Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. Written as: $$\frac{\partial{u}}{\partial{t}} \propto u - u_{a}$$ with $u$ being temp of the object and $u_a$ being the surrounding temp.
Then solving this, I believe this can be written as: $$\frac{\partial{u}}{\partial{t}} = \lambda\left(u - u_{a}\right)$$
Where, (with the boundary conditions under which this law is approx true, and this value is a constant) $\lambda$ is a negative constant loosely representing a heat transfer coefficient that makes this an equation.
I am curious whether or not this $\lambda$ can be viewed as or solved as an eigenvalue. My line of reasoning is that the time differential of $u$ might be able to be viewed as a linear transformation? This would make the corresponding eigenfunction $u(t)$, I believe.