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Abstract.In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain f~ c R n, n >/1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of -Au +/3(u) = f with/3 a nondecreasing function R ~R, /3(0)=0, and f>~0 a.e. in ~ if and only if the integral f(/3(s)s)-l/2dsdiverges at = 0+. We extend the result to general S more equations, in particular to -Ap u +/3 (u) = f where Ap (u) = div( I Du ]P -2Du), 1 < p < ~. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.

The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,We consider data f ∈ L 1 (R N ) and all exponents 0 < σ < 2 and m > 0. Existence and uniqueness of a weak solution is established for m > m * = (N − σ) + /N , giving rise to an L 1 -contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range 0 < m ≤ m * existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above m * . We also study the dependence of solutions on f, m and σ. Moreover, we consider the above questions for the problem posed in a bounded domain.2000 Mathematics Subject Classification. 26A33, 35A05, 35K55, 76S05

We study the well-posedness and describe the asymptotic behavior of solutions of the heat equation with inverse-square potentials for the Cauchy Dirichlet problem in a bounded domain and also for the Cauchy problem in R N . In the case of the bounded domain we use an improved form of the so-called Hardy Poincare inequality and prove the exponential stabilization towards a solution in separated variables. In R N we first establish a new weighted version of the Hardy Poincare inequality, and then show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This work complements and explains well-known work by Baras and Goldstein on the existence of global solutions and blow-up for these equations. In the present article the sign restriction on the data and solutions is removed, the functional framework for well-posedness is described, and the asymptotic rates calculated. Examples of non-uniqueness are also given. Academic Press

Abstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d , d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results.

Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem.Our first goal here is to formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain limit. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the asymptotic Hele-Shaw type problem. The main tools are nonlinear regularizing effects for certain porous medium type equations, regularization techniquesà la Steklov, and a Hilbert duality method for uniqueness. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density.Using this theory as a basis, we go on to consider the more complex model including nutrients. We obtain the equation for the limit of the coupled system; the method relies on some BV bounds and space/time a priori estimates. Here, new technical difficulties appear, and they reduce the generality of the results in terms of the initial data. Finally, we prove uniqueness for the system, a main mathematical difficulty.

The possible continuation of solutions of the nonlinear heat equation in RN × R+ ut = Δum + up with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation ut = Δum − up, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.

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