Simultaneously identifying the thermal conductivity and radiative coefficient in heat equation from Dirichlet and Neumann boundary measured outputs

Let conditions (2.1) hold. Then, for the weak solution of direct problem (1.1), the following estimates hold:

Proof

Multiply both sides of (1.1) by u⁢(x,t), integrate on Ωt:=(0,ℓ)×(0,t) and use the integration by parts formula. Taking into account the initial and boundary conditions, we obtain the energy identity

for a.e. t∈[0,T]. Apply the 𝜀-inequality to the last integral, and use the inequality

to get

Use this inequality in (A.3), and choose ε>0 as ε=k0ℓ, where k0>0 is the constant defined in (2.1). After elementary transformations, we obtain the integral inequality

for a.e. t∈[0,T]. Estimates (A.1) and (A.2) follow from this inequality. ∎

Let conditions (2.1) hold. Assume in addition that f∈H1⁢(0,T). Then, for the regular weak solution u∈L2⁢(0,T;H2⁢(0,ℓ)), with ut∈L2⁢(0,T;V⁢(0,ℓ)), ut⁢t∈L2⁢(0,T;L2⁢(0,ℓ)), of direct problem (1.1), the following estimates hold:

Let conditions (2.1) hold. Assume in addition that f∈H2⁢(0,T). Then, for the regular weak solution with improved regularity u∈L2⁢(0,T;H4⁢(0,ℓ)), with ut∈L2⁢(0,T;H2⁢(0,ℓ)), ut⁢t∈L2⁢(0,T;V⁢(0,ℓ)), ut⁢t⁢t∈L2⁢(0,T;L2⁢(0,ℓ)), of direct problem (1.1), the following estimates hold with the same constants C1,C2>0 defined in Theorem A.1:

Let conditions (2.1) hold. Assume that ν∈L2⁢(0,T). Then, for the weak solution of adjoint problem (A.8), estimates (A.9) and (A.10) hold.

Let conditions (2.1) hold. Assume that f∈H2⁢(0,T) and, in addition, the Neumann measured output satisfies the condition φ∈H1⁢(0,T). Then, for the weak solution of adjoint problem (A.11), the following estimates hold:

and C3>0 is the constant defined in (2.4).

Proof

We use the transformation

where μ⁢(t)=k⁢(0)⁢ux⁢(0,t)+φ⁢(t). Then the function v⁢(x,t) solves the problem

In the same way as in Theorem A.3, we can show that the energy identity corresponding to the transformed problem (A.16) is as follows:

We employ the inequality 2⁢a⁢b≤a2+b2 for the first right-hand side integral and then use the inequality

with the constant CT>0 defined in (A.14) to deduce the main integral inequality

This inequality leads to the estimates

with the constants C^1,C^2>0 defined in (A.14).

In view of transformation (A.15), these estimates imply

where C^3>0 is defined in (A.14). Taking into account that the input μ⁢(t) in the above estimates is defined as μ⁢(0,t)=k⁢(0)⁢ux⁢(0,t)+φ⁢(t) and using estimates (2.4) and (2.5), we arrive at the required estimates (A.12) and (A.13). ∎