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We consider the Cauchy problem for the damped wave equation under the initialstate that the sum of an initial position and an initial velocity vanishes.When the initial position is non-zero, non-negative and compactly supported, westudy the large time behavior of the spatial null, critical, maximum andminimum sets of the solution. The spatial null set becomes a smoothhyper-surface homeomorphic to a sphere after a large enough time. The spatialcritical set has at least three points after a large enough time. The set ofspatial maximum points escapes from the convex hull of the support of theinitial position. The set of spatial minimum points consists of one point aftera large time, and the unique spatial minimum point converges to the centroid ofthe initial position at time infinity.

Movement of time-delayed hot spots in Euclidean space for special initial states