Abstract
We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.
Highlights
In the present paper we are concerned with the inverse problem consisting in the determination of the time dependent coecient σ(t) from Neumann boundary data ∂νu(σ) on Σ, where ∂ν is the derivative in the direction of the unit outward normal vector to Γ
Following [COY], it is quite natural to extend Theorem 1 when the linear parabolic equation is changed to a semi-linear parabolic equation
Combining Lemma 1 and Lemma 2 with some arguments used in the proof of Theorem 1, we prove that f1 and f2 dened respectively by t f1(t) =
Summary
In the present paper we are concerned with the inverse problem consisting in the determination of the time dependent coecient σ(t) from Neumann boundary data ∂νu(σ) on Σ, where ∂ν is the derivative in the direction of the unit outward normal vector to Γ. Yamamoto [CY11] considered the inverse problem of nding a control parameter p(t) that reach a desired temperature h(t) along a curve γ(t) for a parabolic semi-linear equation with homogeneous Neumann boundary data and they established existence, uniqueness as well as Lipschitz stability.
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