Volume 6, Issue 3

If h is a nondecreasing real valued function and 0 ≤ q ≤ 2, we analyse the boundary behaviour of the gradient of any solution u of −Δu + h(u) + |∇u| q = f in a smooth N-dimensional domain Ω with the condition that u tends to infinity when x tends to ∂Ω. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to ∂Ω. Motivated by the blow-up argument in our proof, we also give a symmetry result for some related problems in the half space.

Grigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.

In this paper new estimates on the C 0 -norm of homoclinic orbit are shown for first order convex Hamiltonian systems possessing super-quadratic potentials. Applying these estimates, some new results on the existence of infinitely many geometrically distinct homoclinic orbits are proved, which generalize the main results in [2] and [10].

We prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.

In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝ n , n ≥ 3 and given x 1 , x 2 ∈ Ω, we give a sufficient condition on x 1 , x 2 , for the problem to have a positive weak solution in Ω with 0 boundary data, which is singular at each x i .

We consider positive solutions for a class of non-autonomous equations on a unit ball in R n Δu + f(|x|, u) = 0 for |x| < 1, u= 0 when |x| = 1. We assume that f r (r, u) ≤ 0, so that in view of [8] all positive solutions are radially symmetric. If f(|x|, u) = a(|x|)g(u), we obtain several exact multiplicity results for a class of convex-concave g(u), with g(0) = 0. In another direction, we obtain uniqueness and non-degeneracy of positive solutions for a class of equations, modeled on f(|x|, u) = −a(|x|)u + b(|x|)u p , with .

We show how the theory of finite Morse index solutions for small diffusion problems can be adapted to the shadow system for the Geirer-Meinhardt equation. In particular, we prove that the only possible stable solution of this shadow system is the one peak solution discussed by Wei and Ni, Takagi and Yanagida.