Heat equation with different boundary conditions Consider the heat equation 
$$
u_t=u_{xx}
$$
on an interval $[-L,L]$
with Dirichlet, Neuman and periodic boundary conditions.
Am I Right that with Dirichlet b.c. all solutions are exponentially decaying in $L_2$-Norm (and that this corresponds to a spectrum in the left half-plane) while with the other two boundary conditions solutions are not decaying in the L2- norm (and we have the spectrum in the right half plane)?
 A: The spectrum is determined by the $x$ equation and associated endpoint conditions after performing separation of variables. The equation in $X$ is
$$
               -X''(x)=\lambda X(x)
$$
and there are two general types of conditions:


*

*Separated Conditions, which as described as a two-parameter family in real $\alpha,\beta$:
$$
         \cos\alpha X(a) + \sin\alpha X'(a) = 0 \\
         \cos\beta X(b) + \sin\beta X'(b) = 0.
$$
These include the Dirichlet conditions ($\alpha=\beta=0$) and the Neumann ($\alpha=\beta=\pi/2$) and the more general Robin types of conditions.

*Periodic Conditions
$$
           X(a) = X(b),\;\; X'(a) = X'(b).
$$
There are other variants, but the above is the only practical one.


In all cases there are discrete eigenvalues $\lambda_0 < \lambda_1 < \lambda_2 < \cdots$ which tend to $\infty$ with the index $n$, and the PDE has solution
$$
                u(t,x)=\sum_{n=0}^{\infty}C_n e^{-\lambda_n t}X_n(x)
$$
The constants $C_n$ are determined by the initial condition $u(0,x)=\sum_{n=0}^{\infty}C_nX_n(x)$.
You don't generally get decay in the periodic case because $\lambda_0=0$ is an eigenvalue with constant solution $X(x)\equiv 1$, and that term in the series solution remains stationary throughout time. The Dirichlet condtions $X(a)=X(b)=0$ give strictly positive eigenvalues $\lambda_n$ and, so, you do get exponential decay in the $L^2$ norm and pointwise as well. The Neumann problem $X'(a)=X'(b)=0$ also has a constant solution with $0$ eigenvalue.
For general separated conditions, there can be negative eigenvalues, which gives you an unstable, expanding solution. For example, $e^{x}$ is a solution of $-X''=-1\cdot X(x)$ and satisfies the conditions
$$
                   X(0)-X'(0) = 0,\;\; X(1)-X'(1)=0.
$$
So the heat equation with these conditions has a solution $X(x)=Ce^{t+x}$, which definitely is not stable in time. There can be two negative eigenvalues, depending on the robin conditions imposed.
