关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system

http://cds.cern.ch/record/367814/files/9810070.pdf

Gravitating non-abelian solitons and black holes with yang-mills fields

In this article, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise, a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck [16]. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with or without boundary conditions. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville-type theorems, which unifies and improves many results concerning a priori bounds, decay estimates and initial and final blow-up rates. For example, for the equation u t - Δu = u p on a domain Ω, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form u(x, t) ≤ C(Ω, p)(1 + t -1/(p-1) + (T - t) -1/(p-1) ). Our method is based on rescaling arguments combined with a key "doubling" property, and it is facilitated by parabolic Liouville-type theorems for the whole space or the half-space. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.

/pdf/singularity-and-decay-estimates-in-superlinear-problems-via-52tzq37ywv.pdf

Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations

Unspecified Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22758 Originally published at: Chipot, M; Fila, M; Quittner, P (1991). Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Mathematica Universitatis Comenianae. New Series, 60(1):35-103. Acta Math. Univ. Comenianae Vol. LX, 1(1991), pp. 35–103 35 STATIONARY SOLUTIONS, BLOW UP AND CONVERGENCE TO STATIONARY SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS M. CHIPOT, M. FILA AND P. QUITTNER

Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions

We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces Lpδ(Ω)=Lp(Ωδ(x) dx), where δ(x) is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.

Pavol Quittner

Papers

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations

Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions

A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces