In this section, we consider the diffusion equation (2.1) with an
initial condition (2.2), but now where we seek a solution in a
bounded domain 
. We need to impose certain boundary conditions on in order to obtain a unique solution. Very often, these involve
specifying either u or 
or a combination of the two
on the boundary of (denoted as 
), where is the directional derivative of u along the normal
to the boundary.
For example, consider the following boundary value problem for 
involving the one-dimensional diffusion equation, where t>0
and 
:
Note that pde stands for partial differential equation, and b.c.'s stands for boundary conditions.
Physically, 
represents the initial distribution of at time t=0; specification of the value of u at the
left boundary by 
indicates there is a ``source'' holding
u at some given value on that boundary; and the specification of 
at the right boundary by 
indicates the
``flow'' of u across the boundary is held to a known formula.
Specifically, a boundary condition of the form:
is called an adiabatic boundary condition, and physically corresponds to no net flow of u into or outside of the region across the boundary.
If we assume the boundary conditions specify either u=0 or 
at all points, it can be shown the solution still has the
form 
where 
is a kernel that depends on t,
called the Green's function for the problem. Moreover, the solution
spreads and smooths out as time evolves. Let us examine the special case of
adiabatic boundary conditions in particular.
. Then 
. On the other hand, the divergence theorem implies 
where 
is the outward normal from the boundary of
the region . Hence: 
Next, note that 
, so:
.
Then: 
In particular, 
(strict), unless 
everywhere,
which in turn requires u to be constant.
for some time 
, say at 
. Now, it may be
that 
over some
time interval 
. Then there is a neighborhood of ,
