We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

We prove sharp upper bounds for invariant measures of Markov processes in ℝN associated with second-order elliptic differential operators with unbounded coefficients.

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

We discuss travelling wavefronts of a degenerate and singular parabolic equation in non-divergent form with changing sign sources. Necessary and sufficient conditions will be given for the existence of smooth or non-smooth and non-decreasing or non-increasing solutions. We also study the regularity of such solutions.

Given a bounded domain G ⊂ ℝd, d ≥ 3, we study smooth solutions of a linear parabolic equation with non-constant coefficients in G, which at the boundary have to C1-match with some harmonic function in ℝd \ Ḡ vanishing at spatial infinity.

This problem arises in the framework of magnetohydrodynamics if certain dynamo-generated magnetic fields are considered: for example, in the case of axisymmetry, or for non-radial flow fields, the poloidal scalar of the magnetic field solves the above problem.

We first investigate the Poisson problem in G with the boundary condition described above as well as the associated eigenvalue problem and prove the existence of smooth solutions. As a by-product we obtain the completeness of the well-known poloidal ‘free decay modes’ in ℝ3 if G is a ball. Smooth solutions of the evolution problem are then obtained by Galerkin approximation based on these eigenfunctions.

Building on a recent work, we consider a two-dimensional viscous fluid in the exterior of a thin obstacle shrinking to a curve, proving convergence to a solution of the Navier–Stokes equations in the exterior of a curve. The uniqueness of the limit solution is also shown.>

Assuming that (Ω, Σ, μ) is a complete probability space and that X is a Banach space, we evaluate both the semivariation and the variation norm of a wide class of Pettis integrable functions f : Ω → X.

We study the issues of the reconstruction and stability of the inverse nodal problem for the one-dimensional p-Laplacian eigenvalue problem. A key step is the application of a modified Prüfer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths. Two associated Ambarzumyan problems are also solved.

We describe decompositions of the group of units of a ring and of its subgroups, induced by idempotents with certain properties. The results apply to several classes of rings, most notably to semi-perfect rings.

A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

We study the multiplicity of positive solutions for the following semilinear elliptic equation:

where 1 < q < 2 < p < 2* (2* = 2N/(N − 2) if N ≥ 3, 2* = ∞ if N = 2), the parameters λ, μ ≥ 0, is an infinite strip in ℝN and Θ is a bounded domain in ℝN−1 We assume that fλ(x) = λf+(x) + f−(x) and gμ(x) = a(x) + μb(x), where the functions f±, a and b satisfy suitable conditions.

Let M be a compact connected oriented surface with exactly one boundary component. A pseudo-immersion of M into the plane is a smooth map such that it has only fold singularities and is an orientation-preserving immersion near the boundary of M. For a pseudo-immersion we study the relation between the winding number of the boundary curve and the number of the singular set components.