Volume 9, Issue 1

We study the blow-up set of a solution u(x, t) to the porous medium equation, u t = Δ(u m ), in Ω × (0, T) for two different boundary conditions, u(x, t) = f(x)/(T - t) or (x, t) = g(x)/(T - t) on ӘΩ × (0, T). Here m > 1 and Ω is a bounded smooth domain in ℝ N . We establish point-wise and integral conditions on f and g respectively in order to obtain regional or global blow-up.

In this paper, we study the set of complete noncompact hypersurfaces in ℝ n+1 which have constant mean curvature equal to 1 and a finite number of ends. By means of the application of the implicit function theorem, we will prove that this set has a smooth structure near any nondegenerate element. We start by a review of the functional properties of elliptic operators acting on functions defined on a manifold with cylindrical ends. This will explain the central role played by the indicial roots about an n-Delaunay hypersurface. Then, we will explain how to apply the implicit function theorem in our setting. We partially generalize, in any dimension, the results of R. Kusner, R. Mazzeo and D. Pollack.

We prove existence and comparison results for quasilinear parabolic inclusions with Clarke’s generalized gradient via appropriately defined sub-supersolution only assuming a local L q -boundedness condition on Clarke’s gradient. The developed comparison principle allows us to show the equivalence of the considered parabolic inclusion with an associated evolutionary hemivariational inequality.

This paper is concerned with the following nonlinear Schrödinger equation: where f(x, t)t -1 < μ* := inf σ(-Δ + V) < f(x, t)t -1 < V(∞) = V(x). The existence of a positive solution and a least energy solution for problem (P) are discussed. Here, we assume neither that f(x, t) ≡ f(t), nor that f(x, t)t -1 is non-decreasing on t ≥ 0. Furthermore, we do not require that f(x, t) have a limit as |x| → +∞.

We consider the second order nonlinear ODE uʺ – f(t, u) = 0 and assume that f(·, υ 0 ) ≡ 0; for some υ 0 ∈ ℝ. We prove the existence of closed connected sets Γ ⊆ ℝ 2 of initial points such that for each (α, β) ∈ Γ there exists a solution u(·) of the given differential equation, with (u(t 0 ), uʹ(t 0 )) = (α, β) and (u(t), uʹ(t)) → (υ 0 , 0) as t → –∞ (or as t → +∞). These results are then applied to the search of heteroclinic and homoclinic solutions.

We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ε 2 □ψ + Wʹ(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t, x) = u(x)e- iwt/ε ; for these solutions the Klein Gordon equation becomes We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ε suffciently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.

We study the existence and uniqueness of the positive solutions of the problem (P): Ә t u - Δu + u q = 0 (q > 1) in Ω × (0, ∞), u = ∞ on ӘΩ × (0, ∞) and u(., 0) ∈ L 1 (Ω), when Ω is a bounded domain in ℝ N . We construct a maximal solution, prove that this maximal solution is a large solution whenever q < N/(N - 2) and it is unique if ӘΩ = ӘΩ̅ c . If ӘΩ has the local graph property, we prove that there exists at most one solution to problem (P).

We consider the quasilinear elliptic problem Δ p u = a(x)e u in a smooth bounded domain Ω of ℝ N , with u = +∞ on ӘΩ. Here Δ p u = div(|▽u| p-2 ▽u) is the p-Laplacian operator with p > 1, and a(x) is a continuous positive weight function which can be singular on ӘΩ. Our results include existence, uniqueness and exact boundary behavior of solutions.

We study the following semilinear biharmonic equation: , where 0 ≤ f ≤ 1 and B R ⊂ ℝ N , N ≥ 1, is the ball centered in the origin of radius R. We prove, under Dirichlet boundary conditions u = Әu/Әƞ = 0 on ӘB R , the existence of λ* = λ*(R, f) > 0 such that for λ ∈ (0, λ*) there exists a minimal (classical) solution u̲ λ < which satisfies 0 < u̲ λ < 1. In the extremal case λ = λ*, we prove the existence of a weak solution which has finite energy and which is the unique solution even in a very weak sense. For λ > λ* there are no solutions of any kind. Estimates on λ*, stability properties of solutions and nonexistence results in the whole space are also established.

We prove an existence result for T-periodic retarded functional differential equations of the type xʹ(t) = f(t, x t ), where f is a T-periodic functional tangent vector field on a smooth manifold. As an application we show that any constrained system acted on by a periodic force, possibly with delay, admits a forced oscillation provided that the constraint is a topologically nontrivial compact manifold and the frictional coefficient is nonzero. We conjecture that the same assertion holds true even in the frictionless case.

In this paper we maximize a class of functionals under certain constraints. We find necessary and sufficient conditions for these maximizers to exist and be unique. Moreover, we characterize them and discuss the optimality of our results by constructing counterexamples when one of the hypotheses does not hold.