) and the uniqueness of the limit in 𝐿2(0,) we conclude that 𝜑0.

Definition 3.4. A solution of the equation 𝐴𝑣=(𝑓,𝜑),(3.31) is called a strong solution of problem (2.2), (2.3), (2.4), and (2.5).

Consequence 3.5. Under the conditions of Theorem 3.1, there is a constant 𝑐>0 independent of 𝑣 such that 𝑣𝐸𝑐𝐴𝑣𝐹,𝑣𝐷𝐴.(3.32)

Consequence 3.6. The range 𝑅(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,𝜑)𝐹.