Abstract :
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 the Lrq1 norms 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
