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)


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