Inverse source problems for the polyharmonic heat equation

Abstract In the present paper, we consider the inverse problem for the polyharmonic heat equation, aiming to recover a space/time heat source F along with the temperature distribution <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> {u(x,t)} . The problem is governed by the higher-order heat equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>l</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> {(\partial_{t}+(-\Delta)^{l})u=F} in a finite cylindrical domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {\Omega\times(0,T]} within the half-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{R}^{d}\times[0,+\infty)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>l</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {l,d\geq 1} . This is done through utilizing the values of u and its normal derivatives up to order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>l</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {l-1} on a given lateral surface of the cylinder, as well as the initial value and time-averaged data (or time observation measurement) for space-dependent source and space-averaged data (energy/mass measurement) for time-dependent source. We determine that the integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mfrac> <m:mi>d</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>l</m:mi> </m:mrow> </m:mfrac> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {k=[\frac{d}{4l}]+2} is admissible for the required degree of data regularity. The well-posedness of the classical solution to the inverse problem are established by employing the method of series expansion in terms of eigenfunctions for the Dirichlet poly-Laplacian. Weyl-type eigenvalue inequalities play a key role in our derivations.