The problem of finding coefficients at lower-order terms of a weakly coupled parabolic system (Q2760714)

The article is devoted to studying the following problem: Find functions \(u(x,t)\), \(v(x,t)\), \(q_1(x)\), and \(q_2(x)\) on \(Q = D\times (0,T)\), where \(D\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\Gamma\) using the equations NEWLINE\[NEWLINE \begin{aligned} & u_t - \Delta u + \lambda_1(x,t)u + q_1(x)u + \mu_1(x,t)v = f_1(x,t),\\ & v_t - \Delta v + \lambda_2(x,t)v + q_2(x)u + \mu_2(x,t)u = f_2(x,t) \end{aligned} NEWLINE\]NEWLINE and the initial-boundary conditions NEWLINE\[NEWLINE \begin{gathered} u(x,0) = u_0(x),\quad u(x,T) = u_1(x),\quad x\in D,\\ u(x,t)|_{\Gamma\times (0,T)} = \psi_1(x,t),\\ v(x,0) = v_0(x),\quad v(x,T) = v_1(x),\quad x\in D,\\ v(x,t)|_{\Gamma\times (0,T)} = \psi_2(x,t). \end{gathered} NEWLINE\]NEWLINE This problem arises in mathematical modeling of processes, where the sink (source) functions are unknown in general.NEWLINENEWLINENEWLINEFor solving the problem, the author uses an approach suggested by himself in [\textit{A.~I.~Kozhanov}, Composite type equations and inverse problems, VSP, Utrecht (1999; Zbl 0969.35002)]. The method is based on passing to a special equation of composite type, and then studying a direct problem to the equation obtained.