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• W4225014232 abstract "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. <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>g</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>𝐟</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {g(t)mathbf{f}(mathbf{x})} for the load source and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>g</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {g(t)f(mathbf{x})} for the heat source. In the first inverse source problem, the source <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>𝐟</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathbf{f}(mathbf{x})} has to be determined from the final in time measurement of the displacement <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>𝐮</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathbf{u}(mathbf{x},T)} , or from the time-average measurement <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mn>0</m:mn> <m:mi>T</m:mi> </m:msubsup> <m:mrow> <m:mi>𝐮</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo rspace=4.2pt stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>d</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {int_{0}^{T}mathbf{u}(mathbf{x},t),mathrm{d}t} . In the second inverse source problem, the source <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {f(mathbf{x})} has to be determined from the time-average measurement of the temperature <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mn>0</m:mn> <m:mi>T</m:mi> </m:msubsup> <m:mrow> <m:mi>θ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>𝐱</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo rspace=4.2pt stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>d</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>g</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {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." @default.
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• W4225014232 date "2022-04-28" @default.
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• W4225014232 title "Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems" @default.
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