Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation

The Kuramoto-Sivashinsky equation is a fourth-order partial differential equation used as a model for physical phenomena such as plane flame propagation and phase of turbulence. The inverse problem of recovering the second-order coefficient from the knowledge of the solution in final time, for the linear version of the equation, is studied in this article. The inverse problem is formulated as a regularized nonlinear optimization problem, from which the local uniqueness and the stability are proved. Finally, an algorithm for the reconstruction of the coefficient is proposed and several numerical simulations are presented.