Physical meaning of eigenvalues in the heat equation problem

Let’s consider the heat equation on a $Omega subset mathbb{R}^2$ manifold with a boundary $Gamma$, with initial and boundary conditions
begin{align}
dot{u}(mathbf{r}, t) &= Delta u(mathbf{r}, t)quad mathbf{r}inOmega
lim_{mathbf{r}to Gamma} u &= 0
lim_{tto 0}left<T_u, varphiright> &= left<delta_rho, varphiright>,
end{align}
where $Tin D^star$ is a distribution, $varphiin D$ is a test function, and $rhoin Omega$ is the initial concentration point.

We can convert this problem into an eigenvalue problem by substituting a solution
$$u = sum_{n=1}^infty e^{-lambda_n t} psi_n(mathbf{rho})psi_n(mathbf{r}),$$
where $psi_n$ and $lambda_n$ are the eigenfunctions and eigenvalues respectively. The substitution will yield
$$-sum_{n=1}^infty lambda_n e^{-lambda_n t}psi_n(mathbf{rho})psi_n(mathbf{r}) = sum_{n=1}^infty e^{-lambda_n t}psi_n(mathbf{rho})Delta psi_n(mathbf{r}).$$
For some $n$ we will get
$$-lambda_n psi_n(mathbf{r}) = Delta psi_n(mathbf{r}).$$

The question is:

Do the eigenvalues $lambda_n$ have a clear physical meaning in this scenario?