# Smoothness of solution to transport equation IVP

Consider the following PDE (the unknown is $$u\in C^1(\mathbb{R}^{n+1},\mathbb{R})$$): $$\partial_tu(x,t)+\sum_{k=1}^{n} a_k\partial_k u(x,t)= cu(x,t)$$ $$u(x,0)=g(x)$$

where $$g \in C(\mathbb{R}^n, \mathbb{R})$$, $$a\in \mathbb{R}^n$$ and $$c\in \mathbb{R}$$.

My textbook shows that if $$u$$ is a solution, then we must have $$u(x,t)=g(x-ta)e^{ct}=:v(x,t)$$; and, it says, $$v$$ is a solution.

It seems to me it is assuming $$g\in C^1$$, because (here $$k=1,...,n$$) $$\partial_k v(x,t)= \lim_{s\to 0}{e^{ct}(g(x-ta+se_k)-g(x-ta))/s }$$ exists iff $$\partial_k g(x-ta)$$ exists, and in that case $$\partial_k v(x,t)=e^{ct}\partial_k g(x-ta)$$ so that $$v\in C^1$$ iff $$g \in C^1$$.

After assuming $$g \in C^1$$ I am able to verify that $$v$$ solves the equation.

Am I correct in saying that $$g$$ must be in $$C^1$$ so that the equation can be solved?

PS Is 'it' the correct pronoun when referring to what the text is saying? Or should I use 'he/she' because it is understood that the author is saying that? (Probably a stupid question)