An inverse problem for a quasilinear convection–diffusion equation

We study the inverse problem of recovering a semilinear diffusion term $a(t,\lambda)$ as well quasilinear convection $\mathcal B(t,x,\lambda,\xi)$ in nonlinear parabolic equation $$\partial_tu-\textrm{div}(a(t,u) \nabla u)+\mathcal B(t,x,u,\nabla u)\cdot\nabla u=0, \quad \mbox{in}\ (0,T)\times\Omega,$$ given knowledge flux moving quantity associated with different sources applied at boundary domain. This that is modeled by solution dependent parameters $a$ and B$ has many physical applications related to various classes cooperative interactions or complex mixing processes. Our main result states that, under suitable assumptions, it possible fully recover B$. The recovery based on idea solutions linearized singularities near $\partial \Omega$. proof higher order linearization reduce density property for certain anisotropic products equation. show this constructing sufficiently smooth geometric optic concentrating rays $\Omega$.