×

Inverse problems for parabolic equations. (English) Zbl 1162.35384

Summary: Let \(u_t-\nabla^2u=f(x)\: =\sum^M_{m=1}a_m\delta(x-x_m)\) in \(D\times[0,\infty)\), where \(D\subset {\mathbb R}^3\) is a bounded domain with a smooth connected boundary \(S\), \(a_m=\text{ const}, \delta(x-x_m)\) is the delta-function. Assume that \(u(x,0)=0, u=0\) on \(S\). Given the extra data \(u(y_k,t)\: = b_k(t)\), \(1\leq k\leq K\), can one find \(M, a_m\), and \(x_m\)? Here \(K\) is some number. An answer to this question and a method for finding \(M, a_m\), and \(x_m\) are given.

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35R30 Inverse problems for PDEs
PDF BibTeX XML Cite