Existence and uniqueness of solutions for a degenerate quasilinear parabolic problem

Abstract

We consider the following quasilinear parabolic equation of degenerate type with convection term $u_t=\varphi (u)_{xx}+b(u)_x$ in $(-L,0)\times (0,T)$. We solve the associate initial-boundary data problem, with nonlinear flux conditions. This problem, describes the evaporation of an incompressible fluid from a homogeneous porous media. The nonlinear condition in $x=0$, means that the flow of fluid leaving the porous media depends on variable meteorological conditions and in a nonlinear manner on $u$. In $x=-L$, we have an impervious boundary. For a sufficiently smooth initial data, one proves the existence and uniqueness of the global strong solution in the class of bounded variation functions.