I am studiyng some PDE's for biology and I wish to understand how to interpret the difussion-reaction equation and the Dirichlet problem.
Let $\Omega$ a regular domain of $\mathbb{R}^2$, delimiting an area of the species where $N$ substances diffuse by interacting with each other. Densites are noted $\rho_1,\dots,\rho_N$ and verify the equations:
$$\partial_t \rho_i - D_iΔ\rho_i=F_i(\rho_1,\dots,\rho_N)\; \text{in}\; \Omega,\, i=1,2,\dots,N,$$
where the $D_i>0$ are the diffusion coefficients for each of the species, and the functions
$$F_i:(\rho_1,\dots,\rho_N)\to F_i(\rho_1,\dots,\rho_N)$$
model the terms of reactions between species. We will give initial contions $$\rho_i(t=0)=\rho_i^0\geq 0.$$
We will first consider homogeneous Dirichlet conditions: $$\rho_i =0\;\text{on}\; \partial\Omega.$$
Questions:
- Specify the meaning (in terms of modeling) of the homogeneous Dirichlet condition.
- Write the material balance for the species $\rho_i$, and explain why, if we admit that the $\rho_i$ will remain positive over time, the prescription of this type of condition corresponds to a disappearance of material.
For the question 1.
I thought that the Dirichlet condition means that the environment outside the domain $\Omega$ is so hostile that any organism crossing the border dies immediately.
For the question 2. part of material balance: I thought like this $$\text{quantity of substance contained in}\; \Omega=n(t)=\int_\Omega\rho_i(t,x)dx$$ and we consider a substance which propagates according to the flux vector $J$. Then
$$\frac{d}{dt}\int_\Omega\rho_i(t,x)dx=-\int_{\partial\Omega}J\cdot n =-\int_\Omega\nabla\cdot J$$
and
$$\partial_t\rho_i + \nabla\cdot J =0$$
Can someone help me?
References for books are much appreciated. Thanks