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2011 | Buch

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Blow-up Theories for Semilinear Parabolic Equations

verfasst von: Bei Hu

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

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

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Über dieses Buch

There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Blow-up phenomenon occurs in various types of nonlinear evolution equations. For example, they occur in Schrödinger equations, hyperbolic equations (see the papers Kalantarov–Ladyzhenskaya [81] and Deng [25, 26], Galaktionov–Pohozaev [54]), as well as in parabolic equations.

Bei Hu

Chapter 2. A Review of Elliptic Theories

Abstract

In Chaps. 2–4 we shall review material from first year PDE courses. These theories can be found in many books such as Chen–Wu [21], Chen [20], Gilbarg–Trudinger [67], Lieberman [94]. Compared to others, the books [21] and [20] are not intim- idating even for beginners and therefore are excellent textbooks for beginning graduate students. The book [21] received the textbook excellence award from the education department of China. Some of the more classical theories can also be found in Ladyzenskaja–Solonnikov–Ural’ceva [83], Ladyzenskaja–Ural’ceva [84] and Friedman [43], and the classical maximum principles in Protter–Weinberger [118]. In this chapter we list the elliptic theories that we will need later on.

Bei Hu

Chapter 3. A Review of Parabolic Theories

Abstract

All theorems in Chap. 2 have their parabolic version. As mentioned at the beginning of Chap. 2, the books [21] and [20] are not intimidating even for beginners and therefore are excellent textbooks for beginning graduate students. Elliptic theories are introduced in [21] and the parabolic theories are introduced in [20].

Bei Hu

Chapter 4. A Review of Fixed Point Theorems

Abstract

We collect in this chapter several fixed point theorems, which are useful for proving existence of solutions to nonlinear equations and systems.

Bei Hu

Chapter 5. Finite Time Blow-Up for Evolution Equations

Abstract

As indicated in the preface, we will emphasize the method and techniques for studying blow-up problems. While many of these methods apply to more general equations, we shall use the simplest model in our lectures to avoid lengthy computations.

Bei Hu

Chapter 6. Steady-State Solutions

Abstract

When the solutions of (5.1)–(5.3), or (5.4)–(5.6) are independent of t, they are called steady-state solutions, or stationary solutions. These solutions are the possible limits as t→∞ of the corresponding time-dependent solutions if the time dependent solution is global. There is no panacea for their study. However, many methods were developed to study these types of systems.

Bei Hu

Chapter 7. Blow-Up Rate

Abstract

It is established in Chap. 5 that the nonlinearity causes the blow-up to occur at a finite time in certain situations. If the solution to the ODE $u_t \,= \,f(u)$, blows up at a finite time t = T with $u(T - 0) = +\infty$, then u = G(T - t), where $G(\xi)$ is the inverse function of $\int\nolimits_\infty^u \frac {dn}{f(n)}$

Bei Hu

Chapter 8. Asymptotically Self-Similar Blow-Up Solutions

Abstract

The similarity variables can be used to study the asymptotic behavior (see Barenblatt [9]). Giga−Kohn [64, 65] gave a finer description of the blow-up behavior of the equation

$$u_t\,=\,\triangle u\,=\, |u|^{p-1}u,\,\,\,x\,\epsilon \,\Omega,\,\,\,0\,<\,t\,<T,$$

(8.1)

$$\quad\quad u\,=\,0,\,\,x\,\epsilon\,\partial\,\Omega,\,0\,<\,t<\,T,$$

(8.2)

$$\quad u(x-0)\,=,u_0(x).$$

(8.3)

Their approach also uses the similarity variables, together with the Pohozaev identity; see Theorem 8.5.

Bei Hu

Chapter 9. One Space Variable Case

Abstract

In the one−space−dimensional case, a continuous curve in the x−t plane starting in the left half of the plane {(x, t); x < 0} cannot end up at the right half of the plane {(x, t); x > 0} without crossing the t-axis {x = 0}.

Bei Hu

Backmatter

Titel: Blow-up Theories for Semilinear Parabolic Equations
verfasst von: Bei Hu
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-18460-4
Print ISBN: 978-3-642-18459-8
DOI: https://doi.org/10.1007/978-3-642-18460-4