TD - Black-Scholes, implied volatility and payoff

Question
In this question you may assume a standard Black-Scholes world in which
there are no dividend yields, and hence q = 0.
Suppose that an option, W, is written which is worthless on expiry, but which
pays out a continuous cash-flow of K(S,t)dt during the time interval (t,t+dt),
prior to expiry. Using the standard Black-Scholes analysis, including the
cash-flow Kdt, show that W satisfies the problem
∂W
∂t
+
1
2
σ
2S
2
∂
2W
∂S2
+ rS
∂W
∂S
− rW = −K(S,t),
t < T, W(S, T) = 0.
Use financial arguments to show that if K > 0 for t < T the W > 0 for t < T
and that if K < 0 for t < T then W < 0 for t < T.
What is meant by “implied volatility” of an option V ? What is the vega,
ϕ,of an option, and how does it relate to the “implied volatility”?
Assuming that the payoff for the option is independent of volatility, show
that the vega satisfies the problem
∂ϕ
∂t
+
1
2
σ
2S
2
∂
2ϕ
∂§
2
+ rS
∂ϕ
∂S
− rϕ = −σS
2
∂
2V
∂S2
ϕ(S, T) = 0
Hence deduce that “implied volatility” is well defined if the gamma of the
option does not change sign.