I was reading a paper Section 2, where I found,this line about PDE, Given $T>0$ and $D \subset \mathbb{R}^d$ compact, consider functions $u: [0, T] \times D \rightarrow \mathbb{R}^m$, for $m \geq 1$, that solve the following time-dependent PDE, $$ \label{demo:equation1} \mathcal{L}_a(u)(t, x)=0 \quad \text { and } u(x, 0)=u_0 \quad \forall(t, x) \in[0, T] \times D$$, $\mathcal{L}_a: \mathcal{H} \rightarrow L^2([0, T] \times D)$ is a differential operator that can depend on a parameter (function) $a \in \mathcal{Z} \subset L^2(D)$
Now, I am unable to relate Advection-Reaction PDE with this format,
ADR : $$ \frac{\partial u}{\partial t} - D \frac{\partial^2 u}{\partial x^2} - k u^2 - f(x) = 0, \quad x \in[0,1], t \in[0,1] $$, where $D=0.01$ is the diffusion coefficient, and $k=0.01$ is the reaction rate.
So, We can write it in this way $$ \mathcal{L}_a(u)(t, x) = \frac{\partial u}{\partial t} - D \frac{\partial^2 u}{\partial x^2} - k u^2 - f(x) = 0, \quad x \in[0,1], t \in[0,1]..............(1) $$
I am a little confused about these two questions,
- What is $\mathcal{L}_a$ here?(Equation $1$) 2)what is a?
Edit 1.
Isnt $\mathcal{L}_a$ the Laplacian Operator, in general it is given by (in two dimensions) $\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$. Here in this pde here we are only considering one dimension