An advection–diffusion equation with a generalized advection term: Well-posedness analysis and examples

We consider a Cauchy–Dirichlet problem for a semilinear advection–diffusion equation with a generalized advection term. Specific examples include an incision–diffusion landscape evolution model and a viscous Hamilton–Jacobi equation with an absorbing gradient term. We establish existence and uniqueness of a weak solution and its continuous dependence on initial data and a source term.