Coefficient identification problem with integral overdetermination condition for diffusion equations

Abstract In this paper, we investigate a nonlinear inverse problem aimed at recovering a coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {a(t,x)} , dependent on both time and a subset of spatial variables, in a diffusion equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>x</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>y</m:mi> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> u_{t}-\Delta_{x}u-u_{yy}+a(t,x)u=f(t,x,y) , using an additional measurement given as an integral over the spatial domain. Here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:math> {x\in G\subset\mathbb{R}^{m}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>y</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>π</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {y\in(0,\pi)} . We establish theorems on the existence and uniqueness of both local and global weak solutions. Furthermore, we demonstrate that, under sufficient smoothness of the problem data, there exists a uniquely determined strong solution (both local and global) to the inverse problem. Our approach combines the Fourier method with a priori estimates. Prior studies have either only established the uniqueness of solutions to the inverse problem or demonstrated both existence and uniqueness for the Cauchy problem and under significantly stricter conditions on the data of the problem. For instance, some require the forward problem’s solution to possess up to twelve derivatives.

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