How to get the integral of the nonlinear diffusion equation

Question

The nonlinear diffusion equation is $$\frac{1}{r}\frac{\partial}{\partial r}(rh^n \frac{\partial h}{\partial r})=\frac{\partial h}{\partial t} \ (1)$$ where $h=h(r,t)$, $r\in[0,r_f]$, $t\geq0$. $r_f$ varies with time $t$. The boundary conditions: $$\lim_{r\to 0}(2 \pi r h^n \frac{\partial h}{\partial r})=-Q_0 \ (2)$$ $$h(r_f,t)=0 \ (3)$$ $$\frac{\partial h}{\partial r}(r_f,t)=0 \ (4)$$ The initial conditions: $$h(r,0)=0 \ (5)$$ $$r_f(0)=0 \ (6)$$ The target integral equation is: $${\int _{0} ^{r_f}} hrdr=\frac{Q_0}{2 \pi}t \ (7)$$ How to get Eq.(7) through Eq.(1)-(6). Thanks.

Answer 1

Let's take the time derivative of the integral $(7)$: $$ \frac{d}{dt}\int_0^{r_f}hr\,dr=\frac{dr_f}{dt}h(r_f,t)r_f+\int_0^{r_f}\frac{\partial h}{\partial t}r\,dr. \tag{A} $$ The first term on the RHS vanishes because of $(3)$, and the ODE $(1)$ implies $$ \int_0^{r_f}\frac{\partial h}{\partial t}r\,dr=\int_0^{r_f}\frac{\partial}{\partial r}\left(rh^n \frac{\partial h}{\partial r}\right)dr=rh^n \frac{\partial h}{\partial r}\bigg|_{r_f}-rh^n \frac{\partial h}{\partial r}\bigg|_{0}=\frac{Q_0}{2\pi}, \tag{B} $$ where we've used $(2)$, $(3)$, and $(4)$. Plugging $(\text{B})$ into $(\text{A})$, and integrating with respect to $t$, we obtain $$ \int_0^{r_f}hr\,dr=\frac{Q_0}{2\pi}t+C. \tag{C} $$ The initial conditions $(5)$ and $(6)$ imply $C=0$, hence $$ \int_0^{r_f}hr\,dr=\frac{Q_0}{2\pi}t. \tag{D} $$