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Autor Herrero, Miguel A. |
Documentos disponibles escritos por este autor (100)
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This paper deals with the Cauchy problem u(t)-u(xx)+u(p)=0; -infinity![]()
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We consider here the homogeneous Dirichlet problem for the equation u(t)= u?u - ?|?u|(2) with ? ? R, u ? 0, in a noncylindrical domain in space-time given by |x| ? R(t) = (T - t)(p), with p > 0. By means of matched asymptotic expansion techniqu[...]![]()
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The authors consider the problem ut?div(|?u|p?2?u)=0 in (0,?)×RN, u(x,0)=u0(x). They show that if N?2 and 1![]()
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This paper examines a system first introduced by Keller and Segel in 1970 to model the tendency of slime molds to move towards higher concentrations of a chemical which they themselves secrete. The paper particularly addresses the question of bl[...]![]()
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In this paper the authors study the asymptotic behaviour of solutions u?(x,t) of the Cauchy problems as ? goes to zero: ut???u+up=0, x?RN, t> 0; u(x,0)=u0(x), x?RN, 0![]()
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We consider the Cauchy problem u t -u xx +u p =0,x??,t> 0,u(x,0)=u 0 (x),x??, where 0![]()
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Cappuccio, Antonio ; Herrero, Miguel A. ; Nuñez, Luis | American Association of Physicists in Medicine | 2009-01In tumor radiosurgery, a high dose of radiation is delivered in a single session. The question then naturally arises of selecting an irradiation strategy of high biological efficiency. In this study, the authors propose a mathematical framework [...]![]()
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Consider the Cauchy problem u(t) - u(xx) - F(u) = 0; x is-an-element-of R, t> 0 u(x, 0) = u0(x); x is-an-element-of R where u0 (x) is continuous, nonnegative and bounded, and F(u) = u(p) with p > 1, or F(u) = e(u). Assume that u blows up at x =[...]![]()
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We consider the following nonlinear system of parabolic equations: (1) ut =?u???(u?v), ?vt =?v+u?av for x?B R, t> 0. Here ?,? and a are positive constants and BR is a ball of radius R> 0 in R2. At the boundary of BR, we impose homogeneous Neuma[...]![]()
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We consider the equation (E) u(t) = ?u + u(p) where x ? R(N) (N ? 1), t > 0, p > 1. We show that if N ? 11 and p > N - 2 (N - 1)1/2/(N - 4) - 2(N - 1)1/2 then there exist radial and positive solutions of (E) which blow up at x = 0, t = T![]()
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Let u be a solution of the Cauchy problem ut=uxx+up, x?R, t> 0, u(x,0)=u0(x), x?R, where p> 1 and u0 is continuous, nonnegative, and bounded. Suppose that u blows up at t=T![]()
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The object of this paper is the study of blowing-up phenomena for the initial-boundary value problem (Pa): ut=uxx+?eu for (x,t)?(0,1)×(0,+?), u(0,t)=asin?t and u(1,t)=0 for t?[0,+?), u(x,0)=u0(x) for x?(0,1), where u0(x) is a continuous and boun[...]![]()
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In this work we succintly review the main features of bone formation in vertebrates. Out of the many aspects of this exceedingly complex process, some particular stages are selected for which mathematical modelling appears as both feasible and d[...]![]()
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We consider the semilinear parabolic system (S) { ut-?u=vp ; vt-?v=uq, where x ? R(N) (N ? 1), t > 0, and p, q are positive real numbers. At t=0, nonnegative, continuous, and bounded initial values (u0(x), v0(x)) are prescribed. The correspon[...]![]()
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This work is concerned with the system (S) {u(t)=Delta u-chi del(u del upsilon) for x is an element of Omega, t> 0 Gamma upsilon(t)=Delta upsilon=Delta upsilon+(u-1) for x is an element of Omega, t> 0 where Gamma; chi are positive constants and [...]![]()
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López Alfonso, Juan Carlos ; Butazzo, Giuseppe ; García Archilla, B. ; Herrero, Miguel A. ; Nuñez, L. | American Institute of Mathematical Sciences | 2012-09Radiotherapy is an important clinical tool to fight malignancies. To do so, a key point consists in selecting a suitable radiation dose that could achieve tumour control without inducing significant damage to surrounding healthy tissues. In spit[...]![]()
