Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2p − 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2p − 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

In this paper, we examine a problem of pollutant transport described by a nonlinear parabolic Partial differential equation (PDE) on a planar domain with obstacles. We then establish an existence and uniqueness result for this corresponding problem with Neumann.boundary conditions.