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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

Ebru Ozbilge

Author Affiliations

Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No. 156, Izmir, Balcova, 35330, Turkey

Boundary Value Problems 2013, 2013:2  doi:10.1186/1687-2770-2013-2

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/2


Received:23 November 2012
Accepted:17 December 2012
Published:4 January 2013

© 2013 Ozbilge; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a> in the quasi-linear parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M2">View MathML</a>, with Dirichlet boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a>, by making use of the over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a> separately. The purpose of this study is to identify the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> by using the over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a>. First, by using over measured data as a boundary condition, we define the problem on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M11">View MathML</a>, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>, we reach the result. The main point here is the unique extensions of the solutions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13">View MathML</a> to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a> which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a>.

1 Introduction

Consider the following initial boundary value problem for quasilinear diffusion equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M17">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M18">View MathML</a>. The left boundary value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M19">View MathML</a> is assumed to be constant. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21">View MathML</a> satisfy the following conditions:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M22">View MathML</a>;

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M25">View MathML</a>.

The initial boundary value problem (1) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M27">View MathML</a>[1-4].

In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of biological populations, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M28">View MathML</a> denotes the temperature and density respectively, are given by the equation in the problem (1). Especially <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M29">View MathML</a> represents the density-dependent coefficient in the problems of the spread of biological populations [5-9].

We consider the inverse problems[10] of determining boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> in the problem (1) from Dirichlet type of measured output data at the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M33">View MathML</a>

(2)

and from Neumann type of measured output data at the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M35">View MathML</a>

(3)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M28">View MathML</a> is the solution of the parabolic problem (1). In this context, the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39">View MathML</a>. It is assumed that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a> respectively satisfy the consistency conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M43">View MathML</a>.

The semigroup approach [11] for inverse problems for the identification of an unknown coefficient in a quasi-linear parabolic equation was studied by Demir and Ozbilge [12]. The study in this paper is based on the philosophy similar to that in [12-15].

The paper is organized as follows. In Section 2, the analysis of the semigroup approach is given for the inverse problem with the single measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> given at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a>. The similar analysis is applied to the inverse problem with the single measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a> given at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a> in Section 3. Some concluding remarks are given in Section 4.

2 Analysis of the inverse problem of the boundary condition by Dirichlet type of over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a>

Consider now the inverse problem with one measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a>. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, let us first arrange the parabolic equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M51">View MathML</a>

Then the initial boundary value problem (1) can be rewritten in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M52">View MathML</a>

(4)

In order to determine the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a>, we need to determine the solution of the following parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M54">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55">View MathML</a>. To formulate the solution of the above problem in terms of a semigroup, we need to define a new function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M56">View MathML</a>

(6)

which satisfies the following parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M57">View MathML</a>

(7)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M58">View MathML</a> is a second-order differential operator and its domain is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M59">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M61">View MathML</a> are Sobolev spaces. Obviously, by completion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M62">View MathML</a>, since the initial value function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M21">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M64">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M65">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M66">View MathML</a>, which is a necessary condition for being an infinitesimal generator.

In the following, despite doing the calculations in the smooth function space, by completion they are valid in the Sobolev space.

Let us denote the semigroup of linear operators by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a> generated by the operator A[8,9]. We can easily find the eigenvalues and eigenfunctions of the differential operator A. Moreover, the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a> can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator A. Hence, we first consider the following eigenvalue problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M69">View MathML</a>

(8)

We can easily determine that the eigenvalues are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M70">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M71">View MathML</a> and the corresponding eigenfunctions are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M72">View MathML</a>. In this case, the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a> can be represented in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M74">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M75">View MathML</a>. Under this representation, the null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a> of the linear operators can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M77">View MathML</a>

From the definition of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a>, we can say that the null space of it consists of only zero function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M79">View MathML</a>. This result is very important for the uniqueness of the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a>.

The unique solution of the initial-boundary value problem (7) in terms of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M67">View MathML</a> can be represented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M82">View MathML</a>

Now, by using the identity (6) and taking the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M83">View MathML</a> into account, the integral equation for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26">View MathML</a> of the parabolic problem (5) in terms of a semigroup can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M85">View MathML</a>

(10)

In order to arrange the above integral equation, let us define the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M86">View MathML</a>

Then we can rewrite the integral equation in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M88">View MathML</a> in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M89">View MathML</a>

(11)

This is the integral representation of a solution of the initial-boundary value problem (5) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55">View MathML</a>. It is obvious from the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M91">View MathML</a>, the domain of eigenfunctions can be extended to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>. Moreover they are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>. Under this extension, the uniqueness of the solutions of the initial-boundary value problems (4) and (5) imply that the integral representation (11) becomes the integral representation of a solution of the initial-boundary value problem (4) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M94">View MathML</a>.

At this stage, it is obvious that the solution of the inverse problem can easily be obtained by substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> into the integral representation (11) of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M97">View MathML</a>

(12)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39">View MathML</a> can be determined analytically.

The right-hand side of the identity (12) defines the semigroup representation of the unknown boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a>.

3 Analysis of the inverse problem of the boundary condition by Neumann type of over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a>

Consider now the inverse problem with one measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M32">View MathML</a>. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, we arrange the parabolic equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M104">View MathML</a>

Then the initial boundary value problem (1) can be rewritten in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M105">View MathML</a>

(13)

In order to determine the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a>, we need to determine the solution of the following parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M107">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55">View MathML</a>.