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We consider the Cauchy problem (1) ut=uxx+up, x?R, t> 0, p> 1, (2) u(x,0)=u0(x),x?R, where u0(x) is continuous, nonnegative and bounded. Assume that the solution u(x,t) of (1), (2) blows up at x=0, t=T. We describe here the generic asymptotic be[...]![]()
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In the following the reader will find a short description of some issues related to the modeling, analysis and simulation of large populations of living systems, a research field which is currently deserving a considerable interest, and that has[...]![]()
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Background: The choice of any radiotherapy treatment plan is usually made after the evaluation of a few preliminary isodose distributions obtained from different beam configurations. Despite considerable advances in planning techniques, such fin[...]![]()
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Herrero, Miguel A. ; Oleaga Apadula, Gerardo Enrique ; Echeverri, L.F. ; López, J. M. | Springer | 2014-12This work is concerned with the sequence of events taking place during the first stages of bone fracture healing, from bone breakup until the formation of early fibrous callus (EFC). The latter provides a scaffold over which subsequent remodelin[...]![]()
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We obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one-dimensional motion of a non-newtonian fluid. Precise estimates for the support of these soluti[...]![]()
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This note is an account of results obtained by the author [Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 75 (1981), no. 5, 1165–1183; MR0649591 (83m:35076)], and the author and J. L. Vázquez ["On a class of nonlinear parabolic equations'', to[...]![]()
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The authors consider the system, defined for t> 0, -?![]()
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The authors consider blow-up for the equation (1) ut=?u+up (x?RN, t> 0), where p> 1 and N> 1. For N> 11and (2) p> (N?2(N?1)1/2)/(N?4?2(N?1)1/2)=p1(N) there exist some radial positive solutions that blow up at x=0, t=T![]()
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A simple model of chemical kinetics with two concentrations u and v can be formulated as a system of two parabolic variational inequalities with reaction rates v(p) and u(q) for te diffusion processes of u and v, respectively. It is shown that i[...]![]()
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Consider the initial-boundary value problem for ut=?u-?u(q) with ?> 0, 0![]()
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We consider the semilinear heat equation with critical power nonlinearity. Using formal. arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign-changing solutions, which blow-up at x[...]![]()
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Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealt[...]![]()
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We consider the following system: [GRAPHICS] which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial [...]![]()
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Consider the Cauchy problem ut=uxx+up, x?R, t> 0, u(x,0)=u0(x), x?R, where p> 1 and u0(x) is continuous, nonnegative and bounded. Assume that u(x,t) blows up at x=0, t=T and set u(x,t)=(T?t)?1/(p?1)?(y,?), y=x/T?t?????, ?=?ln(T?t). Here we show [...]![]()
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Bru Espino, Antonio ; Casero Díaz-Cano, David ; De Franciscis, Sebastiano ; Herrero, Miguel A. | Pergamon-Elsevier Science Ltd | 2008-03Tumour growth can be described in terms of mathematical models from different points of view due to its multiscale nature. Dynamic scaling is a heuristic discipline that exploits the geometrical features of growing fronts using different concept[...]![]()
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Bellomo, Nicola ; Bellouquid, Abdelghani ; Herrero, Miguel A. | Pergamon-Elsevier Science Ltd | 2007-02This paper presents an asymptotic theory for a large class of Boltzmann-type equations suitable to model the evolution of multicellular systems in biology. The mathematical approach described herein shows how various types of diffusion phenomena[...]![]()
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We shall briefly review some early nucleation models, and then examine some aspects of the subsequent evolution of their solutions. Such situation is characterised by the onset of comparatively large clusters that can diffuse into the medium and[...]![]()
