Relation between Diffusion equation and 1st order linear ODE

Question

I'm studying PDE with a book written by Walter A Strauss.

In sec 3.3, we tried to guess what is the solution of diffusion equation with a source.

In analogy, they brought 1st order linear ODE

$u_t+Au(t)=f(t),\ \ \ u(0)=\phi$.

And with its solution, they guessed the solution of diffusion equation with a source(with source operator).

I don't know why the solution of 1st order linear ODE is a hint for solution of diffusion equation.

Is there any physical meaning? or some other theory?

Answer 1

in the reaction-diffusion equation $Au = u_{xx}+u_{yy},$ the laplacian of $u.$ if $A$ were a scalar,the differential equation is just first order differential equation. you can use, for example, variation of parameters to represent the solution as an integral and then use estimates. there is book by henry, in the springer lecture notes series, i remember using this method to good advantage.