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A note on the unique solvability of an inverse problem with integral overdetermination. (English) Zbl 1161.35514

Summary: In this paper we study the unique solvability of the inverse problem of determining a pair of functions \(\{u, f\}\) satisfying the equation
\[ u_t-\Delta_u + \sum^n_{i=1} b_i(x)u_{x_i} + \alpha_u = f(t)g(x, t),\quad (x,t)\in Q_T\equiv \Omega\times (0, T), \]
the initial condition
\[ u(x, 0) = u_0(x),\quad x\in\Omega, \]
the boundary condition
\[ u(x, t) = 0,\quad x\in \partial\Omega,\quad t\in(0,T) \]
and the overdetermination condition
\[ \int_\Omega u(x, t)w(x)\,dx =\xi(t),\quad t\in (0,T), \]
where is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial \Omega\).

MSC:

35R30 Inverse problems for PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35A35 Theoretical approximation in context of PDEs
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