Decay of a weak gap and wave propagation in a nonlocal environment with a source
A spatially non-local fourth-order wave equation with a source is considered. The results are set out in terms of the heat transfer theory. The temperature derivative of the source function is positive (a technical source) or negative (a source in biological tissue). The wave velocity (subsonic, sonic, supersonic) is determined with respect to the velocity of propagating heat perturbations. We give examples of the exact solving the problem of disintegration of a weak discontinuity in the temperature field. This problem is set as follows. In the initial state the continuous thermal field contains a point of a weak discontinuity; in that point the first coordinate derivative undergoes a first-order rupture. Further the weak discontinuity disintegrates into two waves which propagate in opposite directions. Initiation of such waves is discussed in detail. A technical source: two subsonic, sonic or supersonic waves; the non-uniform space in front of the waves is spatially periodic; in a particular case spatial non-uniformity is localized on both sides of a weak discontinuity. A source in biological tissue: the thermal field between the waves is a superposition of two running waves for which the product of velocity moduli is equal to the square of propagating thermal perturbations velocity. The non-uniform space in front of the waves is spatially periodic and is displayed as spatial coordinate beating. An example is built for disintegration of a weak discontinuity when time evolution in the perturbed region leads to forming a standing wave.