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NDSolve[{D[c[x, y, t], t] ==
    e/r D[c[x, y, t], x, x] + h/r D[c[x, y, t], y, y] -
     u/r D[c[x, y, t] x] - g c[x, y, t], c[x, y, 0] == 0,
   c[0, -a < y, t > 0] == b,
   c[x > Infinity, y < Infinity, t > 0] == 0} c, {x, 0, Infinity}, {y,
   Infinity, 0}, {t, 0, Infinity}]
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