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This work is concerned with a model which has been proposed to describe the growth of solid tumors. More precisely, the model under consideration provides a procedure to extract information about the growth dynamics from the analysis of the geom[...]![]()
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This paper concerns the Cauchy problem ut?uxx=up, x?R, t> 0, u(x,0)=u0(x), x?R, where p> 1 and u0(x) is a continuous, nonnegative and bounded function. It has been previously proved that if x=x¯, t=T is a blow-up point, then there are three case[...]![]()
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We consider the Cauchy problem (1) u(t) = u(xx) + u(p); x ? R, t > 0, p > 1 (2) u(x, 0) = u0 (x), x ? R, where u0(x) is continuous, nonnegative and bounded. Assume that the solution u(x, t) of (1), (2) blows up at x = 0, t = T. We describe her[...]![]()
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The pair of parabolic equations u(t) = a ? u + f(u,v), (1) v(t) = b ? b - f(u, v), (2) with a > 0 and b > 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. [...]![]()
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We shall recall some reaction-difussion models which have been used to describe the growth of net-like structures, mainly in a biological context. In particular, a modified activator-inhibitor system proposed by Hans Meinhardt in 1976 will be co[...]![]()
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The authors consider the initial-boundary value problem for the porous medium equation ut =(um)xx in (0,?)×(0,T), where m> 1, 0![]()
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Arias, Clemente F. ; Herrero, Miguel A. ; Acosta Salmerón, Francisco Javier ; Fernández Arias, Clemente | Elsevier | 2014-05-21We formulate and analyze an algorithm of cell fate decision that describes the way in which division vs. apoptosis choices are made by individual T cells during an infection. Such model involves a minimal number of known biochemical mechanisms: [...]![]()
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Fasano, A. ; Herrero, Miguel A. ; López, J. M. ; Medina Reus, Elena | Italian Society for Applied and Industrial Mathematics (SIMAI) | 2010A mathematical model to describe the process of formation of bone tissue by replacement of cartilage tissue is presented and discussed. This model is based on an absorption-diffusion system which describes the interaction of two key signalling m[...]![]()
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Guria, G. T. ; Herrero, Miguel A. ; Zlobina, K. E. | American Institute of Mathematical Sciences | 2009-09In this work a mathematical model for blood coagulation induced by an activator source is presented. Blood coagulation is viewed as a process resulting in fibrin polymerization, which is considered as the first step towards thrombi formation. We[...]![]()
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We briefly review some classical models of aggregate formation with regard to their elementary monomeric components. Particular attention is paid to the role played by explicit solutions in the overall evolution of the theory, for which some rel[...]![]()
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This article considers a mathematical model for tumour growth based on an acid-mediated hypothesis, i.e. the assumption that tumour progression is facilitated by acidification of the region around the tumour-host interface. The resulting destruc[...]![]()
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In this work we present a comprehensive account of our current knowledge on vascular morphogenesis, both from a biological and a mathematical point of view. To this end, we first describe the basic steps in the known mechanisms of blood vessel m[...]![]()
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This work is concerned with some aspects of the social life of the amoebae Dictyostelium discoideum (Dd). In particular, we shall focus on the early stages of the starvation-induced aggregation of Dd cells. Under such circumstances, amoebae are [...]![]()
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Arias, Clemente F. ; Herrero, Miguel A. ; Stern, Claudio D. ; Bertocchini, Federica | Nature Publishing Group | 2017The first obvious sign of bilateral symmetry in mammalian and avian embryos is the appearance of the primitive streak in the future posterior region of a radially symmetric disc. The primitive streak marks the midline of the future embryo. The m[...]![]()
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From the introduction: "We continue here the investigation initiated earlier by us [Trans. Amer. Math. Soc. 314 (1989), no. 1, 187–224] concerning the solvability of the Cauchy problem and the existence of initial traces for nonnegative weak sol[...]![]()
