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In this paper, we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source-type solutions to the fast diffusion or porous medium equations, that are also known as Barenblatt solutions. As functions of time, these functionals have convexity properties for generic solutions, so that their asymptotic slopes are extremal for Barenblatt profiles. The method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp Gagliardo–Nirenberg–Sobolev inequalities in scale invariant form. The method also gives refined estimates of the growth of the second moment and, as a consequence, establishes the monotonicity of the delay corresponding to the best matching Barenblatt solution compared to the Barenblatt solution with same initial second moment. Here the notion of best matching is defined in terms of a relative entropy. [ABSTRACT FROM AUTHOR]

Abstract: Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be built, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy–Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities. [Copyright &y& Elsevier]

Abstract: To study the problem of the assigned Gauss curvature with conical singularities on Riemannian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce the problem to a Liouville equation on the Euclidean plane. We prove new multiplicity results for bounded radial solutions, which improve on earlier results of C.-S. Lin and his collaborators. Based on numerical computations, we also present various conjectures on the number of unbounded solutions. Using symmetries, some multiplicity results for non-radial solutions are also stated. [Copyright &y& Elsevier]

Abstract: This paper is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. Under regularity assumptions on the obstacle and on the coefficients, we prove that the time derivative of the solution is continuous for almost every time. When the solution is nondecreasing in time this result holds for every time. We also give an energy criterion which characterizes the continuity of the time derivative of the solution at a point of the free boundary. Such a problem arises in the pricing of American options in generalized Black–Scholes models of finance. Our results apply in financial mathematics. [Copyright &y& Elsevier]

We consider the problem of finding positive solutions of Δu+λu+uq=0 in a bounded, smooth domain Ω in R3, under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ1. We analyze the role of Green''s function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if λ*<λ<λ1, where λ* is the Brezis–Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q-5 is sufficiently small. [Copyright &y& Elsevier]

We prove local monotonicity and symmetry properties for nonnegative solutions of scalar field equations with nonlinearities which are not Lipschitz. Our main tools are a local moving plane method and a unique continuation argument. [Copyright &y& Elsevier]

We consider radial solutions of Δu+u−|u|−2θu=0 in Rd with d>1, θ∈(0,1/2) and prove by a shooting method the existence of compactly supported solutions with any given number of nodes. [Copyright &y& Elsevier]

In this paper, we find optimal constants of a special class of Gagliardo–Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations. [Copyright &y& Elsevier]

Abstract: In “On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients” our statement on Harnack''s inequality is incorrect. This statement was used to establish a priori estimates. In this erratum we give a direct proof of these a priori estimates. [Copyright &y& Elsevier]

Abstract: This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in self-similar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy/entropy-production methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations; see Bonforte et al. (2009) . The results extend to the case of a Fokker–Planck equation with a general confining potential. [ABSTRACT FROM AUTHOR]