In a laboratory experiment, a small spherical tumour of radius $a$ composed of living cells is immersed in a large, nutrient-rich water bath.
Inside the tumour, the concentration satisifies the equation
$0= −k +\frac1{r^2}\frac{d}{dr}(r^2 \frac{dc}{dr})$.
In the bath, the concentration satisfies the equation
$0= \frac1{r^2}\frac{d}{dr}(r^2 \frac{dc}{dr})$ in a ≤ r < ∞.
The nutrient flux at the tumour surface $r = a$ is continuous so that $\frac {dc}{dr}\vert_{r=a^-} = \frac {dc}{dr}\vert_{r=a^+}$
In the far-field the condition $c → c_0$ as $r → ∞$ applies, for constant $c_0$.
(a) Write down a condition on c to be satisfied at $r = a$.
(b) Find the solution for the nutrient concentration in 0 ≤ r ≤ a and a ≤ r < ∞.
(c) Cells die if $c < \hat c$ for a constant threshold value $\hat c$ . Obtain a condition on the tumour size a for a necrotic core to develop.
I am very new to tumour models, I understand the general idea is to integrate to find $c(r)$ and apply boundary conditions, but given my lack of understanding of the science behind it (the maths I'm fine with), I am struggling to come up with suitable boundary conditions to solve this. This is a past exam question I found online.
To expand on my last comment, $$\frac1{r^2}\frac d{dr}\left(r^2 \frac{dc}{dr}\right) = 0$$ gives $$c = C - \frac Ar$$ and $$\frac1{r^2}\frac d{dr}\left(r^2 \frac{dc}{dr}\right) = k$$ gives $$c = D - \frac Br + \frac k{2r^2}$$ for some constants $A, B, C, D$. $c \to c_0$ as $r\to \infty$ yields $C = c_0$, and assuming $c$ is both continuous (presumably the answer to part (a)) and has continuous derivative yields $$D = c_0 + \dfrac k{2a^2}$$ and $$B = A + \frac ka$$
which gives $$c(r) = \begin{cases} \dfrac {c_0r - A}r + k\dfrac {r^2 - 2ar + a^2}{2a^2r^2},&r < a\\\dfrac {c_0r - A}r,& r\ge a\end{cases}$$
Regardless of the value chosen for $A$, this function meets all the conditions of the problem. Thus we do not know what $A$ is. The best that can be done for the rest of the problem is to solve it in terms of $A$.