On upper bounds for waiting times for doubly nonlinear parabolic equations

  • Über obere Schranken für Wartezeiten bei doppelt nichtlinearen parabolischen Gleichungen

Djie, Kianhwa Colin; Wiegner, Michael (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2008)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2008

Abstract

It is the aim of this thesis to derive quantitative upper bounds for waiting time phenomena. Herein the waiting time of a nonnegative function u: R^n x [0, infty) -> [0, infty) denotes the time when u starts leaving the initial support supp(u(.,0)) for the first time. We will examine how the waiting time of nonnegative solutions for degenerated diffusion equations depends on the growth of the initial value. The situation of the porous medium equation is fully understood thanks to the works of Aronson, Alikakos, Caffarelli, Chipot, Kamin, Sideris (1983-1985). Although lower bounds, e. g. by Giacomelli, Grün (2006), are also known for further classes of equations, for example the doubly degenerate parabolic differential equation $u_t - Delta_p u^m = 0$, nothing was known about quantitative upper bounds in this more general setting. An upper bound will be derived for this situation in the first chapter. This will be done (roughly sketched) in the following steps: * Existence of weak solutions which preserve radial symmetry, sign of the radial derivative and comparability of initial values (Theorem 1.1.3),* derivation of a superlinear ordinary differential equation for an energy functional of a specific nonlinear solution (proof of Theorem 1.4.2),* analysis of the blow-up time for differential inequalities of this type (Lemma 1.4.1).Those energy functionals can be used in order to derive upper bounds in the more general situation $u_t - Delta_p u^m pm lambda u^alpha = 0$, i. e. with reaction terms resp. absorption terms. It will be shown that the value of alpha (depending on p, m) is essential for the property whether this additional term can be neglected or whether it has significant influence on the waiting time, see the second chapter. Eventually one has to switch to an indirect argument together with a functional inequality (instead of a differential inequality). This chapter ends with a discussion for the variant $(u^{q-1})_t - (|u_x|^{p-2}u_x)_x pm lambda(u^alpha)_x = 0$ with convection terms resp. advection terms. Finally it will be shown in the third chapter (Theorem 3.1) that these energy methods also work for the coupled system $u_t - Delta u^m - v^alpha = 0, v_t - Delta v^n - u^eta = 0$. Parts of the first chapter were already published in the following article: * Djie, Kianhwa Colin, An upper bound for the waiting time for doubly nonlinear parabolic equations, Interfaces and Free Boundaries, 9 No. 1, 2007, 95-105.

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