Superlinear Parabolic Problems

Blow-up, Global Existence and Steady States

verfasst von: Pavol Quittner, Philippe Souplet

Verlag: Birkhäuser Basel

Buchreihe : Birkhäuser Advanced Texts / Basler Lehrbücher

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

1. Preliminaries

Abstract

We denote by B_r(x) or B(x, R) the open ball in ℝⁿ with center x and radius R. We set B_r:= B_r(0). The (n—1)-dimensional unit sphere is denoted by S^n-1. The characteristic function of a given set M is denoted by χ_M. We write D′ ⊂⊂ D for D′, D ⊂ ℝⁿ if the closure of D′ is a compact subset of D. For any real number s, we set s₊:= max(s, 0) and s_:= max(s, 0). We also denote ℝ₊:=[0, ∞)

Chapter I. Model Elliptic Problems

Abstract

In Chapter I, we study the problem

$$ \left. {\begin{array}{*{20}c} { - \Delta u = f\left( {x,u} \right),{\mathbf{ }}x \in \Omega } \\ {u = 0,{\mathbf{ }}x \in \partial \Omega ,} \\ \end{array} {\mathbf{ }}} \right\} $$

where f: Ω × ℝ → ℝ is a Carathéodory function (i.e. f(·, u) is measurable for any u ∈ ℝ and f(x, ·) is continuous for a.e. x ∈ Ω). Of course, the boundary condition in (2.1) is not present if ω = ℝⁿ. We will be mainly interested in the model case

$$ f\left( {x,u} \right) = |u|^{p - 1} u + \lambda u,{\mathbf{ }}where{\mathbf{ }}p > 1{\mathbf{ }}and{\mathbf{ }}\lambda {\mathbf{ }} \in \mathbb{R} $$

Denote by p_s the critical Sobolev exponent,

$$ ps: = \left\{ {\begin{array}{*{20}c} {\infty {\mathbf{ }}if{\mathbf{ }}n \leqslant 2,} \\ {\left( {n + 2} \right)/\left( {n - 2} \right){\mathbf{ }}if{\mathbf{ }}n > 2,} \\ \end{array} } \right. $$

We shall refer to the cases p p_s, p = p_s or p p_s as to (Sobolev) subcritical, critical or supercritical, respectively.

Chapter II. Model Parabolic Problems

Abstract

In Chapter II, we mainly consider semilinear parabolic problems of the form

$$ \left. {\begin{array}{*{20}c} {u_t - \Delta u = f\left( u \right),{\mathbf{ }}x \in \Omega ,t > 0} \\ {u = 0,{\mathbf{ }}x \in \partial \Omega ,t > 0,} \\ {u\left( {x,0} \right) = u_0 \left( x \right),{\mathbf{ }}x \in \Omega ,} \\ \end{array} {\mathbf{ }}} \right\} $$

where f is a C¹-function with a superlinear growth. For simplicity, we formulate most of our assertions for the model case f(u) = |u|^p-1u with p 1, but the methods of our proofs can be applied to more general parabolic problems (not necessarily of the form (14.1)). Some of possible generalizations and modifications will be mentioned as remarks, other can be found in the subsequent chapters.

Chapter III. Systems

Abstract

Chapter III is devoted to systems of elliptic and parabolic types. In Section 31, we study the questions of a priori estimates and existence for weakly coupled elliptic systems which are natural extensions of the scalar equations considered in Chapter I. In Section 32, we study a simple parabolic system which is the analogue of the scalar model problem (15.1) studied in Chapter II. For this system, we treat the questions of well-posedness, global existence and blow-up. In Section 33, we discuss the different possible effects of adding linear diffusion (and some boundary conditions) to a system of ODE’s. It will turn out that quite opposite effects can be observed. This will lead us to consider some systems arising in biological or physical contexts, such as mass-preserving and Gierer-Meinhardt systems.

Chapter IV. Equations with Gradient Terms

Abstract

In Chapter IV, we consider problems with nonlinearities depending on u and its space derivatives:

$$ \left. {\begin{array}{*{20}c} {u_t - \Delta u = F\left( {u,\nabla u} \right),{\mathbf{ }}x \in \Omega ,t > 0,} \\ {u = 0,{\mathbf{ }}x \in \partial \Omega ,t > 0,} \\ {u\left( {x,0} \right) = u_0 \left( x \right),{\mathbf{ }}x \in \Omega .} \\ \end{array} {\mathbf{ }}} \right\} $$

Here F = F(u, ξ):ℝ×ℝⁿ → lo is a C¹-function (except for problem (34.4) with 1 < q > 2, see below).

Chapter V. Nonlocal Problems

Abstract

In this chapter, we study various problems with nonlocal nonlinearities. The equations that we consider involve nonlocal terms taking the form of an integral in space, or in time. These terms may also be combined with local ones, either in an additive or in a multiplicative way.

Backmatter

Titel: Superlinear Parabolic Problems
verfasst von: Pavol Quittner
Philippe Souplet
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8442-5
Print ISBN: 978-3-7643-8441-8
DOI: https://doi.org/10.1007/978-3-7643-8442-5