diffusionC = 1;
charge = 1;
mobility = 1;

equation =
  D[u[t, x], t] == -D[-diffusionC*D[u[t, x], x] -
    charge*mobility*u[t, x]*D[[CapitalPhi][t, x], x], x];

potential = [CapitalPhi][t, x] == -2.5*x + 5;

ic = {(D[[CapitalPhi][t, x], t] /. t -> 0) ==  0, [CapitalPhi][0, x] == 5 - 2.5 x}

bc = {u[t, 0] == 8, u[t, 2] == 0}

sol = NDSolve[{equation, potential, ic, bc}, {u, [CapitalPhi]}, {t, 0, 10}, {x, 0, 2}]

Plot3D[u[t, x] /. sol, {x, 0, 2}, {t, 0, 10}, PlotRange -> All, AxesLabel -> {x, t, u}]

Plot3D[[CapitalPhi][t, x] /. sol, {x, 0, 2}, {t, 0, 10},
       PlotRange -> All, AxesLabel -> {x, t, [CapitalPhi]}]

NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. >>

NDSolve::bcart: Warning: an insufficient number of boundary conditions have
been specified for the direction of independent variable x. Artificial
boundary effects may be present in the solution. >>

bc = {u[t, 0] == 8-7*Exp[-1000t], u[t, 2] == 0}

NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. >>

NDSolve::bcart: Warning: an insufficient number of boundary conditions have
been specified for the direction of independent variable x. Artificial
boundary effects may be present in the solution. >>

NDSolve::mconly: For the method IDA, only machine real code is available.
Unable to continue with complex values or beyond floating-point exceptions. >>

NDSolve::eerr: Warning: scaled local spatial error estimate of
652895.9781327343` at t = 0.14272838549715564` in the direction of
independent variable x is much greater than the prescribed error tolerance.
Grid spacing with 25 points may be too large to achieve the desired accuracy
or precision. A singularity may have formed or a smaller grid spacing can be
specified using the MaxStepSize or MinPoints method options. >>