Partial data inverse problems for reaction-diffusion and heat equations

TL;DR

This paper establishes uniqueness and injectivity results for partial data inverse problems in reaction-diffusion and heat equations, using asymptotic quasimodes and weighted Laplace transform analysis.

Contribution

It introduces new techniques involving spherical quasimodes and complex analysis to solve partial data inverse problems for parabolic equations.

Findings

Proved uniqueness for partial data inverse problems in semilinear reaction-diffusion equations.

Established injectivity of the Fréchet derivative of the partial Dirichlet-to-Neumann map.

Developed asymptotic analysis methods using spherical quasimodes and weighted Laplace transforms.

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.