| Abstract: |
| We study a linear second-order parabolic equation in divergence form with drift and potential terms on a bounded domain with initial data and non-homogeneous Dirichlet boundary conditions. The analysis focuses on situations where the coefficients and data may exhibit singular or distributional behavior.
First, assuming bounded measurable diffusion, drift, and potential coefficients, we prove existence, uniqueness, and a priori estimates for weak solutions in the natural energy space associated with the divergence-form operator. The proof is based on energy methods and Galerkin approximations.
We then extend the framework to allow singular coefficients and data, including distributional potentials, source terms, initial values, and boundary conditions. In this setting we introduce a notion of very weak solutions obtained via regularization of the coefficients and data. Existence and uniqueness of such solutions are established under suitable ellipticity and positivity assumptions. Finally, we prove a consistency result showing that, in the regular case, representatives of the very weak solution converge in the energy space to the classical weak solution. |
|