The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

Quenching and Blowup Problems for Reaction Diffusion Equations

Timo Salin

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium E at Helsinki University of Technology (Espoo, Finland) on the 26th of March, 2004, at 12 o'clock noon.

Overview in PDF format (ISBN 951-22-6987-2)   [395 KB]
Dissertation is also available in print (ISBN 951-22-6943-0)

Abstract

In this thesis we study quenching and blowup problems for reaction diffusion equations with Cauchy-Dirichlet data. We give sufficient conditions for certain reaction terms under which quenching or blowup can occur. Furthermore we show that the set of quenching points is finite for certain nonlinearities. The main results concern the asymptotic behavior of the solution in a neighborhood of a quenching or blowup point. We prove two kinds of asymptotic theorems. First we study quenching or blowup rate results and then give precise asymptotic expressions for solutions in a backward space-time parabola near a quenching point for certain reaction terms.

This thesis consists of an overview and of the following 4 publications:

  1. Salin T., 2003. On quenching with logarithmic singularity. Nonlinear Analysis TMA 52, number 1, pages 261-289.
  2. Salin T., Quenching-rate estimate for a reaction diffusion equation with weakly singular reaction term. Dynamics of Continuous, Discrete and Impulsive Systems (Series A: Mathematical Analysis), to appear.
  3. Salin T., 2003. On a refined asymptotic analysis for the quenching problem. Helsinki University of Technology, Institute of Mathematics, Research Report A457.
  4. Salin T., 2003. The quenching problem for the N-dimensional ball. Helsinki University of Technology, Institute of Mathematics, Research Report A459.

Keywords: reaction-diffusion equation, quenching, quenching set, quenching rate, asymptotic behavior of solutions, refined asymptotics, blow-up, blow-up set, blow-up rate

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© 2004 Helsinki University of Technology

Last update 2011-05-26