The one-dimensional impulsive Barenblatt–Zheltov–Kochina equation with a transition layer

The initial-boundary value problem for the one-dimensional impulsive pseudoparabolic equation is studied. As a coefficient in the second-order diffusion term, this equation contains the smoothed Dirac delta-function concentrated at some time moment. From a physical viewpoint, such term allows to describe impulsive pressure drop phenomena in filtration problems. Existence and uniqueness of solutions for fixed values of the small parameter of smoothing is proved. After this, the limiting passage as the small parameter tends to zero is fulfilled and rigorously justified. As the result, the limit instantaneous impulsive microscopic-macroscopic model is established. This model is well-posed and involves the additional equation on a transition layer posed on a ‘very fast’ timescale.