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An inverse problem of finding a parameter in a semi-linear heat equation. (English) Zbl 0727.35137

The authors consider the following problem: Find a pair (u,p) such that \[ u_ t=u_{xx}+p(t)u+F(x,t,u,u_ x,p(t)),\quad 0<x<1,\quad 0<t\leq T, \]
\[ u(x,0)=u_ 0(x),\quad 0<x<1;\quad u_ x(0,t)=f(t),\quad u_ x(1,t)=g(t),\quad 0<t\leq T, \] and \(\int^{1}_{0}\phi (x,t)u(x,t)dx=E(t)\), \(0<t\leq T,\)
where \(u_ 0\), f, g, \(\phi\), E, and F are known functions. In two previous papers [ISNM 77, 31-49 (1986; Zbl 0613.35032), Inverse Probl. 4, No.1, 35-45 (1988; Zbl 0697.35162)] the authors have shown that the above equations have a unique local solution. In this paper uniqueness of a global solution is demonstrated provided that certain assumptions are imposed on the data. Moreover, the existence, continuous dependence upon the data, and the regularity of (u,p) are discussed in detail.

MSC:

35R30 Inverse problems for PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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