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Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. (English) Zbl 0849.35146

The paper considers specific coefficient-to-data mappings associated with nonlinear parabolic partial differential equations and the inverse problems of coefficient determination. In particular, one seeks to determine the coefficients \(a(u)\) and \(b(u)\) in the initial boundary value problem \[ \partial_t a(u)= \partial_{xx} b(u)\text{ for } 0< x< 1,\;0< t< T,\;u(x, 0)= u_0,\;\partial_x u(0, t)= 0\text{ and } u(1, t)= f(t). \] For this purpose the overspecifications \(h(t)= u(0, t)\) and \(g(t)= \partial_x b(u(1, t))\) are considered. It is shown that under appropriate assumptions on the input data \(f(t)\) the coefficient-to-data mapping for these overspecifications is invertible. Integral identities are derived that show also the monotonicity of this mapping in some situations.

MSC:

35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
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