Volume 4, Issue 2

Using a topological invariant, due to Bahri [4], and dynamical methods, we prove some existence results for the problem prescribing the scalar curvature under zero boundary conditions on the standard half sphere.

We consider the superlinear boundary value problem uʺ + a μ (t)u γ+1 = 0, u(0) = 0, u(1) = 0, where γ > 0 and a μ (t) is a sign indefinite weight of the form a + (t)−μa − (t). We prove, for μ positive and large, the existence of 2 k − 1 positive solutions where k is the number of positive humps of a μ (t) which are separated by k − 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.

Shadowing is a means of characterizing global errors in the numerical solution of initial value differential equations by allowing for small perturbations in the initial conditions. The method presented in this paper provides a technique for efficient estimation of the shadowing global error for systems that have a large number of exponentially decaying modes. The method is formulated for one-step methods and is applied to the spatial discretization of some dissipative PDEs.

We introduce the notion of a generalized attractor-repeller pair for ẋ = A(t)x which in fact is an attractor-repeller pair for the induced (nonlinear) flow on the projective space, and use it to characterize diagonalizability. We show that integral separation implies the existence of generalized attractor-repeller pairs, and use topological dynamics, more precisely the hull of A(t), to prove a converse theorem. The relation to exponential separation is also discussed.

We obtain blow-up results for a wide class of nonlinear parabolic problems with nonlinearity of the Chipot-Weissler type in the gradient term. Some of these answer an open question concerning the nonexistence of positive solutions to the problem where λ > 0 is small, u 0 ∈ L loc (ℝ N ) with u 0 ≥ 0, when . For this purpose we use the method of “test function”.