## 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

### Keywords:

uniqueness; global solution; existence; continuous dependence

### Citations:

Zbl 0613.35032; Zbl 0697.35162
Full Text:

### References:

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