I am currently studying the following nonlinear PDE model for cells migrating under the influence of diffusion and chemotaxis:
\begin{align} \hspace{2cm} \frac{\partial m}{\partial t} & = \frac{\partial^2 m}{\partial x^2} - \chi \frac{\partial}{\partial x} \left(m \frac{\partial c}{\partial x} \right) && \hspace{2cm} (1) \\[5pt] \hspace{2cm} \frac{1}{a} \frac{\partial c}{\partial t} &= \frac{\partial^2 c}{\partial x^2} + m - c && \hspace{2cm} (2) \end{align}
Here $m(x,t)$ and $c(x,t)$ denote the concentrations of cells and chemoattractant (a chemical to which the cells are attracted), respectively, and $a,\chi$ are positive constants. Let $(\overline{m},\overline{c})$ denote a homogeneous steady state of (1)-(2), that is,
- $(\overline{m},\overline{c})$ is a solution of (1)-(2), and
- $ \dfrac{\partial \overline{m}}{\partial t} = \dfrac{\partial \overline{c}}{\partial t} = 0$ and $ \dfrac{\partial^2 \overline{m}}{\partial x^2} = \dfrac{\partial^2 \overline{c}}{\partial x^2} = 0$.
My goal is to now linearize (1)-(2) about the homogeneous steady state $(\overline{m},\overline{c})$ so that I can determine conditions under which the HSS is unstable. The only nonlinear term in the model is the chemotaxis term $\frac{\partial}{\partial x} \big(m \frac{\partial c}{\partial x} \big)$.
My question. How do I "linearize" the term $\frac{\partial}{\partial x} \big(m \frac{\partial c}{\partial x} \big)$ ?
I have been consulting Mathematical Models in Biology by Leah-Edelstein Keshet as a reference, where in section 11.4 the author studies a reaction-diffusion system of the form
\begin{align*} \frac{\partial C_1}{\partial t} = D_1 \frac{\partial C_1}{\partial x^2} + R_1(C_1,C_2) \\[5pt] \frac{\partial C_2}{\partial t} = D_2 \frac{\partial C_2}{\partial x^2} + R_2(C_1,C_2). \\[5pt] \end{align*}
Given a solution $(C_1,C_2)$ of the RD-system, the author defines inhomogeneous perturbation terms $C_1'$ and $C_2'$ by \begin{align*} C_1'(x,t) := C_1(x,t) - \overline{C}_1 \\[2pt] C_1'(x,t) := C_1(x,t) - \overline{C}_1 \\[2pt] \end{align*}
and then linearizes the system by writing the Taylor series expansions for $R_1(\overline{C}_1,\overline{C}_2)$ and $R_2(\overline{C}_1,\overline{C}_2)$ about $(\overline{C}_1,\overline{C}_2)$, and then dropping the nonlinear terms: \begin{align*} R_1(\overline{C}_1,\overline{C}_2) &= R_1(\overline{C}_1 + C_1', \overline{C}_2 + C_2') \\[4pt] & = R_1(\overline{C}_1, \overline{C}_2) + \frac{\partial R_1}{\partial C_1}(\overline{C}_1, \overline{C}_2) C_1' + \frac{\partial R_1}{\partial C_2}(\overline{C}_1, \overline{C}_2) C_2' + \text{H.O.T.} \\[10pt] R_2(\overline{C}_1,\overline{C}_2) &= R_2(\overline{C}_1 + C_1', \overline{C}_2 + C_2') \\[4pt] & = R_2(\overline{C}_1, \overline{C}_2) + \frac{\partial R_2}{\partial C_1}(\overline{C}_1, \overline{C}_2) C_1' + \frac{\partial R_2}{\partial C_2}(\overline{C}_1, \overline{C}_2) C_2' + \text{H.O.T.} \end{align*}
Dropping the higher-order terms and using the fact that $R_1(\overline{C}_1,\overline{C}_2) = R_1(\overline{C}_1,\overline{C}_2) = 0$ (which follows immediately from the fact that $(\overline{C}_1,\overline{C}_2)$ is a HSS) leads to the following linear PDEs for the perturbations: \begin{align*} \frac{\partial C_1'}{\partial t} = D_1 \frac{\partial C_1'}{\partial x^2} + a_{11} C_1' + a_{12} C_2' \\[5pt] \frac{\partial C_2'}{\partial t} = D_2 \frac{\partial C_2'}{\partial x^2} + a_{21} C_1' + a_{22} C_2' \end{align*}
where $a_{ij} := \frac{\partial R_i}{\partial C_j}(\overline{C}_1,\overline{C}_2)$. I am not sure if this method is applicable to equations (1)-(2) however because the term $\frac{\partial}{\partial x}(m \frac{\partial c}{\partial x})$ is not of the form $f(m,c)$. How then would I go about "linearizing" this term?