Consider the heat equation on the interval $a \leq x \leq b$ with Newmman boundary conditions: $$u_{t}(x,t)-u_{xx}(x,t)=0 \quad \textit{ for } a \leq x \leq b, t>0$$ $$u(x,0)=g(x) \quad \textit{ for } a \leq x \leq b$$ $$u_{x}(a,t)=0, u_{x}(b,t)=0 \quad \textit{ for } t>0$$

Show that if $g(x)$ is an increasing, differentiable function, then $u(x,t)$ is lso an increasing function of $x$ for each fixed $t>0$

I have no idea how to do this one since there is no solution for heat equation with interval domain [a,b].

Any help would be apprecitated!