Communications in Contemporary Mathematics

A notion of vertex bialgebra and a notion of nonlocal vertex module-algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra V satisfying a suitable condition, a canonical bialgebra B(V) is constructed such that primitive elements of B(V) are essentially pseudo-derivations and group-like elements are essentially pseudo-endomorphisms. As an application, vertex algebras associated with the Heisenberg Lie algebras as well as those associated with the nondegenerate even lattices are reconstructed through smash products, and furthermore, a different approach to the construction of modules for the lattice vertex algebras is given.

The equation -Δu = λe^u posed in the unit ball B ⊆ ℝ^N, with homogeneous Dirichlet condition u|_∂B = 0, has the singular solution when λ = 2(N - 2). If N ≥ 4 we show that under small deformations of the ball there is a singular solution (u,λ) close to (U,2(N - 2)). In dimension N ≥ 11 it corresponds to the extremal solution — the one associated to the largest λ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when N ≥ 10, the extremal solution remains bounded in many cases.

We present a simple way to construct an infinite family of simply connected, nonspin, smooth 4-manifolds with one basic class which do not admit a symplectic structure with either orientation.

The structure of the group Aut(ℂⁿ) of biholomorphisms of ℂⁿ is largely unknown if n > 1. In stark contrast Aut(ℂ) is rather small, consisting of the non-constant affine linear maps. The description of Aut(ℂ) follows from the observation that an injective holomorphic function f : ℂ → ℂ satisfying f(0) = 0 and f′(0) = 1 must be the identity. These considerations suggest that similar characterizations of the identity might be useful in understanding the structure of Aut(ℂⁿ). Using geometric methods we prove that an injective holomorphic map f : ℂⁿ → ℂⁿ is the identity I if and only if the power series at 0 of f - I has no terms of order ≤ 2n + 1 and the function |det Df(z)| |z|²ⁿ |f(z)|^-2n is subharmonic throughout ℂⁿ.

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation

We study the long-time behavior of the solutions of the Porous Medium Equation u_t = Δ u^m, m > 1, posed in a tube Ω = ℝ × D, where D is a bounded domain in ℝⁿ, and with homogeneous Dirichlet conditions on the lateral boundary. We show that the asymptotic behavior of general nonnegative solutions follows the KPP pattern in suitable rescaled variables. We proceed as follows: we pass to the renormalized problem and show that this problem admits a wave solution that travels along the tube with constant speed with respect to the new time (which is logarithmic in the old time scale). This solution has a bounded free boundary as a forward front.

We also show that the universal asymptotic pattern for solutions with compactly supported data is described in the renormalized form by two of these traveling waves going out in different directions, joined by a stationary profile in the middle region.

We compare this situation with the heat equation case u_t = Δu that behaves quite differently.

The purpose of this paper is to establish the exponential decay properties of the solutions for the nonlinear biharmonic equation