  The paper deals with a third order semilinear equation which char- acterizes
exponentially shaped Josephson junctions in superconductivity. The
initial-boundary problem with Dirichlet conditions is analyzed. When the source
term F is a linear function, the problem is explicitly solved by means of a
Fourier series with properties of rapid convergence. When F is nonlin- ear,
appropriate estimates of this series allow to deduce a priori estimates,
continuous dependence and asymptotic behaviour of the solution.
