Volume 14, Issue 1

It is well known from the seminal paper by Fujita [22] for 1 < p < p 0 , and Hayakawa [36] for the critical case p = p 0 , that all the solutions u ≥ 0 of the semilinear heat equation u t = Δu + |u| p−1 u in ℝ N × ℝ + , in the range , (0.1) with arbitrary initial data u 0 (x) ≥ 0, ≢ 0, blow-up in finite time, while for p > p 0 there exists a class of sufficiently “small” global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p 0 , as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p 0 . Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ℝ N or in the unit ball B 1 ⊂ℝ N were mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected: , l = 0, 1, 2, ... .. See [47, 48] for earlier interesting contributions on sign changing solutions.

We give some explicit formulas for the minimizers of the Rudin-Osher-Fatemi functionals and on BV(Ω), where Ω ⊂ ℝ 2 is a bounded domain containing the ball B R (0) and f = χ R (x) is the characteristic function of B R (0).

Let . We prove that any positive solution of (E) ∂ t u − Δu + u q = 0 in ℝ N × (0, ∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity in ℝ N (qʹ = q/(q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in ℝ N with the above properties we construct a positive solution u of (E) with initial trace ν and we prove that this solution is the unique σ-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is σ-moderate.

In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

This paper is a continuation of the work begun in [6] on superposition operators, (N g u) (x) = g(u(x)), between two arbitrary Sobolev spaces. Sufficient conditions which ensure the well-definedness, the continuity and the validity of the higher-order chain rule for such operators are given in the supercritical case (see Remark 1.1). As a consequence of these properties, it is proved that N g (W m,p (Ω) ∩ W 0 k,p (Ω)) ⊂ W 0 l,q (Ω).

We investigate the behavior of weak solutions to the nonlocal Robin problem for quasilinear elliptic divergence second order equations in a neighborhood of the boundary corner point. We find an exponent of the solution decreasing rate near the boundary corner point.

This paper deals with the existence of solutions to the elliptic equation Δu = k(x) f (u) in D satisfying the boundary blow-up condition u| ∂D = ∞, where D ⊂ ℝ N is an open bounded domain with smooth boundary ∂D. Under suitable assumptions on k and f , we show the existence of at least one solution u. Under additional hypotheses on k, we give asymptotic estimate for u near the boundary.

This paper introduces new nonlinear heat approximation and L ∞ preserving homotopy techniques to investigate regularity properties of bounded weak solutions of strongly coupled p-Laplacian parabolic systems which consist of more than one equation defined on a domain of any dimension. The main results imply everywhere Hölder continuity of bounded weak solutions and the global existence of strong solutions to nonlinear p-Laplacian systems.

We find the behavior of the solution of the optimal transport problem for the Euclidean distance (and its approximation by p−Laplacian problems) when the involved measures are supported in a domain that is contracted in one direction.