Let me elaborate on Muhammad's excellent answer. Another way to think about it is in the context of convectively unstable systems and absolutely unstable systems. In the former case, a disturbance perturbation grows with respect to coordinates moving with the disturbance, but decays in the laboratory frame of reference because the structures are carried downstream. For example, the appearance of 2-D waves in inclined film flow. In an absolutely unstable system, the disturbance grows with respect to the fixed laboratory frame, and quickly contaminates the complete flow field. For example, the appearance of Marangoni flows in an evaporating film. Once the instability is initiated it contaminates the whole flow region.

Well, the stability of the linearized system is given only in time not in the space variables, which means that a perturbation in the space variables will give different solution at the end and makes thus not useful statement about the original system.

From stability point of view in a linearized system having plane wave solution, there occur two type of situations one is with stable in space but unstable in time which results in damping/growing of wave in time. On the other hand, stable in time but unstable in space which results in damping in space i.e., skin effects.......

Let me elaborate on Muhammad's excellent answer. Another way to think about it is in the context of convectively unstable systems and absolutely unstable systems. In the former case, a disturbance perturbation grows with respect to coordinates moving with the disturbance, but decays in the laboratory frame of reference because the structures are carried downstream. For example, the appearance of 2-D waves in inclined film flow. In an absolutely unstable system, the disturbance grows with respect to the fixed laboratory frame, and quickly contaminates the complete flow field. For example, the appearance of Marangoni flows in an evaporating film. Once the instability is initiated it contaminates the whole flow region.

Temporal stability refers to an initial wave like disturbance. The wave amplitude should decrease over time (as the wave propagates). If it increases instead, that is the amplitude blows up, we have an instability. The energy content of the wave increases over time. This is for initial value problems. The Burnett equations have modes that are unstable in this sense. Some modifications of the Burnett equations, however, remove the problems, that is all modes are stable in time (no blow up).

In a boundary value problem, the wave originates at a wall, for instance a loudspeaker. The expectation is that the further away you are from the speaker, the amplitude of the wave is smaller. The Burnett equations, and their modifications, have modes where the amplitude grows in direction of travel, which certainly is unphysical. Calling it "unstable" in space might not be the best name, but emphasizes the analogy to the time case. These modes make it difficult, or impossible, to solve the equations for certain problems. I have a publication where you can see this, link is below.

https://www.researchgate.net/publication/226583574_Resonance_in_rarefied_gases?ev=prf_pub

Article Resonance in rarefied gases

Another way to think about this is in how we choose our guesses when making the initial approximations that lead to a closed solution (e.g., linearize by assuming ∂/∂t --> -i w). Technically, a truly general dispersion relation would allow for finite real and imaginary parts for both frequency and wave numbers, but choosing one or the other is often too difficult to solve let alone both. Typically people assume that the frequency is composed of real (w) and imaginary (Y) parts, or:

Ω = w + i Y

In this case, if we also assume that the wave numbers are purely real (i.e., Im[k] = 0), then the wave can only grow/damp in time. If we assume the converse, namely that Im[Ω]=0, then we have:

K = k + i ∆

which implies that the wave can only grow/damp as a function of space. As Muhammad eluded to, a wave whose amplitude depends upon spatial changes often refers to situations where a wave propagates into a region with vastly different dispersive properties (e.g., different index of refraction that causes gain/loss of amplitude). An example would be certain frequencies of radio waves that propagate into or out of the ionosphere. At the right frequencies, these waves can reach a region where they strongly interact with the background particles, either giving energy to (damping) or taking energy from (growth) these particles. Some of the frequencies reflect off this region (typically lower frequencies) while others are directly transmitted without loss (typically higher frequencies).

Does this help?