Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems

Abstract

We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g ⁢ ( t ) ⁢ 𝐟 ⁢ ( 𝐱 ) {g(t)\mathbf{f}(\mathbf{x})} for the load source and g ⁢ ( t ) ⁢ f ⁢ ( 𝐱 ) {g(t)f(\mathbf{x})} for the heat source. In the first inverse source problem, the source 𝐟 ⁢ ( 𝐱 ) {\mathbf{f}(\mathbf{x})} has to be determined from the final in time measurement of the displacement 𝐮 ⁢ ( 𝐱 , T ) {\mathbf{u}(\mathbf{x},T)} , or from the time-average measurement ∫ 0 T 𝐮 ⁢ ( 𝐱 , t ) ⁢ d t {\int_{0}^{T}\mathbf{u}(\mathbf{x},t)\,\mathrm{d}t} . In the second inverse source problem, the source f ⁢ ( 𝐱 ) {f(\mathbf{x})} has to be determined from the time-average measurement of the temperature ∫ 0 T θ ⁢ ( 𝐱 , t ) ⁢ d t {\int_{0}^{T}\theta(\mathbf{x},t)\,\mathrm{d}t} . We show the uniqueness of a solution to these problems under suitable assumptions on the function g ⁢ ( t ) {g(t)} . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources.