In this section we formulate sufficient conditions for the existence of a
solution to the problems (P) and (FBP) and discuss cases of non-existence.
The process described by these problems is, in general, driven by forces
acting in opposite directions.
A typical example is characterized by the boundary conditions 
and 
at 
(see [38,39,44]).
Whereas the homogeneous boundary condition for 
and the positive velocity
Q cause a
dissolution effect, the positive inflow of 
prevents or hinders that
process. This interaction of opposite influences distinguishes the considered
problem from classical Stefan-like problems.
Depending on the relation between the boundary conditions, pathological cases
of degeneration or ill-posedness of the problem are possible as well as
a well-posed dissolution problem. In the transformed problem (FBP), the
right-hand side
of the differential equation, g, represents the effect of the chemical
reaction causing a consumption of u, since g is nonnegative.
The flux function 
represents the difference, 
,
between the imposed boundary conditions and the inflow, 
,
which would be necessary to keep the
chemical equilibrium in the whole domain.
If 
in the special
case 
or if 
then 
is negative,
i.e., the inflow is high enough to prevent the dissolution.
In this case, the maximum principle forces the solution, u, to be negative in
and the free boundary condition requires a monotone decreasing
boundary function, 
. This means that zone I does not appear. Moreover,
the boundary conditions are incompatible with the equilibrium state in zone II,
i.e. the problem is ill-posed in this case.
In the other case the maximum principle does not yield any global statements
about the sign of u or the monotony of s. The condition
is sufficient
to ensure the local solvability of the problem. If
,
the dissolution process is continuous. We prove the well-posedness
of the problem in a more general case, admitting a non-monotonicity of the
free boundary as a result of the interaction between dissolution and
precipitation.
In the following we present some existence, uniqueness and stability results.
These results can be proved using a classical embedding technique and applying
Schauders fixed point theorem to a integral representation for the free
boundary function, 
The main idea of these methods was first presented
in 1951 by Evans [13] and was employed and extended in numerous
papers and monographes, e.g., [23], [24], [9],
[40].