In this paper we consider, for 1 ≤ m < p < 2, the generalized KPZ equation ut = (u) − |∇u|p. For m = 1, we show existence and uniqueness of the so called very singular solution which is self-similar. A complete classification of self-similar solutions is also given. For m > 1, we establish the existence of very singular self-similar solution and prove that such a solution must have compact supp...

Numerical simulations show that channel waveguides can be self-written in photosensitive materials. As the waveguide evolves, its shape remains approximately constant, even though its depth and width change. We find an exact solution that describes this evolution, which we show to be self-similar. A wide variety of single-peaked beams form waveguides that converge to this solution.

ABSTRACT In order to understand which mechanism is responsible for accretion in protoplanetary discs, a robust knowledge of the observed disc radius using gas tracers such as 12CO and other CO isotopologues pivotal. Indeed, two main theories proposed, viscous wind-driven accretion, predict different time evolution radii. this letter, we present an analytical solution radii viscously evolving di...