Let $p(x,t)$ denote the number density of bacteria growing in a straight pipe (straight along the x-axis) through which water is streaming with a velocity $v(x)$ to the right. The growth may be modelled by a term linear in density. The growth rate $r$ depends on the illumination, hence $r = r(x)$. The initial density of bacteria is $p(x,0) = p_0 (x) $.
(a) Write down the continuity equation for the density $p(x,t) $ and a general current $j = j(p)$ with the growth term. Solve this PDE for vanishing velocity, $v = 0$, and constant rate $r = r_0$.
I have a final examination on PDE next month, was solving past exam papers and
tried the following:
Since bacteria growing in a pipe line can be represented as a flux flowing.
Rate that $q$ is flowing through the imaginary surface $S$ can be written as
$$\int \int_s j \space ds$$
which in a differential form can be written as
$$ \partial_t p + \nabla j = \delta$$
$\delta$ is the generation of $q$ per unit volume per unit time. In this question $\delta$ is a growth term
hence $$\partial_t p + \nabla j = r$$
which is just the one-way wave equation, I am guessing the solution should look like
something like $p(x,t) = p_0 (x-v_0 t)$......
I am having some trouble finding the correct continuity equation and its solution, any help will be appreciated.