Volume 8, Issue 3

The semilinear heat equation with absorption and a power-type potential, u t = Δu - |x| β |u| p-1 u in ℝ N × ℝ + , with p > 1, β > -2, is considered. In the subcritical range , this problem is shown to admit a finite set of self-similar very singular solutions (VSSs), where f(y) has exponential decay at infinity and solves the semilinear elliptic equation . Some local bifurcation results are extended to the fourth-order equation u t = -Δ 2 u - |x| β |u| p-1 u, with p > 1, β > -4.

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations.

We establish , within the framework of the variational problem and the homology for contact forms defined in [3], partial compactness results for flow-lines originating at periodic orbits. These compactness results are stronger when the index of the dominating periodic orbit is even.

In this paper, we compute precisely Morse indices of iterates of the two closed geodesics on the 2-sphere with Katok’s metric. This information is used to give a direct proof for the Poincaré series of the free loop space on S 2 .

We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

We give two short and elementary proofs of a characterization of constants function by Brezis. Whereas the original proof involves some refined arguments on Sobolev spaces and BV functions, ours are based either on convolutions or on a sort of nonsmooth Mean Value Theorem which is new to our knowledge.

This paper deals with a doubly degenerate parabolic equation with absorption (u n ) t = (|u x | m-1 u x ) x - ηu p in (0, 1) × (0, T) subject to the boundary source (|u x | m-1 u x )(1, t) = u q (1, t). The critical boundary source exponent q c is determined under different dominations to identify global and non-global solutions. A complete classification for all of the four nonlinear exponents m, n, p, q with the coefficient η draws a very clear picture of interactions among the multi-nonlinearities. It is shown that q c depends on the absorption exponent p if and only if p > n. It is found as well that q c is related to small m only: q c relies on m if and only if m < p when p > n, also q c relies on m if and only if m < n when p ≤ n. The behavior of solutions in the critical case of q = q c is interesting. The absorption coefficient η affects the critical property of solutions with q = q c if and only if p > n: the case q = q c belongs to the global existence situation (regardless of the absorption coefficient η) if p ≤ n; while for q = q c with p > n, the solutions may be both global and non-global, depending on the size of the absorption coefficient.

Assume that the linear part at a singular point p 0 of a C 4 differential system Y 0 in ℝ 4 has eigenvalues ±αi and ±βi such that β/α = 1/3. In the main result of the paper we exhibit a one-parameter family of systems Y ε for ε ∈ (-δ 0 ;+δ 0 ) where is shown that the original vector field around p 0 can bifurcate in 0, 1, 2, 3 or 4 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory for non-C 1 differentiable system. Moreover, assuming now that Y ε is a one-parameter family of ℤ 2 - reversible polynomial vector fields of degree 5, we show that it can bifurcate in 0 or 2 one-parameter families of periodic orbits.