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ISRN Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 138308, 9 pages
http://dx.doi.org/10.5402/2012/138308
Research Article

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

1Department of Mathematics, University of LomΓ©, P.O. Box 1515, LomΓ©, Togo
2Faculty of Mechanics and Mathematics, Belarusian State University, Minsk 220050, Belarus

Received 23 July 2012; Accepted 14 August 2012

Academic Editors: C.Β Lu and E.Β Skubalska-Rafajlowicz

Copyright Β© 2012 Moussa Zakari Djibibe et al. 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.

Abstract

The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation: πœ•π‘£/πœ•π‘‘βˆ’(π‘Ž(𝑑)/π‘₯2)(πœ•/πœ•π‘₯)(π‘₯2πœ•π‘£/πœ•π‘₯)+𝑏(π‘₯,𝑑)𝑣=𝑔(π‘₯,𝑑), 𝑣(π‘₯,0)=πœ“(π‘₯), 0≀π‘₯≀ℓ, 𝑣(β„“,𝑑)=𝐸(𝑑), 0≀𝑑≀𝑇, βˆ«β„“0π‘₯3𝑣(π‘₯,𝑑)𝑑π‘₯=𝐺(𝑑), 0≀𝑑≀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.

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 β„“>0,𝑇>0, and Ξ©={(π‘₯,𝑑)βˆˆβ„2∢0<π‘₯<β„“,0<𝑑<𝑇}. We will determine a solution 𝑣, in Ξ© of the differential equation πœ•π‘£βˆ’πœ•π‘‘π‘Ž(𝑑)π‘₯2πœ•ξ‚€π‘₯πœ•π‘₯2πœ•π‘£ξ‚πœ•π‘₯+𝑏(π‘₯,𝑑)𝑣=𝑔(π‘₯,𝑑),(π‘₯,𝑑)∈Ω,(1.1) satisfying the initial condition 𝑣(π‘₯,0)=πœ“(π‘₯),0≀π‘₯≀ℓ,(1.2) the classical condition 𝑣(β„“,𝑑)=𝐸(𝑑),0≀𝑑≀𝑇,(1.3) and the integral condition ξ€œβ„“0π‘₯3𝑣(π‘₯,𝑑)𝑑π‘₯=𝐺(𝑑),0≀𝑑≀𝑇.(1.4) For consistency, we have ξ€œβ„“0π‘₯3πœ“(π‘₯)=𝐺(0),πœ“(β„“)=𝐸(0),(1.5) where β„“ and 𝑇 are fixed but arbitrary positive numbers, π‘Ž(𝑑) and 𝑏(π‘₯,𝑑) are the known fuctions satisfying the following condition.

Condition 1. For π‘‘βˆˆ[0,𝑇] and π‘₯∈[0,β„“], we assume that (i)𝑑0β‰€π‘Ž(𝑑)≀𝑑1, (ii)𝑏(π‘₯,𝑑)≀𝑑2, (iii)π‘‘π‘Ž(𝑑)/𝑑𝑑≀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 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 𝑒 defined by 𝑣(π‘₯,𝑑)=𝑒(π‘₯,𝑑)+𝑀(π‘₯,𝑑), where 𝑀(π‘₯,𝑑)=5π‘₯β„“ξ‚βˆ’4𝐸(𝑑)βˆ’20β„“5(π‘₯βˆ’β„“)𝐺(𝑑).(2.1) Then, problem becomes πœ•π‘’βˆ’πœ•π‘‘π‘Ž(𝑑)π‘₯2πœ•ξ‚€π‘₯πœ•π‘₯2πœ•π‘’ξ‚ξ€œπœ•π‘₯+𝑏(π‘₯,𝑑)𝑒=𝑓(π‘₯,𝑑),(2.2)𝑒(π‘₯,0)=πœ‘(π‘₯),0≀π‘₯≀ℓ,(2.3)𝑒(β„“,𝑑)=0,0≀𝑑≀𝑇,(2.4)β„“0π‘₯3𝑒(π‘₯,𝑑)𝑑π‘₯=0,0≀𝑑≀ℓ,(2.5) where πœ‘(π‘₯)=πœ“(π‘₯)+20β„“1(π‘₯βˆ’β„“)𝐺(0)βˆ’β„“(5π‘₯βˆ’4β„“)𝐸(0),𝑓(π‘₯,𝑑)=𝐹(π‘₯,𝑑)+20β„“5ξ€·(π‘₯βˆ’β„“)𝑏(π‘₯,𝑑)𝐺(𝑑)+πΊξ…žξ€Έβˆ’1(𝑑)β„“ξ€·(5π‘₯βˆ’4β„“)𝑏(π‘₯,𝑑)𝐸(𝑑)+πΈξ…žξ€Έ+(𝑑)10β„“5ξ€·β„“π‘₯π‘Ž(𝑑)4ξ€Έ.𝐸(𝑑)βˆ’πΊ(𝑑)(2.6) We introduce appropriate function spaces. Let 𝐿2(Ξ©) 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)=πœ•π‘’,1πœ•π‘‘π‘₯2πœ•π‘’,πœ•πœ•π‘₯2π‘’πœ•π‘₯2∈𝐿2ξ‚Ό(Ξ©),(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 π‘’βˆˆπΏ2(Ξ©) satisfying the boundary conditions (2.4) and (2.5) and having the finite norm: ‖𝑒‖2=ξ€œΞ©π½2π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑+sup0β‰€π‘‘β‰€π‘‡ξ‚»ξ€œβ„“0π‘₯2𝑒2(ξ€œπ‘₯,𝑑)𝑑π‘₯+β„“0ξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘2ξ€œπ‘‘π‘₯+β„“0ξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘₯2ξ‚Ό,𝑑π‘₯(2.8) and 𝐹 is the Hilbert space of vector-value function β„±=(𝑓,πœ‘) having the norm β€–β€–(𝑓,πœ‘)2=ξ€œΞ©π‘₯2𝑓2ξ€œ(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+β„“0π‘₯2πœ‘2ξ€œ(π‘₯)𝑑π‘₯+β„“0ξ‚΅π‘₯πœ•πœ‘(π‘₯)ξ‚Άπœ•π‘₯2𝑑π‘₯,(2.9) where 𝐽π‘₯βˆ«β„Ž=β„“π‘₯πœƒ2β„Ž(πœƒ,𝑑)π‘‘πœƒ.

3. A Priori Estimate and Its Consequences

Theorem 3.1. Under Condition 1, for any function π‘£βˆˆπ·(𝐴), one has the following a priori estimate ‖𝑣‖𝐸≀𝑐‖𝐴𝑣‖𝐹,(3.1) where 𝑐 is a positive constant independent of the solution 𝑣.

