Inverse source problem for recovering a space-dependent factor in a Caputo–Hadamard pseudo-parabolic equation

We study a terminal inverse source problem for a pseudo-parabolic equation with the Caputo–Hadamard fractional derivative, where a space-dependent factor in <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:mo>⁢</m:mo> <m:mi>q</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> {f(x)q(t)} is recovered from final data. By spectral decomposition of the Dirichlet Laplacian, we derive an explicit solution formula and characterize the forward operator. The forward coefficients satisfy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>≍</m:mo> <m:msubsup> <m:mi>λ</m:mi> <m:mi>n</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:math> {Hₙλₙ⁻¹} , yielding a two-order spatial smoothing effect. Under a positivity condition on q , we prove uniqueness, a Picard-type range characterization, well-posedness in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="script">ℋ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H²(Ω)} , ill-posedness in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L²(Ω)} , and a conditional stability estimate. For noisy data, we analyze Tikhonov regularization and Landweber iteration under a priori and Morozov-type a posteriori parameter choices, and establish convergence and Hölder-type error estimates under standard source conditions. Numerical experiments illustrate the theory and compare the two reconstruction methods.