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I have a reaction-diffusion equation in 1-dimensions of the typical form:

$$\frac{\partial }{\partial t} u(x,t)= \frac{\partial^2 }{\partial x^2} u(x,t)+ \alpha(x) u(x,t), \,\qquad (x,t)\in (0,1)\times (0,T)$$ with homogeneous Dirichlet boundary conditions. I want to stress that the parameter $\alpha(\cdot)$, is not a constant, but depends on the spatial variable $x$.

In theory, we can, for example, take $\alpha(x)=x,\; u(x,0)=\sin(x)...$

My question is about the concrete interpretation of such an equation. How can be interpreted the parameter $\alpha,$ how is its form (dependance on $x$), how can we take the initial state in order to be interpretable. What is the interpretation of the solution $u(x,t)$.

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What such an reaction-diffusion equation is often meant to describe is a quantity $u$ which spreads itself out through space, and grows or decays at the same time due to the 'reaction terms' in the equation. The term $a u$ is such a reaction term. The most direct way to interpret this term is to neglect the diffusion term $\frac{\partial^2 u}{\partial x^2}$ for the moment. Then, you obtain the ODE \begin{equation} \frac{\text{d} u}{\text{d} t} = a u, \end{equation} which just describes the exponential growth of the quantity $u$, with growth rate $a$. The only difference a space-dependent $a$ makes, is that this growth rate now depends on where you are. Imagine some animal population which eats a plant which is only found at certain places, for example. You can imagine that in that case, the birth rate of the species is higher at places where this plant grows a lot, and that the birth rate is lower at places where this plant cannot be found (because the animals need more energy to stay alive, and cannot spare that energy to produce offspring).

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