Simultaneous recoveries for semilinear parabolic systems

Abstract

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general $C^1$ class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special CGO (Complex Geometrical Optics) solutions for the parabolic system.

arXiv comment34 pages

References in corpus (19)

• Phase-Field Model of Mode III Dynamic Fracture
• Determining both sound speed and internal source in thermo- and photo-acoustic tomography
• Identification of nonlinearity in conductivity equation via Dirichlet-to-Neumann map
• Reconstruction for the coefficients of a quasilinear elliptic partial differential equation
• Determining a random Schrödinger operator: both potential and source are random
• Determining a random Schrödinger equation with unknown source and potential
• On identifying magnetized anomalies using geomagnetic monitoring
• On identifying magnetized anomalies using geomagnetic monitoring II. A Magnetohydrodynamic Model
• Detection of Hermitian connections in wave equations with cubic non-linearity
• An inverse boundary value problem for certain anisotropic quasilinear elliptic equations
• Determination of vacuum space-times from the Einstein-Maxwell equations
• Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term
• Determination of convection terms and quasi-linearities appearing in diffusion equations
• Uniqueness and stability of an inverse problem for a semi-linear wave equation
• Partial Data Inverse Problems for Nonlinear Magnetic Schrödinger Equations
• Determining a nonlinear hyperbolic system with unknown sources and nonlinearity
• Partial data inverse problems for quasilinear conductivity equations
• Inverse problem for Einstein-scalar field equations
• Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities

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