Let the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}, \ t \geq 0 , \ x= (x_1, x_2, x_3)$$ admits a exponential function $\exp(i(k\cdot x+ wt))$ as its solution, where $k$ is a non-zero constant real vector and $w$ is a constant. Then, the solution
remains constant on certain planes in $\mathbb{R}^3$.
repeats itself after a certain length $L$.
has, in general, an amplitude decaying exponentially with time $t$.
is bounded uniformly for $x \in \mathbb{R}^3$ for a fixed $t$.
As- 1, 2, 3, 4
We have
$$e^{i(k\cdot x+ wt)} = \cos(k\cdot x+ wt)+ i \sin (k\cdot x+ wt)$$
Clearly after $L = 2\pi$, i.e. $x \to x + 2\pi$ it repeats itself.
Holds. Because as $t \to \infty$, heat released by the wave $\to 0$. As solution we are taking is exponential, so result.
For a fixed $t$, both $\sin$ and $\cos$ are uniformly bounded functions, and sum of two uniformly bounded functions is uniformly bounded.
How to look for (1), and other options are correct?