We show the existence of noncircular, self-similar solutions to the three-dimensional Hele-Shaw suction problem with surface tension regularisation up to complete extinction. In an appropriate scaling, these solutions are found as bifurcation solutions to a nonlocal elliptic equation of order three. The bifurcation parameter is the ratio of the suction speed and the surface tension coefficient....

p∗ := ⎧⎨ ⎩ pN N − 2p if p < N 2 , +∞, if p ≥ N2 . Regarding the function a : Ω × R → R, we assume that A : Ω̄× R → R, A(x, ξ) is continuous in Ω̄ × R, with continuous derivative with respect to ξ, a = DξA = A′, having the following properties: Received November 9, 2014, accepted March 5, 2015. Communicated by Eiji Yanagida. 2010 Mathematics Subject Classification: 35J35, 35J60.

We prove that if a solution of an equation of KdV type is bounded above by a traveling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the traveling wave.

We describe in superspace a classical theory of two dimensional (1, 1) dilaton supergravity with a cosmological constant, both with and without coupling to a massive superparticle. We give general exact non-trivial superspace solutions for the compensator superfield that describes the supergravity in both cases. We then use these compensator solutions to construct models of two-dimensional supe...

We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. In particular, we obtain the exact number of bifurcation branches of non trivial solutions at every eigenvalue of a square and at the second eigenvalue of a cube. We also compute the Morse index of the solutions in those branches.

In a recently published book, it is shown that there are essentially seven different non– trivial systems of three intertwined second–order recurrence relations, but the solutions are not given. In this note we present explicit solutions to all seven systems. AMS Classification: 11B39, 11B37