Backward problems in time for fractional diffusion-wave equation
Giuseppe Floridia, Masahiro Yamamoto
Abstract
Abstract In this article, for a time-fractional diffusion-wave equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>∂</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , 0 < t < T with fractional order α ∈ (1, 2), we consider the backward problem in time: determine u (⋅, t ), 0 < t < T by u (⋅, T ) and ∂ t u (⋅, T ). We prove that there exists a countably infinite set Λ ⊂ (0, ∞) with a unique accumulation point 0 such that the backward problem is well-posed for T ∉ Λ.