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The initial value problem for the equation (?2 / ?t2 ? ?2 / ?x2) ?T / ?t = (? ?2 / ?t ? ?2 / ?x2) eT, ?> 1, is considered. It is proved that under some restrictions on the initial data there is a curve, denoted by t=??(x), which is positive, [...]![]()
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We consider an infinite system of reaction-diffusion equations that models aggregation of particles. Under suitable assumptions on the diffusion coefficients and aggregation rates, we show that this system can be reduced to a scalar equation, fo[...]![]()
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We consider here the radial Stefan problem with Gibbs-Thomson law, which is a classical model describing growth or melting of a spherical crystal in a surrounding liquid. We shall specialize to the cases of two and three space dimensions and dis[...]![]()
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The paper is a nice brief review of fundamental results about the initial value problem for one-dimensional nonlinear degenerate parabolic equations of "porous media'' type: ut=[?(ux)]x, x?R, t> 0, with ? continuous, nondecreasing, ?(0)=0, |?(s)[...]![]()
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Velázquez, J.J. L. ; Galaktionov, V. A. ; Posashkov, S. A. ; Herrero, Miguel A. | Pergamon-Elsevier Science | 1993The authors study asymptotic behaviour of positive solutions of equations of the type ut =??(u)±Q(u), where ?? and Q are given positive functions. By determining an auxiliary function F(u) appearing in an expression posed by A. Friedman and B. M[...]![]()
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The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p?2 Du)(p![]()
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This paper deals with the following initial value problem: (1) ?u/?t??u+u p=0 on RN, N?1, 0 0,u(x,t)> 0}, u(x,t) being the solution of (1). The following results are established. (i) If u0(x)?A(|x|)+B|x?a|2/(1?p) for some a?RN where A(|x|)=o(|x[...]![]()
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Bellomo , Nicola ; Herrero, Miguel A. ; Tosin, Andrea | American Institute of Mathematical Sciences | 2013-09This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled [...]![]()
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In this work a mathematical model for the interaction of two key signalling molecules in rat tibia ossification is presented and discussed. The molecules under consideration are Indian hedgehog (Ihh) and parathyroid hormone-related peptide (PTHr[...]![]()
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Herrero, Miguel A. ; Oleaga Apadula, Gerardo Enrique ; Velázquez, J.J. L. | American Mathematical Society | 2006This work deals with the linear wave equation considered in the whole plane R2 except for a rectilinear moving slit, represented by a curve ? (t) = {(x1, 0) : ??![]()
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We analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a g[...]![]()
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The author deals with the propagation of the support of the initial function in the following problem: ut?(um)xx+cun=0 on R×(0,+?), u(x,0)=u0(x) on R with n?m> 1, c> 0; u0 is bounded and has bounded support, and u0?0. The author proves the follo[...]![]()
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Bénilan, Philippe ; Boccardo, L. ; Herrero, Miguel A. | Libreria Editrice Universitaria Levrotto & Bella | 1989Let f?L1(RN), N?1, f?0, and consider the Cauchy problem ut=?um on ]0,?[×RN, u(0,?)=f on RN. The authors prove that as m??, the corresponding solutions um(t)?u_=f+?w in L1(RN), uniformly for t in a compact set in ]0,?[, where 0?w_?L1(Rn) is the s[...]![]()
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We consider here the problem of describing the melting of an ice ball surrounded by water. The corresponding mathematical model consists of the Stefan problem with radial symmetry. We obtain asymptotic expansions for the radius of the melting ba[...]![]()
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In this expository paper the equation (1) ut=div(|?u|m?1 ?u) is discussed in one space dimension (N=1). L?-gradient bounds have been established for arbitrary N by the reviewer and R. Rostamian [Math. Ann. 259 (1982), no. 1, 53–70; Proc. Roy. So[...]![]()
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Oleaga Apadula, Gerardo Enrique ; Herrero, Miguel A. ; Velázquez, J.J. L. | Royal Society of London | 2004-02-08Consider a crack propagating in a plane according to a loading that results in anti-plane shear deformation. We show here that if the crack path consists of two straight segments making an angle psi not equal 0 at their junction, examples can be[...]![]()
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Herrero, Miguel A. ; Oleaga Apadula, Gerardo Enrique ; Velázquez, J.J. L. | Royal Society of London | 2004Consider a crack propagating in a plane according to a loading that results in anti-plane shear deformation. We show here that if the crack path consists of two straight segments making an angle psi not equal 0 at their junction, examples can be[...]![]()
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We consider the problem (1) u(t) = u(xx) + e(u) when x is-an-element-of R, t > 0, (2) u (x, 0) = u0(x) when x is-an-element-of R, where u0(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combus[...]![]()
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Arias, C. ; Herrero, Miguel A. ; Acosta, F. J. ; Fernandez Arias, c. | Nature Publishing Group | 2017Unlike other cell types, T cells do not form spatially arranged tissues, but move independently throughout the body. Accordingly, the number of T cells in the organism does not depend on physical constraints imposed by the shape or size of speci[...]![]()
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Some propagation properties, including a discussion on the existence of compactly supported solutions and the asymptotics of the interfaces thereby determined, are considered for the Cauchy problem u t =(|u x | m-1 ·u x ) x in S=?×(0,?); u(x,0)=[...]![]()
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Arias, C. F. ; Bertocchini, F. ; Herrero, Miguel A. ; Oleaga Apadula, Gerardo Enrique | Cambridge University Press | 2020-12-23One of the most remarkable aspects of human homeostasis is bone remodeling. This term denotes the continuous renewal of bone that takes place at a microscopic scale and ensures that our skeleton preserves its full mechanical compliance during ou[...]![]()
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We analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equation -?u + u(p) = f in R(N), N ? 1, where 0![]()
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We study in this paper the asymptotic behaviour of solutions of a nonlinear Fokker-Plank equation. Such an equation describes the evolution of radiation for a gas of photons, which interacts with electrons by means of Compton scattering and Brem[...]![]()
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This book is addressed at curricula in applied mathematics, bio-physics, bio-mathematics, and theoretical biology and medicine. It is proposed as an advanced textbook for graduate interdisciplinary courses having as a common point the interest i[...]![]()
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In this paper we consider some systems of ordinary differential equations which are related to coagulation-fragmentation processes. In particular, we obtain explicit solutions {c(k)(t)} of such systems which involve certain coefficients obtained[...]![]()
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This work is concerned with the following system: [GRAPHICS] which is a model to describe several phenomena in which aggregation plays a crucial role as, for instance, motion of bacteria by chemotaxis and equilibrium of self-attracting clusters.[...]![]()
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Consider the system (S) {ut–?u=v(p),inQ={(t,x),t> 0, x??}, vt–?v=u(q), inQ, u(0,x)=u0(x)v(0,x)=v0(x)in?, u(t,x)=v(t,x)=0, whent?0, x???, where ? is a bounded open domain in ?N with smooth boundary, p and q are positive parameters, and functions[...]![]()
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It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T![]()
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The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, na[...]![]()
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This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show [...]![]()
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We consider an infinite system of reaction-diffusion equations which describes the dynamics of cluster growth, and show that there are solutions which exist for all times and exhibit a sol-gel transition in a finite time. The manner in which suc[...]![]()
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The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)> 0 in R, where f(u)=up, p> 1, or f(u)=eu. A complete description of the types of blow-up patterns and of the[...]![]()