To formulate the solution of the above problem in terms of a semigroup, we need to define a new function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M109">View MathML</a>

(15)

which satisfies the following parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M110">View MathML</a>

(16)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M111">View MathML</a> is a second-order differential operator, its domain is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M112">View MathML</a>. It is clear from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M113">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M114">View MathML</a>.

Let us denote the semigroup of linear operators by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a> generated by the operator B[8,9]. We can easily find the eigenvalues and eigenfunctions of the differential operator B. Moreover, the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a> can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator B. Hence, we first consider the following eigenvalue problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M117">View MathML</a>

(17)

We can easily determine that the eigenvalues are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M118">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M119">View MathML</a> and the corresponding eigenfunctions are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M120">View MathML</a>. In this case, the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a> can be represented in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M122">View MathML</a>

(18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M123">View MathML</a>. Under this representation, the null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a> of the linear operators can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M125">View MathML</a>

From the definition of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a>, we can say that the null space of it consists of only zero function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M127">View MathML</a>. This result is very important for the uniqueness of the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a>.

The unique solution of the initial-boundary value problem (16) in terms of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M115">View MathML</a> can be represented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M130">View MathML</a>

Now, by using the identity (15) and taking the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M83">View MathML</a> into account, the integral equation for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26">View MathML</a> of the parabolic problem (14) in terms of a semigroup can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M133">View MathML</a>

(19)

In order to arrange the above integral equation, let us define the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M134">View MathML</a>

Then we can rewrite the integral equation in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M88">View MathML</a> in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M137">View MathML</a>

(20)

This is the integral representation of a solution of the initial-boundary value problem (14) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M55">View MathML</a>. It is obvious from the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M91">View MathML</a>, the domain of eigenfunctions can be extended to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>. Moreover, they are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>. Under this extension, the uniqueness of the solutions of the initial-boundary value problems (13) and (14) imply that the integral representation (20) becomes the integral representation of a solution of the initial-boundary value problem (13) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M94">View MathML</a>.

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> into the integral representation (20) of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M26">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M145">View MathML</a>

(21)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M39">View MathML</a> can be determined analytically.

The right-hand side of the identity (21) defines the semigroup representation of the unknown boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a>. …

4 Conclusion

The goal of this study is to identify the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> by using the over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a>. The key point here is the unique extensions of solutions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13">View MathML</a> to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a> which are implied by the uniqueness of the solutions. This key point leads to the integral representation of the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> obtained analytically. …

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Dedicated to my father and mother Yusuf/Sevim Ozbilge.

The research was supported by parts by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics. …

References

  1. DuChateau, P: Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. SIAM J. Math. Anal.. 26, 1473–1487 (1995). PubMed Abstract | Publisher Full Text OpenURL

  2. Isakov, V: On uniqueness in inverse problems for quasilinear parabolic equations. Arch. Ration. Mech. Anal.. 124, 1–13 (1993). Publisher Full Text OpenURL

  3. Pilant, MS, Rundell, W: A uniqueness theorem for conductivity from overspecified boundary data. J. Math. Anal. Appl.. 136, 20–28 (1988). Publisher Full Text OpenURL

  4. Renardy, M, Rogers, R: An Introduction to Partial Differential Equations, Springer, New York (2004)

  5. Cannon, JR: The One-Dimensional Heat Equation, Addison-Wesley, Reading (1984)

  6. DuChateau, P, Thelwell, R, Butters, G: Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Probl.. 20, 601–625 (2004). Publisher Full Text OpenURL

  7. Showalter, R: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Am. Math. Soc., Providence (1997)

  8. Hasanov, A, Demir, A, Erdem, A: Monotonicity of input-output mappings in inverse coefficient and source problem for parabolic equations. J. Math. Anal. Appl.. 335, 1434–1451 (2007). Publisher Full Text OpenURL

  9. Hasanov, A, DuChateau, P, Pektas, B: An adjoint approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equations. J. Inverse Ill-Posed Probl.. 14, 435–463 (2006). Publisher Full Text OpenURL

  10. DuChateau, P, Gottlieb, J: Introduction to Inverse Problems in Partial Differential Equations for Engineers, Physicists and Mathematicians, Kluwer Academic, Netherlands (1996)

  11. Ashyralyev, A, San, ME: An approximation of semigroup method for stochastic parabolic equations. Abstr. Appl. Anal.. 2012, (2012) Article ID 684248. doi:10.1155/2012/684248

  12. Demir, A, Ozbilge, E: Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation. Math. Methods Appl. Sci.. 30, 1283–1294 (2007). Publisher Full Text OpenURL

  13. Ozbilge, E: Identification of the unknown diffusion coefficient in a quasi-linear parabolic equation by semigroup approach with mixed boundary conditions. Math. Methods Appl. Sci.. 31, 1333–1344 (2008). Publisher Full Text OpenURL

  14. Demir, A, Ozbilge, E: Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient. Math. Methods Appl. Sci.. 31, 1635–1645 (2008). Publisher Full Text OpenURL

  15. Demir, A, Hasanov, A: Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach. J. Math. Anal. Appl.. 340, 5–15 (2008). Publisher Full Text OpenURL