The Boltzmann and Landau equations are the classical models in statistical physics and plasma physics. We obtain the satisfactory existence of global classical solutions to the Boltzmann and Landau equations with soft potentials in the whole space. The convergence rate to Maxwellians for such solutions is also obtained.

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.

We analyse the positive solutions to classes of p-Laplacian systems with Dirichlet boundary conditions, in particular, when the reaction terms tend to −∞ at the origin and satisfy a combined sublinear condition at ∞. We use the method of sub- and supersolutions to establish our results.

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.

Consider a scattering problem for the Dirac equation with a non-local term including the Hartree type, say the cubic convolution term. We show the existence of scattering operators for small initial data in the subcritical and critical Sobolev spaces.

We study the monotonicity for certain combinations of generalized elliptic integrals, thus generalizing analogous well-known results for classical complete elliptic integrals, and prove a conjecture put forward by Heikkala, Vamanamurthy and Vuorinen.

We investigate real convex-transitive Banach spaces X, which admit a one-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. For example, if uSX is a big point such that there is a bicontractive linear projection P : X → [u] and X* is weak*-locally uniformly rotund, then X is a Hilbert space. The results obtained here are motivated by the well-known Banach—Mazur rotation problem, as well as a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

We establish some regularity criteria in terms of the velocity field with weight for the Navier—Stokes equations in ℝ3. It is proved that if the weak solution satisfies

or

then the weak solution actually is regular up to time T.

The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

We study the following non-autonomous singularly perturbed Neumann problem:

where the index p is subcritical and a(x) is a positive smooth function in . We show that, given ε small enough, there exists a K(ε) such that, for any positive integer K ≤ K(ε), there always exists a solution with K interior peaks concentrating at a strict sth-order local minimum point of a.