Partial data inverse problems for reaction-diffusion and heat equations

Ali Feizmohammadi; Yavar Kian; Gunther Uhlmann

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arXiv · 2406.01387

Partial data inverse problems for reaction-diffusion and heat equations

Ali Feizmohammadi·Yavar Kian·Gunther Uhlmann

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Abstract

We study partial data inverse problems for linear and nonlinear parabolic equations with unknown time-dependent coefficients. In particular, we prove uniqueness results for partial data inverse problems for semilinear reaction-diffusion equations where Dirichlet boundary data and Neumann measurements of solutions are restricted to any open subset of the boundary. We also prove injectivity of the Fr\'{e}chet derivative of the partial Dirichlet-to-Neumann map associated to heat equations. Our proof consists of two crucial ingredients; (i) we introduce an asymptotic family of spherical quasimodes that approximately solve heat equations modulo an exponentially decaying remainder term and (ii) the asymptotic study of a weighted Laplace transform of the unknown coefficient along a straight line segment in the domain where the weight may be viewed as a semiclassical symbol that itself depends on the complex-valued frequency. The latter analysis will rely on Phragm\'{e}n-Lindel\"{o}f principle and Gr\"{o}nwall inequality.

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Ali Feizmohammadi, Yavar Kian, Gunther Uhlmann. 2024-06-03. Partial data inverse problems for reaction-diffusion and heat equations. https://arxiv.org/abs/2406.01387

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