A new kind of the solutions of the convection-diffusion equation related to the
p(x)-Laplacian is introduced. The equation is degenerate on the boundary,
accordingly, the usual boundary value condition cannot be imposed in Dirichlet’s
way. The test function chosen to verify the uniqueness of the solutions should be
independent of the boundary value condition. By the new definition, one can study
the stability of the weak solutions without any boundary value condition. The main
results of the paper show that the usual homogeneous boundary value condition can
be replaced by the degeneracy of the diffusion coefficient and the degeneracy of the
convection term on the boundary.

• A new kind of the solutions of a convection-diffusion equation related to the p(x)-Laplacian