Uniqueness for inverse source problems for fractional diffusion-wave equations by data during not acting time
Masahiro Yamamoto
Abstract
Abstract We consider fractional diffusion-wave equations with source term which is represented in a form of a product of a temporal function and a spatial function. We prove the uniqueness for inverse source problem of determining spatially varying factor by decay of data as the time tends to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> , provided that the source does not work during the observation. Our main result asserts the uniqueness if data decay more rapidly than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:math> with any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> . Data taken not from the initial time are realistic but the uniqueness is not known in general. The proof is based on the analyticity and the asymptotic behavior of a function generated by the solution.