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This paper introduces a family of transportation costs between non-negativemeasures. This family is used to obtain parabolic and reaction-diffusionequations with drift, subject to Dirichlet boundary condition, as the gradientflow of the entropy functional$\int_{\Omega}\rho\log\rho+V\rho+1\hspace{1mm}dx$. In 2010, Alessio Figalli andNicola Gigli introduced a transportation cost that can be used to obtainparabolic equations with drift subject to Dirichlet boundary condition.However, the drift and the boundary condition are coupled in their work. Thecosts in this paper allow the drift and the boundary condition to be detached.

A new family of transportation costs with applications to reaction-diffusion and parabolic equations with boundary conditions