Let the heat equation \(\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x_1^2}\) + \(\frac{\partial^2 u}{\partial x_2^2}+\frac{\partial^2 u}{\partial x_3^2}\), t ≥ 0, x = (x1, x2, x3) ∈ ℝ3 admit an exponential function exp(i(kx + wt)) as its solution, where k is a non-zero constant real vector and w is a constant. Then, the solution
1
Remains constant on certain planes in ℝ3 .
2
Repeats itself after a certain length L.
3
Has, in general, an amplitude decaying exponentially with time t.
4
Is bounded uniformly for x ∈ ℝ3 for a fixed t.