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Under Boussinesq and hydraulic engineering approximations, convection in a closed loop under a given external heat flux is governed by an initial-boundary value problem for a first-order nonlinear PDE and an integral equation in two unknown func[...]![]()
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The paper surveys some recent results for the one-dimensional nonlinear heat equation with absorption subject to continuous nonnegative initial data with compact support. Without absorption, the equation becomes the porous media equation. (Gener[...]![]()
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This paper deals with the Cauchy problem for the nonlinear diffusion equation ?u/?t - ? (u|u|m+1) = 0 on (0, ?) x RN,u(0, .) = u0 when 0![]()
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It is well known that the ice-water phase transition can be modelled by the following set of equations: ?T ?t =?Tfor x?R N ??(t),t> 0, (1) T=0for x???(t),t> 0, (2) vn =??T ?n for x???(t),t> 0, (3) where T(x,t) stands for the te[...]![]()
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Arias, Clemente F. ; Herrero, Miguel A. ; Cuesta, José A. ; Acosta Salmerón, Francisco Javier ; Fernández Arias, Cristina | Royal Society (London) | 2015-07-01Adaptive immune responses depend on the capacity of T cells to target specific antigens. As similar antigens can be expressed by pathogens and host cells, the question naturally arises of how can T cells discriminate friends from foes. In this w[...]![]()
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This chapter provides a description of some of the mathematical approaches that have been developed to account for quantitative and qualitative aspects of chemotaxis. This last is an important biological property, consisting in motion of cells i[...]![]()
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In this paper we consider the limit m?+? of solutions of the porous-medium equation ut = ?•(um?u)(x?RN), with N > 1. We conjecture that, for initial data with a unique maximum, the evolution is characterized by the onset of a ‘mesa’ region, in [...]![]()
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We consider the equation ut=(Um)xx-?un with m> 1, ?> 0, n?m as a model for heat diffusion with absorption. Hence we assume that u?0 for x?R, t?0. We study the regularity of the solution to the Cauchy problem for this degenerate parabolic equatio[...]![]()
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Positive solutions of the semilinear parabolic equations u(t) - u(xx) = u(p), p > 1 and u(t) - u(xx) = e(u) for - ? 0 which blow up at a single point x = 0 at a finite instant of time t = T > 0 are considered. Using formal me[...]![]()
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We consider nonnegative solutions of the semilinear parabolic equation u t —u xx + u p = 0, -? 0, 0![]()
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We discuss the existence of travelling-wave solutions with interfaces for the nonlinear heat equation with absorption ut = a(um)xx – bu(n) with a, b> 0 and m, n ? R. Several situations occur depending on the relative strength of the di[...]![]()
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We consider the semilinear system (S) ut?uxx+vp=0, vt?vxx+uq=0(?? 0 and q> 0. We seek nonnegative and nontrivial travelling wave solutions to (S): u(x,t)=?(ct?x), v(x,t)=?(ct?x) possessing sharp fronts, i.e., such that ?(?)=?(?)=0 for ???0 and [...]![]()
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This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, [...]![]()
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Radiotherapy consists in the delivery of ionizing radiation with curative goals. It represents a major modality for treating solid malignancies in any anatomical site. It requires extremely precise dosimetry and delivery to control the lesion wh[...]![]()
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A mathematical model for the formation of microaggregates (microthrombi) of fibrin polymers in blood flow is considered. It is assumed that the former are induced by an external source (which may be of inflammatory or tumor nature) located in a [...]![]()
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Let (u(t, x), v(t, x)) and (uBAR(t, x), vBAR(t, x)) be two nonnegative classical solutions of (S)[GRAPHICS:{ut=?u+vp, p> 0 ; vt=?v+uq, q> 0] in some strip S(T) = (0, T) x R(N), where 0![]()
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We show the existence of instantaneous shrinking of the support and finite extinction time for a nonlinear parabolic problem (see (P) below) corresponding to a nonlinear diffusion with absorption phenomenon