Would it be correct that the more explicit version of your equation would be
diff(T(x, t), t) = -A(t)*(diff(T(x, t), x))-B*T(x, t)^(3/2)*(T(x, t)-C(t))
If so then Maple's pdsolve() believes there is no solution.
Consider your boundary conditions at x=0, t=0. According to your first boundary condition, T(0,0)=498, but according to your second boundary condition, T(0,0)=520 . That is a contradiction and hence there is no solution.