Under the influence of a sufficiently “weak” nonlinear source term, it is by now well known that a degenerate diffusion equation is globally solvable. A similar result is known when the nonlinear source is present as a forcing term at the boundary. Such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle. However, these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present. In this work, we obtain the global existence of solutions for such problems by deriving a differential inequality involving theL^r+1norms of solutions. The result is similar to that described above for equations with a nonlinear source term and, furthermore, establishes the global existence of “small” solutions even for “strong” forcings. The technique also applies to a more general class of problems involving nonlinear reaction, diffusion, and convection.