Identifying the unknown heat source in a parabolic partial differential equation from the over-specified data plays an important role in applied mathematics, physics and engineering. These problems are widely encountered in the modeling of physical phenomena. A typical example is groundwater pollutant source estimation in cities with large population [1]. Now many scholars have used different methods to identify various types of heat sources. In [2, 3], the authors used the method of fundamental solutions and radial basis functions to identify the unknown heat source. In [4, 5], the authors used the Fourier truncation method and the wavelet dual least squares method to identify the spatial variable heat source. In [6], the authors used the simplified Tikhonov method to identify the spatial variable heat source. In [7, 8], the authors determined the heat source which depends on one variable in a bounded domain using the boundary-element method and an iterative algorithm. In [9], the authors identified the heat source which depends only on time variable using the Lie-group shooting method (LGSM). In [10], the authors used the truncation method based on Hermite expansion to identify the unknown source in a space fractional diffusion equation. In [11], the authors identified the point source with some point measurement data. In [12], the authors proved the existence and uniqueness for identifying the heat source which depends only on time variable. In [13], the authors used the variational method to identify the heat source which has the form \(F(x,t)\). In [14], the authors used the variational method to identify the heat source which has the form of \(F(x,t)=F(x)H(t)\) for the variable coefficient heat conduction equation. As far as we know, most of the researches on heat source identification problem mainly concentrated on one-dimensional case. But for a high dimensional case, there are few research results. In [15], the authors used the spectral method to identify the heat source in a columnar symmetric domain. In [16], the authors used the spectral method to identify the heat source in a spherically symmetric parabolic equation. But the regularization parameters is selected by the *a priori* rule. There is a defect for any *a priori* method, *i.e.*, the *a priori* choice of the regularization parameter depends seriously on the *a priori* bound *E* of the unknown solution. However, the *a priori* bound *E* cannot be known exactly in practice, and working with a wrong constant *E* may lead to a badly regularized solution. In this paper, we not only give the *a posteriori* choice of the regularization parameter which depends only on the measurable data, but also we give some different examples to compare the effectiveness between the *posterior* choice rule and the *priori* choice rule. Moreover, we find the truncation regularization method is better than the other regularization methods, such as Tikhonov regularization and the quasi-boundary value regularization method for solving this problem. To the best of the authors’ knowledge, there are few papers to choose the regularization parameter under the *a posteriori* rule for this problem.

In this paper, we consider the following heat source identification problem in spherical symmetric domain:

$$ \textstyle\begin{cases} u_{t}-\frac{2}{r}u_{r}-u_{rr}=f(r), & 0< t< T,0< r< r_{0},\\ u(r,0)=0, & 0\leq r\leq r_{0},\\ u(r_{0},t)=0, & 0\leq t\leq T,\\ \lim_{r\rightarrow0}u(r,t) \text{ is bounded}, & 0< t< T,\\ u(r,T)=\varphi(r), & 0\leq r\leq r_{0}, \end{cases} $$

(1)

where \(r_{0}\) is the radius, \(f(r)\) is the unknown heat source. Our purpose is to identify \(f(r)\) from the additional data \(u(r,T)=\varphi(r)\). Since the data \(\varphi(r)\) is based on (physical) observation, there must be measurement errors, and we assume the measured data function \(\varphi^{\delta}(r)\in L^{2}[0,r_{0};r^{2}]\), and it satisfies

$$ \bigl\Vert \varphi(\cdot)-\varphi^{\delta}(\cdot) \bigr\Vert \le \delta, $$

(2)

where \(\delta>0\) is the measurable error level.

Using the separation variable method, we get the solution of problem (1) as follows:

$$ u(r, t)=\sum_{n=1}^{\infty}f_{n} \biggl( \int^{t}_{0}e^{-(\frac{n \pi}{r_{0}})^{2}(T-\tau)}\,d\tau\biggr)\psi _{n}(r), $$

(3)

where \(\psi_{n}(r)\) defined as follows are the characteristic functions:

$$ \psi_{n}(r)=\frac{\sqrt{2}n\pi}{\sqrt{r^{3}_{0}}}j_{0} \biggl( \frac{n\pi r}{r_{0}} \biggr)=\frac{\sqrt{2}n\pi}{\sqrt{r^{3}_{0}}}\frac {\sin(\frac{n\pi r}{r_{0}})}{\frac{n\pi r}{r_{0}}}, \quad n=1,2,3,\ldots, $$

(4)

\(j_{0}(x)\) is the zero Bessel function. \(\psi_{n}(r)\in L^{2}[0,r_{0};r^{2}]\) is an orthonormal system in the \(Hilbert\) space \(L^{2}[0,r_{0};r^{2}]\). \(f_{n}\) is the Fourier coefficient of \(f(r)\), which is defined by

$$ f_{n}= \int^{r_{0}}_{0}r^{2}f(r)\psi _{n}(r)\,dr. $$

(5)

Using \(u(r,T)=\varphi(r)\), we obtain

$$ \varphi(r)=\sum_{n=1}^{\infty}f_{n} \biggl( \int^{T}_{0}e^{-(\frac{n\pi}{r_{0}})^{2}(T-\tau)}\,d\tau\biggr)\psi _{n}(r). $$

(6)

Due to the mean value theorem of integrals, we obtain

$$ \varphi(r)=\sum_{n=1}^{\infty}f_{n} \bigl(Te^{-(\frac{n\pi}{r_{0}})^{2}(T-t_{n})} \bigr)\psi_{n}(r),\quad 0< t_{n}< T. $$

(7)

Define the operator \(K: f(\cdot)\rightarrow\varphi(\cdot)\), then we have

$$ \varphi(r)=K f(r)=\sum_{n=1}^{\infty} \bigl(Te^{-(\frac{n\pi}{r_{0}})^{2}(T-t_{n})} \bigr) (f,\psi_{n})\psi_{n}. $$

(8)

It is easy to see that *K* is a linear compact operator, and the singular values \(\{\sigma_{n}\}^{\infty}_{n=1}\) of *K* satisfy

$$ \sigma_{n}=Te^{-(\frac{n\pi}{r_{0}})^{2}(T-t_{n})} $$

(9)

and

$$ (\varphi,\psi_{n})=(f,\psi_{n})Te^{-(\frac{n\pi}{r_{0}})^{2}(T-t_{n})}, $$

(10)

*i.e.*,

$$ (f,\psi_{n})=\sigma_{n}^{-1}(\varphi,\psi _{n}). $$

(11)

So

$$ f(r)=K^{-1}\varphi(r)=\sum_{n=1}^{\infty} \sigma_{n}^{-1} \bigl(\varphi(r),\psi_{n}(r) \bigr)\psi_{n}(r). $$

(12)

From equation (12), we can see \(\sigma^{-1}_{n}\rightarrow\infty \)
\((n\rightarrow\infty)\). Thus, the exact data function \(\varphi(r)\) must decrease rapidly. But the measured data function \(\varphi^{\delta}(r)\) only belongs to \(L^{2}[0,r_{0};r^{2}]\), we cannot expect it has the same decay rate in \(L^{2}[0,r_{0};r^{2}]\). Thus the problem (1) is ill-posed. It is impossible to solve this problem using a classical method. We will use the truncated regularization method to deal with the ill-posed problem. Before doing that, we impose an *a priori* bound on the unknown heat source, *i.e.*,

$$ \bigl\Vert f(\cdot) \bigr\Vert _{H^{p}(0,\pi)}\le E,\quad p>0, $$

(13)

where \(E>0\) is a constant and \(\Vert \cdot \Vert _{H^{p}(0,\pi)}\) denotes the norm in Sobolev space which is defined as follows:

$$ \bigl\Vert f(\cdot) \bigr\Vert _{H^{p}(0,\pi)}:= \Biggl(\sum _{n=1}^{\infty} \bigl(1+n^{2} \bigr)^{p} \bigl\vert \bigl(f(\cdot),\psi_{n}(\cdot) \bigr) \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$

(14)

This paper is organized as follows. In Section 2, under the *a posteriori* parameter choice rule, we give the convergence error estimate. In Section 3, three numerical examples are used to verify the effectiveness for the proposed method. In Section 4, the conclusion of this paper is given.