Diffusion phenomena for the wave equation with structural damping in the Lp−Lq framework

Abstract In this paper, we study diffusion phenomena for the wave equation with structural damping u t t − Δ u + 2 a ( − Δ ) σ u t = 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , with a > 0 and σ ∈ ( 0 , 1 / 2 ) . We show that the solution u behaves like the solution v + to v t + + 1 2 a ( − Δ ) 1 − σ v + = 0 , v + ( 0 , x ) = v 0 + ( x ) , for suitable choice of initial data v 0 + . More precisely, we derive L p − L q decay estimates for the difference u − v + and its time and space derivatives, where 1 ⩽ p ⩽ q ⩽ ∞ , possibly not on the conjugate line, satisfying some additional condition related to σ. In particular, we show that, under suitable assumptions on p , q , σ , a double diffusion phenomenon appears, that is, the difference u − v + behaves like the solution to v t − + 2 a ( − Δ ) σ v − = 0 , v − ( 0 , x ) = v 0 − ( x ) , for a suitable choice of initial data v 0 −

The influence of oscillations on energy estimates for damped wave models with time-dependent propagation speed and dissipation

The aim of this paper is to derive higher order energy estimates for solutions to the Cauchy problem for damped wave models with time-dependent propagation speed and dissipation. The model of interest is \begin{equation*} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u +\rho(t)\omega(t)u_t=0, \quad u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x). \end{equation*} The coefficients $\lambda=\lambda(t)$ and $\rho=\rho(t)$ are shape functions and $\omega=\omega(t)$ is an oscillating function. If $\omega(t)\equiv1$ and $\rho(t)u_t$ is an "effective" dissipation term, then $L^2-L^2$ energy estimates are proved in [2]. In contrast, the main goal of the present paper is to generalize the previous results to coefficients including an oscillating function in the time-dependent coefficients. We will explain how the interplay between the shape functions and oscillating behavior of the coefficient will influence energy estimates.Comment: 37 pages, 2 figure

Mechanisms and role of microRNA deregulation in cancer onset and progression

MicroRNAs are key regulators of various fundamental biological processes and, although representing only a small portion of the genome, they regulate a much larger population of target genes. Mature microRNAs (miRNAs) are single-stranded RNA molecules of 20–23 nucleotide (nt) length that control gene expression in many cellular processes. These molecules typically reduce the stability of mRNAs, including those of genes that mediate processes in tumorigenesis, such as inflammation, cell cycle regulation, stress response, differentiation, apoptosis and invasion. MicroRNA targeting is mostly achieved through specific base-pairing interactions between the 5′ end (‘seed’ region) of the miRNA and sites within coding and untranslated regions (UTRs) of mRNAs; target sites in the 3′ UTR diminish mRNA stability. Since miRNAs frequently target hundreds of mRNAs, miRNA regulatory pathways are complex. Calin and Croce were the first to demonstrate a connection between microRNAs and increased risk of developing cancer, and meanwhile the role of microRNAs in carcinogenesis has definitively been evidenced. It needs to be considered that the complex mechanism of gene regulation by microRNAs is profoundly influenced by variation in gene sequence (polymorphisms) of the target sites. Thus, individual variability could cause patients to present differential risks regarding several diseases. Aiming to provide a critical overview of miRNA dysregulation in cancer, this article reviews the growing number of studies that have shown the importance of these small molecules and how these microRNAs can affect or be affected by genetic and epigenetic mechanisms

The complete genome sequence of Chromobacterium violaceum reveals remarkable and exploitable bacterial adaptability

Chromobacterium violaceum is one of millions of species of free-living microorganisms that populate the soil and water in the extant areas of tropical biodiversity around the world. Its complete genome sequence reveals (i) extensive alternative pathways for energy generation, (ii) ≈500 ORFs for transport-related proteins, (iii) complex and extensive systems for stress adaptation and motility, and (iv) wide-spread utilization of quorum sensing for control of inducible systems, all of which underpin the versatility and adaptability of the organism. The genome also contains extensive but incomplete arrays of ORFs coding for proteins associated with mammalian pathogenicity, possibly involved in the occasional but often fatal cases of human C. violaceum infection. There is, in addition, a series of previously unknown but important enzymes and secondary metabolites including paraquat-inducible proteins, drug and heavy-metal-resistance proteins, multiple chitinases, and proteins for the detoxification of xenobiotics that may have biotechnological applications

Methods for Partial Differential Equations

This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area. The book is organized in five parts: In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation. Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes Laplace equation, heat equation and wave equation as well as the different features of each theory. It also discusses the notion of energy of solutions, a highly effective tool for the treatment of non-stationary or evolution models and shows how to define energies for different models. Part 3 demonstrates how phase space analysis and interpolation techniques are used to prove decay estimates for solutions on and away from the conjugate line. It also examines how terms of lower order (mass or dissipation) or additional regularity of the data may influence expected results. Part 4 addresses semilinear models with power type non-linearity of source and absorbing type in order to determine critical exponents: two well-known critical exponents, the Fujita exponent and the Strauss exponent come into play. Depending on concrete models these critical exponents divide the range of admissible powers in classes which make it possible to prove quite different qualitative properties of solutions, for example, the stability of the zero solution or blow-up behavior of local (in time) solutions. The last part features selected research projects and general background material

Systematics and biodiversity of sharks, rays, and chimaeras (Chondrichthyes) of Taiwan

De Carvalho, Marcelo R., Ebert, David A., Ho, Hsuan-Ching, White, William T., Ebert, David A., Ho, Hsuan-Ching, White, William T., De Carvalho, Marcelo R. (2013): Systematics and biodiversity of sharks, rays, and chimaeras (Chondrichthyes) of Taiwan. Zootaxa 3752 (1): 3-4, DOI: http://dx.doi.org/10.11646/zootaxa.3752.1.

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), given by L = partial derivative/partial derivative t + (a(x) + ib(x))partial derivative/partial derivative x, b not equivalent to 0, near the characteristic set Sigma = {0} x S(1). We show that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. (C) 2008 Elsevier Inc. All rights reserved.CNPqConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESPFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Introduction to the systematics and biodiversity of sharks, rays, and chimaeras (Chondrichthyes) of Taiwan

Ebert, David A., Ho, Hsuan-Ching, White, William T., De Carvalho, Marcelo R. (2013): Introduction to the systematics and biodiversity of sharks, rays, and chimaeras (Chondrichthyes) of Taiwan. Zootaxa 3752 (1): 5-19, DOI: http://dx.doi.org/10.11646/zootaxa.3752.1.