Proof. Firstly, applying operator 𝐽π‘₯ to (2.1), multiplying the obtained result with 𝐽π‘₯(πœ•π‘’/πœ•π‘‘), and integrating over Ω𝜏=(0,β„“)Γ—(0,𝜏), oberve that ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯π‘‘π‘‘βˆ’Ξ©πœπ½π‘₯ξ‚΅π‘Ž(𝑑)π‘₯2πœ•ξ‚€π‘₯πœ•π‘₯2πœ•π‘’ξ‚ξ‚Άπ½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚+βˆ«πœ•π‘‘π‘‘π‘₯π‘‘π‘‘Ξ©πœπ½π‘₯(𝑏(π‘₯,𝑑)𝑒)𝐽π‘₯ξ‚€πœ•π‘’ξ‚βˆ«πœ•π‘‘π‘‘π‘₯𝑑𝑑=Ω𝜏𝐽π‘₯(𝑓(π‘₯,𝑑))𝐽π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.2) Integrating by parts of the second integral on the left-hand side of (3.2), we get βˆ’ξ€œΞ©πœπ½π‘₯ξ‚΅π‘Ž(𝑑)π‘₯2πœ•ξ‚€π‘₯πœ•π‘₯2πœ•π‘’ξ‚ξ‚Άπ½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑=Ω𝜏π‘₯2π‘Ž(𝑑)πœ•π‘’π½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.3) Substituting (3.3) into (3.2), we get ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑+Ω𝜏π‘₯2π‘Ž(𝑑)πœ•π‘’π½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚+ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑.Ω𝜏𝐽π‘₯(𝑏(π‘₯,𝑑)𝑒)𝐽π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑=Ω𝜏𝐽π‘₯(𝑓(π‘₯,𝑑))𝐽π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.4) In the second time, multiplying the equality (2.1) with π‘₯2πœ•π‘’/πœ•π‘‘, and integrating the obtained equality over Ω𝜏, we get ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘2ξ€œπ‘‘π‘₯π‘‘π‘‘βˆ’Ξ©πœπœ•π‘Ž(𝑑)ξ‚€π‘₯πœ•π‘₯2πœ•π‘’ξ‚πœ•π‘₯πœ•π‘’+ξ€œπœ•π‘‘π‘‘π‘₯π‘‘π‘‘Ξ©πœπ‘₯2𝑏(π‘₯,𝑑)π‘’πœ•π‘’ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑=Ω𝜏π‘₯2𝑓(π‘₯,𝑑)πœ•π‘’πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.5) The standard integration by parts of the second term on the left-hand side of (3.5), leads to βˆ’ξ€œΞ©πœπœ•π‘Ž(𝑑)ξ‚€π‘₯πœ•π‘₯2πœ•π‘’ξ‚πœ•π‘₯πœ•π‘’1πœ•π‘‘π‘‘π‘₯𝑑𝑑=2ξ€œβ„“0π‘Ž(𝜏)π‘₯2ξ‚€πœ•π‘’ξ‚πœ•π‘₯(π‘₯,𝜏)21𝑑π‘₯βˆ’2ξ€œβ„“0π‘Ž(0)π‘₯2ξ‚΅πœ•πœ‘ξ‚Άπœ•π‘₯2βˆ’12ξ€œΞ©πœπ‘Žξ…ž(𝑑)π‘₯2ξ‚€πœ•π‘’ξ‚πœ•π‘₯2𝑑π‘₯𝑑𝑑.(3.6) Substituting (3.6) into (3.5), we get ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘21𝑑π‘₯d𝑑+2ξ€œβ„“0π‘Ž(𝜏)π‘₯2ξ‚€πœ•π‘’ξ‚πœ•π‘₯(π‘₯,𝜏)21𝑑π‘₯βˆ’2ξ€œβ„“0π‘Ž(0)π‘₯2ξ‚΅πœ•πœ‘ξ‚Άπœ•π‘₯2βˆ’12ξ€œΞ©πœπ‘Žξ…ž(𝑑)π‘₯2ξ‚€πœ•π‘’ξ‚πœ•π‘₯2ξ€œπ‘‘π‘₯𝑑𝑑+Ω𝜏π‘₯2𝑏(π‘₯,𝑑)π‘’πœ•π‘’ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑=Ω𝜏π‘₯2𝑓(π‘₯,𝑑)πœ•π‘’πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.7) Finally, adding (3.4) to (3.7), we have ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑+Ξ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘21𝑑π‘₯𝑑𝑑+2ξ€œβ„“0π‘Ž(𝜏)π‘₯2ξ‚€πœ•π‘’(ξ‚πœ•π‘₯π‘₯,𝜏)2=ξ€œπ‘‘π‘₯Ω𝜏𝐽π‘₯(𝑓(π‘₯,𝑑))𝐽π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑+Ω𝜏π‘₯2𝑓(π‘₯,𝑑)πœ•π‘’1πœ•π‘‘π‘‘π‘₯𝑑𝑑+2ξ€œβ„“0π‘Ž(0)π‘₯2ξ‚΅πœ•πœ‘ξ‚Άπœ•π‘₯2βˆ’ξ€œΞ©πœπ‘₯2𝑏(π‘₯,𝑑)π‘’πœ•π‘’ξ€œπœ•π‘‘π‘‘π‘₯π‘‘π‘‘βˆ’Ξ©πœπ½π‘₯(𝑏(π‘₯,𝑑)𝑒)𝐽π‘₯ξ‚€πœ•π‘’ξ‚+1πœ•π‘‘π‘‘π‘₯𝑑𝑑2ξ€œΞ©πœπ‘Žξ…ž(𝑑)π‘₯2ξ‚€πœ•π‘’ξ‚πœ•π‘₯2ξ€œπ‘‘π‘₯π‘‘π‘‘βˆ’Ξ©πœπ‘₯2π‘Ž(𝑑)πœ•π‘’π½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑.(3.8) In the light of Cauchy inequality, certain terms of (3.8) are then majorized as follows: ξ€œΞ©πœπ½π‘₯(𝑓(π‘₯,𝑑))𝐽π‘₯ξ‚€πœ•π‘’ξ‚π›Όπœ•π‘‘π‘‘π‘₯𝑑𝑑≀12ξ€œΞ©πœπ½2π‘₯(1𝑓(π‘₯,𝑑))𝑑π‘₯𝑑𝑑+2𝛼1ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑,(3.9)Ω𝜏π‘₯2𝑓(π‘₯,𝑑)πœ•π‘’π›Όπœ•π‘‘π‘‘π‘₯𝑑𝑑≀22ξ€œΞ©πœπ‘₯2𝑓2(1π‘₯,𝑑)𝑑π‘₯𝑑𝑑+2𝛼2ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘2βˆ’ξ€œπ‘‘π‘₯𝑑𝑑,(3.10)Ω𝜏π‘₯2π‘Ž(𝑑)πœ•π‘’π½πœ•π‘₯π‘₯ξ‚€πœ•π‘’ξ‚π›Όπœ•π‘‘π‘‘π‘₯𝑑𝑑≀32ξ€œΞ©πœπ‘Ž2ξ‚€π‘₯(𝑑)πœ•π‘’ξ‚πœ•π‘₯21𝑑π‘₯𝑑𝑑+2𝛼3ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚βˆ’ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑,(3.11)Ω𝜏π‘₯2𝑏(π‘₯,𝑑)π‘’πœ•π‘’π›Όπœ•π‘‘π‘‘π‘₯𝑑𝑑≀42ξ€œΞ©πœπ‘₯2𝑏2(π‘₯,𝑑)𝑒21(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+2𝛼4ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘2βˆ’ξ€œπ‘‘π‘₯𝑑𝑑,(3.12)Ω𝜏𝐽π‘₯(𝑏(π‘₯,𝑑)𝑒)𝐽π‘₯ξ‚€πœ•π‘’ξ‚π›Όπœ•π‘‘π‘‘π‘₯𝑑𝑑≀52ξ€œΞ©πœπ½2π‘₯1(𝑏(π‘₯,𝑑)𝑒)𝑑π‘₯𝑑𝑑+2𝛼5ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑.(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 πœ†1ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑+πœ†2ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘21𝑑π‘₯𝑑𝑑+2ξ€œβ„“0ξ‚€π‘₯π‘Ž(𝜏)πœ•π‘’(ξ‚πœ•π‘₯π‘₯,𝜏)2≀1𝑑π‘₯2ξ€œΞ©πœξ€·π›Ό3π‘Ž2(𝑑)+π‘Žξ…žξ€Έξ‚€π‘₯(𝑑)πœ•π‘’ξ‚πœ•π‘₯2𝛼𝑑π‘₯𝑑𝑑+42ξ€œΞ©πœπ‘₯2𝑏2(π‘₯,𝑑)𝑒2+𝛼(π‘₯,𝑑)𝑑π‘₯𝑑𝑑52ξ€œΞ©πœπ½2π‘₯𝛼(𝑏(π‘₯,𝑑)𝑒)𝑑π‘₯𝑑𝑑+12ξ€œΞ©πœπ½2π‘₯+𝛼(𝑓(π‘₯,𝑑))𝑑π‘₯𝑑𝑑22ξ€œΞ©πœπ‘₯2𝑓21(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+2ξ€œβ„“0π‘Žξ‚΅π‘₯(0)πœ•πœ‘ξ‚Άπœ•π‘₯2,(3.14) where πœ†11=1βˆ’2ξ‚΅1𝛼1+1𝛼3+1𝛼5ξ‚Άπœ†21=1βˆ’2ξ‚΅1𝛼2+1𝛼4ξ‚Ά.(3.15)

Lemma 3.2. For π‘₯∈(0,β„“), the following inequalities hold: ξ€œΞ©πœπ½2π‘₯(ℓ𝑒)𝑑π‘₯𝑑𝑑≀22ξ€œβ„“0π‘₯2𝑒2πœ†π‘‘π‘₯,2ξ€œβ„“0π‘₯2𝑒2𝑑π‘₯β‰€πœ†2ξ€œβ„“0π‘₯2πœ‘2(π‘₯)𝑑π‘₯+πœ†2ξ€œΞ©πœπ‘₯2𝑒2(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+πœ†2ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘‘2𝑑π‘₯𝑑𝑑.(3.16) It follows by using Lemma 3.2 and (3.18) that πœ†1ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚1πœ•π‘‘π‘‘π‘₯𝑑𝑑+2ξ€œβ„“0ξ‚€π‘₯π‘Ž(𝜏)πœ•π‘’(ξ‚πœ•π‘₯π‘₯,𝜏)2πœ†π‘‘π‘₯+22ξ€œβ„“0π‘₯2𝑒2≀1𝑑π‘₯2ξ€œΞ©πœξ€·π›Ό3π‘Ž2(𝑑)+π‘Žξ…žξ€Έξ‚€π‘₯(𝑑)πœ•π‘’ξ‚πœ•π‘₯2𝑑π‘₯𝑑𝑑+2𝛼4+𝛼5β„“2ξ€Έ4ξ€œΞ©πœπ‘₯2𝑏2(π‘₯,𝑑)𝑒2+𝛼(π‘₯,𝑑)𝑑π‘₯𝑑𝑑1β„“2+2𝛼24ξ€œΞ©πœπ‘₯2𝑓2πœ†(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+22ξ€œβ„“0π‘₯2πœ‘21(π‘₯)𝑑π‘₯+2ξ€œβ„“0π‘Žξ‚΅π‘₯(0)πœ•πœ‘ξ‚Άπœ•π‘₯2.(3.17) Therefore, by formula (3.17) and Condition 1, we obtain ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑+β„“0ξ‚€π‘₯πœ•π‘’(ξ‚πœ•π‘₯π‘₯,𝜏)2ξ€œπ‘‘π‘₯+β„“0π‘₯2𝑒2𝑑π‘₯β‰€πœ†3ξƒ©ξ€œΞ©πœπ‘₯2𝑓2ξ€œ(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+β„“0π‘₯2πœ‘2ξ€œ(π‘₯)𝑑π‘₯+β„“0ξ‚΅π‘₯πœ•πœ‘ξ‚Άπœ•π‘₯2ξƒͺ𝑑π‘₯+πœ†4ξ‚΅ξ€œΞ©πœξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘₯2ξ€œπ‘‘π‘₯𝑑𝑑+Ω𝜏π‘₯2𝑒2ξ‚Ά,(π‘₯,𝑑)𝑑π‘₯𝑑𝑑(3.18) where πœ†3=𝛼maxξ€·ξ€·1β„“2+2𝛼2ξ€Έ/4,πœ†2/2,𝑑1ξ€Έ/2ξ€·πœ†min1,πœ†2/2,𝑑0ξ€Έ/2,πœ†4=𝛼maxξ€·ξ€·5β„“2+2𝛼4𝑑22/4,𝑑3+𝛼3𝑑21ξ€Έ/2ξ€·πœ†min1,πœ†2/2,𝑑0ξ€Έ./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 ξ€œΞ©πœπ½2π‘₯ξ‚€πœ•π‘’ξ‚ξ€œπœ•π‘‘π‘‘π‘₯𝑑𝑑+β„“0ξ‚€π‘₯πœ•π‘’(ξ‚πœ•π‘₯π‘₯,𝜏)2ξ€œπ‘‘π‘₯+β„“0π‘₯2𝑒2𝑑π‘₯β‰€πœ†5ξƒ©ξ€œΞ©π‘₯2𝑓2ξ€œ(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+β„“0π‘₯2πœ‘2ξ€œ(π‘₯)𝑑π‘₯+β„“0ξ‚΅π‘₯πœ•πœ‘ξ‚Άπœ•π‘₯2ξƒͺ,𝑑π‘₯(3.20) where πœ†5=πœ†3π‘’πœ†4𝑇.
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 ξ€œΞ©π½2π‘₯ξ‚€πœ•π‘’ξ‚πœ•π‘‘π‘‘π‘₯𝑑𝑑+sup0β‰€π‘‘β‰€π‘‡ξ‚»ξ€œβ„“0π‘₯2𝑒2(ξ€œπ‘₯,𝑑)𝑑π‘₯+β„“0ξ‚€π‘₯πœ•π‘’ξ‚πœ•π‘₯2ξ‚Όξƒ©ξ€œπ‘‘π‘₯≀𝑐Ωπ‘₯2𝑓2ξ€œ(π‘₯,𝑑)𝑑π‘₯𝑑𝑑+β„“0π‘₯2πœ‘2ξ€œ(π‘₯)𝑑π‘₯+β„“0ξ‚΅π‘₯πœ•πœ‘(π‘₯)ξ‚Άπœ•π‘₯2ξƒͺ,𝑑π‘₯(3.21) where βˆšπ‘=πœ†5=βˆšπœ†3π‘’πœ†4𝑇/2. This completes the proof of Theorem 3.1.

Lemma 3.3. The operator π΄βˆΆπΈβ†’πΉ with domain 𝐷(𝐴) has a closure 𝐴.

Proof of Lemma 3.2. Suppose that π‘’π‘›βˆˆπ·(𝐴) is a sequence such that lim𝑛→+βˆžπ‘’π‘›=0,in𝐸,(3.22)lim𝑛→+βˆžπ΄π‘’π‘›=(𝑓,πœ‘),in𝐹,(3.23) we must show that 𝑓≑0 and πœ‘β‰‘0. Equality (3.22) implies that lim𝑛→+βˆžπ‘’π‘›=0,inπ’Ÿξ…ž(Ξ©).(3.24) By virtue of the condition of derivation of π’Ÿξ…ž(Ξ©) in π’Ÿξ…ž(Ξ©), we get lim𝑛→+βˆžξ‚Έπœ•π‘’π‘›πœ•πœ•π‘‘βˆ’π‘Ž(π‘₯,𝑑)2π‘’π‘›πœ•π‘₯2+𝑏(π‘₯,𝑑)πœ•π‘’π‘›πœ•π‘₯+𝑐(π‘₯,𝑑)𝑒𝑛=0,inπ’Ÿξ…ž(Ξ©).(3.25) Then from equality (3.23) it follows that lim𝑛→+βˆžξ‚Έπœ•π‘’π‘›βˆ’πœ•π‘‘π‘Ž(𝑑)π‘₯2πœ•ξ‚΅π‘₯πœ•π‘₯2πœ•π‘’π‘›ξ‚Άπœ•π‘₯+𝑏(π‘₯,𝑑)𝑒𝑛=𝑓,in𝐿2(Ξ©).(3.26) therefore lim𝑛→+βˆžξ‚Έπœ•π‘’π‘›βˆ’πœ•π‘‘π‘Ž(𝑑)π‘₯2πœ•ξ‚΅π‘₯πœ•π‘₯2πœ•π‘’π‘›ξ‚Άπœ•π‘₯+𝑏(π‘₯,𝑑)𝑒𝑛=𝑓,inπ’Ÿξ…ž(Ξ©).(3.27) By virtue of the uniqueness of the limit in π’Ÿξ…ž(Ξ©), the identies (3.25) and (3.27) conduct to 𝑓≑0.
By analogy, from (3.23), we get lim𝑛→+βˆžπ‘’π‘›(π‘₯,0)=πœ‘(π‘₯),in𝐿2(0,β„“).(3.28) We see via (3.22) and the obvious inequality ‖‖𝑒𝑛‖‖(π‘₯,0)𝐿2(0,β„“)≀‖‖𝑒𝑛‖‖(π‘₯,𝑑)𝐸,βˆ€π‘›βˆˆβ„•(3.29) that lim𝑛→+βˆžπ‘’π‘›(π‘₯,0)=0,in𝐿2(0,β„“).(3.30) By virtue of (3.28), (3.30) and the uniqueness of the limit in 𝐿2(0,β„“) we conclude that πœ‘β‰‘0.

Definition 3.4. A solution of the equation 𝐴𝑣=(𝑓,πœ‘),(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 ‖𝑣‖𝐸‖‖≀𝑐‖‖𝐴𝑣𝐹,βˆ€π‘£βˆˆπ·π΄ξ‚.(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 β„±=(𝑓,πœ‘)∈𝐹.

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