An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

Abstract

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

“…where 0 < µ ≪ 1 -the turbulent diffusion coefficient, u(x, t) -the dimensionless pollutant density, x -the spatial coordinate, t -the time variable, k -the positive coefficient of distribution of a pollutant in the environment, f (x) -a positive function, representing the intensity and location of the pollutant source. Model (1) assumes that the propagation speed does not depend on the water flow rate and depends only on the amount of the pollutant and that pollutant dissipates quickly (for example it can be noise pollution of the water or spread of electricity in the water). In this work, our focuses are on the speed, location, as well as the width of the border between two regions -a region with a high concentration of a pollutant and a region with an acceptable concentration and we assume that in the middle point on this border (named as the transition point), which define the location of the border, the concentration of the pollutant is equal to the critical value of pollutant density in the medium.…”

“…In this work, our focuses are on the speed, location, as well as the width of the border between two regions -a region with a high concentration of a pollutant and a region with an acceptable concentration and we assume that in the middle point on this border (named as the transition point), which define the location of the border, the concentration of the pollutant is equal to the critical value of pollutant density in the medium. Note that besides the presented model (1), the framework proposed in this paper can also be applied to various linear and nonlinear inverse problems in singularly perturbed PDEs, e.g. inverse source problems in parabolic or hyperbolic singularly perturbed PDEs [1], parameter identification problems in singularly perturbed PDEs [2,3,4,5], etc.…”

“…Note that besides the presented model (1), the framework proposed in this paper can also be applied to various linear and nonlinear inverse problems in singularly perturbed PDEs, e.g. inverse source problems in parabolic or hyperbolic singularly perturbed PDEs [1], parameter identification problems in singularly perturbed PDEs [2,3,4,5], etc. Advection-diffusion-reaction PDEs with small parameters arise in a wide range of scientific disciplines such as astrophysics [6], biology [7,8,9,10], liquid chromatography [11,12,13,14], industrial and environmental problems [15,16,17,18].…”

“…Specific examples of our interest here are the reaction-diffusion-advection models for predicting the spread of environmental pollutions as shown in [19,20]. But since in this paper we are investigating rapidly dispersing pollutants, we suggest using an autowave approach with the model (1). The application of the autowave model to the reaction-diffusion problem was considered in [21], where the authors proposed a model to predict the growth of the city of Shanghai in subsequent years.…”

Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force

“…Among these papers, we cite [20,21,22,24,34], where the authors obtain results concerning well-posedness, controllability and Carleman estimates. For inverse issues related to this type of equations we refer to [2].…”