Two coupled linear diffusion equations $$\begin{split}\partial_ta&=\nabla(\nabla a)\\ \partial_tb&=\nabla(\nabla b)\end{split}$$ can be split into blocks by putting everything onto one side, and writing it as $$\begin{pmatrix}\partial_t -\Delta & 0\\\ 0 & -\Delta + \partial_t\end{pmatrix}\begin{pmatrix}a \\ b\end{pmatrix}=0$$ How can I do the same for the coupled non-linear diffusion equations $$\begin{split}\partial_ta&=\nabla(f(b)\nabla a)\\ \partial_tb&=\nabla(g(a)\nabla b)\end{split}$$ with $f$, $g$ functions of $a$, $b$? When trying to create the blocks as above, I have no solution for implementing $f(b)$ and $g(a)$ in the matrix, after they depend on the solutions of the vector the matrix is multiplied with.
The same problem arises when extending the non-linear equations to $$\begin{split}\partial_t(h(b)\cdot a)&=\nabla(f(b)\nabla a)\\ \partial_t(i(a)\cdot b)&=\nabla(g(a)\nabla b)\end{split}$$ with $h$, $i$ two additional functions. Is it still possible to create a block matrix as done for the linear equations, and if yes, how?


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