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

where ${\mathrm{\Omega }}_T=\{(x,t)\in R^2:0<x<1,0<t\le T\}$. The left boundary value ${\psi }_0$ is assumed to be constant. The functions $c_1>k(u(x,t))\ge c_0>0$ and $g(x)$ satisfy the following conditions:

(C 1) $|k_u(u_1)-k_u(u_2)|<d|u_1-u_2|$;

(C 2) $g(x)\in C^2[0,1]$, $g(0)={\psi }_0$, $g(1)=f(0)$.

The initial boundary value problem (1) has a unique solution $u(x,t)$ satisfying $u(x,t)\in H^{2,2}[0,1]\cap H^{1,2}[0,1]$ [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 $u=u(x,t)$ denotes the temperature and density respectively, are given by the equation in the problem (1). Especially $k=k(u(x,t))$ represents the density-dependent coefficient in the problems of the spread of biological populations [5–9].

Here $u=u(x,t)$ 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 $g(x)$, $k(u(x,t))$ and $f(t)$. It is assumed that the functions $u(x_0,t)={\psi }_1$ and $u_x(x_0,t)={\psi }_2$ respectively satisfy the consistency conditions ${\psi }_1=g(x_0)$ and ${\psi }_2=g^{\prime }(x_0)$.

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 $u(x_0,t)={\psi }_1$ given at $x=x_0$. The similar analysis is applied to the inverse problem with the single measured output data $u_x(x_0,t)={\psi }_2$ given at the point $x=x_0$ in Section 3. Some concluding remarks are given in Section 4.

Here, $A[\cdot ]:=-k(u(0,0))\frac{d^2[\cdot ]}{d{x}^2}$ is a second-order differential operator and its domain is $D_A=\{v(x)\in H_0^{2,2}(0,1)\cap H_0^{3,2}[0,1]:v(0)=0=v(1)\}$, where $H_0^{2,2}(0,1)=\overline{C_0^2(0,1)}$ and $H_0^{1,2}[0,1]=\overline{C_0^1[0,1]}$ are Sobolev spaces. Obviously, by completion $g(x)\in D_A$, since the initial value function $g(x)$ belongs to $C^3[0,1]$. Hence, $D_A$ is dense in $H_0^{2,2}[0,1]$, 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.

From the definition of the semigroup $T(t)$, we can say that the null space of it consists of only zero function, i.e., $N(T)=\{0\}$. This result is very important for the uniqueness of the unknown boundary condition $u(1,t)$.

This is the integral representation of a solution of the initial-boundary value problem (5) on ${\mathrm{\Omega }}_{T_0}=\{(x,t)\in R^2:0<x<x_0,0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi }_n(x)$, the domain of eigenfunctions can be extended to the closed interval $[0,1]$. Moreover they are continuous on $[0,1]$. 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 ${\mathrm{\Omega }}_T=\{(x,t)\in R^2:0<x<1,0<t\le T\}$.

which implies that $f(t)$ can be determined analytically.

The right-hand side of the identity (12) defines the semigroup representation of the unknown boundary condition $u(1,t)$ at $x=1$.

where ${\mathrm{\Omega }}_{T_0}=\{(x,t)\in R^2:0<x<x_0,0<t\le T\}$.

Here $B[\cdot ]:=-k(u(0,0))\frac{d^2[\cdot ]}{d{x}^2}$ is a second-order differential operator, its domain is $D_B=\{v\in C^2(0,x_0)\cap C^1[0,x_0]:v(0)=v_x(x_0)=0\}$. It is clear from the definition of $D_B$ that $D_B\subset C^2[0,x_0]$.

From the definition of the semigroup $S(t)$, we can say that the null space of it consists of only zero function, i.e., $N(S)=\{0\}$. This result is very important for the uniqueness of the unknown boundary condition $u(1,t)$.

This is the integral representation of a solution of the initial-boundary value problem (14) on ${\mathrm{\Omega }}_{T_0}=\{(x,t)\in R^2:0<x<x_0,0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi }_n(x)$, the domain of eigenfunctions can be extended to the closed interval $[0,1]$. Moreover, they are continuous on $[0,1]$. 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 ${\mathrm{\Omega }}_T=\{(x,t)\in R^2:0<x<1,0<t\le T\}$.

which implies that $f(t)$ can be determined analytically.

The right-hand side of the identity (21) defines the semigroup representation of the unknown boundary condition $u(1,t)$ at $x=1$. …

The goal of this study is to identify the unknown boundary condition $u(1,t)$ at $x=1$ by using the over measured data $u(x_0,t)={\psi }_1$ and $u_x(x_0,t)={\psi }_2$. The key point here is the unique extensions of solutions on $[0,x_0]$ to the closed interval $[0,1]$ which are implied by the uniqueness of the solutions. This key point leads to the integral representation of the unknown boundary condition $u(1,t)$ at $x=1$ obtained analytically. …