Global Sign-changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range

Abstract A detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita range with bounded integrable initial data u(x, 0) = u0(x). This study is continued and extended here for the 2mth-order heat equation, for m ≥ 2, with non-monotone nonlinearity with the same initial data u0. The fourth order biharmonic case m = 2 is studied in greater detail. The blow-up Fujita-type result for (0.2) now reads as follows: blow-up occurs for any initial data u0 with positive first Fourier coefficient: ∫ u0(x) dx > 0, i.e., as for (0.1), any such arbitrarily small initial function u0(x) leads to blow-up. The construction of two countable families of global sign changing solutions is performed on the basis of bifurcation/branching analysis and a further analytic-numerical study. In particular, a countable sequence of bifurcation points of similarity solutions is obtained

Classification of global and blow-up sign-changing solutions of a semilinear heat equation in the subcritical fujita range:Second-order diffusion

AbstractIt is well known from the seminal paper by Fujita [22] for 1 p0there 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 p0, 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 ≤ p0. 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 ℝNor in the unit ball B1⊂ℝNwere 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

La computación electrónica aplicada a la selección de los aceros

El presente trabajo tiene por objeto demostrar cómo, mediante la utilización de "una computadora electrónica, y aplicando ciertos principios técnico-científicos, se puede llegar a establecer una equivalencia entre los aceros al carbono y aleados destinados a la construcción mecánica. Para ello se ha adoptado el criterio de tres parámetros fundamentales: (Di) diámetro crítico ideal, (Ms) temperatura de transformación austenítica-mar-tensítica, y (Hr) dureza a distintas temperaturas y tiempos de revenido. Luego de un estudio crítico del sistema de equivalencia, se presentan las tablas de valores obtenidas por la computadora electrónica y se analiza la trascendencia de estas determinaciones.The purpose of the present report is to demonstrate how employing and electronic computer and certain technical-scientifical principles, it is possible to establish an equivalence between carbon and alloy steels used in mechanical constructions. To accomplish this objective it was adopted the criterium of three fundamental parameters: (Dl) ideal critical diameter, (Ms) transformation temperature austenite martensite and (Hr) hardness corresponding to different temperatures and tempering times. After a critical study of the equivalence system, tables and values obtained by electronic computer are shown and the importance of these determinations are analyzed

A new critical curve for the Lane-Emden system

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.Comment: 13 pages, 1 figur

Enanismo y podredumbre basal de Eustoma grandiflorum y su relación con la densidad de Fusarium solani en el suelo

En cultivos comerciales de lisianthus de La Plata y alrededores (Argentina), se observó que Fusarium solani afecta al 100% de los establecimientos productivos provocando enanismo y podredumbre basal. Se realizó un ensayo para analizar la densidad de F. solani del suelo bajo el efecto de distintos tratamientos y relacionarla con la evolución de ambas patologías. Los análisis del suelo se hicieron en 3 épocas: antes del transplante (octubre), al inicio (enero) y al final (marzo) de la floración. En enero y marzo se evaluó la incidencia de cada enfermedad y se cuantificaron las plantas cosechadas. La población de F. solani no varió en las 3 épocas y se redujo significativamente con los fumigantes y el vapor. Esto determinó una incidencia del enanismo (enero) de 0 a 9,5% con los primeros y de 31,4% con el vapor. Para el testigo y los fungicidas la incidencia varió entre el 87,9 y el 100%, disminuyendo las plantas cosechadas. La podredumbre basal comenzó con el ingreso del patógeno a través de las heridas producidas por la cosecha. Ésta fue significativamente mayor con los fumigantes (90,7 a 99%) y también con el vapor (76,8%), produciéndose la mayor incidencia en marzo (15,7 a 22,2 y 10,1% respectivamente), reduciéndose el período productivo. Se destaca la importancia de disminuir la densidad de F. solani en el momento del transplante. Entre los métodos de control el Metam sodio y Dazomet podrían ser eficaces sustitutos del Bromuro de metilo, seguidos por el vapor.Fusarium solani was recorded in 100% of the commercial crops of lisianthus (Eustoma grandiflorum) located around La Plata (Argentina). The pathogen causes stunt and basal rot. A trial was carried out to analyse the F. solani soil density and its relation to the development of both diseases. Soil samples were analysed at three times: before transplant (October), at the beginning of flowering (January) and at the ending of flowering (March) of flowering. The incidence of each disease was evaluated in January and March, and the plants harvested were quantified. The soil density of F. solani was not altered during the three times for each treatment, and fumigants and steam decrease it significantly. In January the stunt incidence reached 0 to 9.5% with the fumigants and 31.4% with the steam treatment. In the control and fungicide plots, the incidence reached 87.9 to 100%, and a consequent reduction of flower production was observed. Stem rot began after the flowers were harvested as a result of pathogen entry through the wounds. The major values of harvested plants were obtained with fumigants (90.7 to 99%) and steam (76.8%), so the greatest stem rot incidence was 15.7 to 22.2 and 10.1%, respectively, in March, and the productive period was shortened. The need of reducing F. solani soil density before lisianthus transplant is emphasized. Among soil control treatments, Metham sodium and Dazomet could be effective substitutes for Methil bromide followed by steam treatment

5-FU targets rpL3 to induce mitochondrial apoptosis via cystathionine-b-synthase in colon cancer cells lacking p53

Recent findings revealed in cancer cells novel stress response pathways, which in response to many chemotherapeutic drugs causing nucleolar stress, will function independently from tumor protein p53 (p53) and still lead to cell cycle arrest and/or apoptosis. Since it is known that most cancers lack functional p53, it is of great interest to explore these emerging molecular mechanisms. Here, we demonstrate that nucleolar stress induced by 5-fluorouracil (5-FU) in colon cancer cells devoid of p53 leads to the activation of ribosomal protein L3 (rpL3) as proapoptotic factor. rpL3, as ribosome-free form, is a negative regulator of cystathionine-β-synthase (CBS) expression at transcriptional level through a molecular mechanism involving Sp1. The rpL3-CBS association affects CBS stability and, in addition, can trigger CBS translocation into mitochondria. Consequently apoptosis will be induced through the mitochondrial apoptotic cell death pathway characterized by an increased ratio of Bax to Bcl-2, cytochrome c release and subsequent caspase activation. It is noteworthy that silencing of CBS is associated to a strong increase of 5-FU-mediated inhibition of cell migration and proliferation. These data reveal a novel mechanism to accomplish p53-independent apoptosis and suggest a potential therapeutic approach aimed at upregulating rpL3 for treating cancers lacking p53

Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio