Theoretical and Numerical Aspects of Blow-up-Effects in a special Class of nonlinear parabolic Differential Equations

I investigated blow-up-effects in nonlinear parabolic initial-boundary value problems of the form

u_t = Δu + Q(u).

The thesis is divided into a theoretical and a numerical part.

Theoretical part:

For the case Q(u) = u^p and Q(u) = δe^u I stated theorems about the existence of global solutions.

For the general case Q(u) I stated theorems about existence, uniqueness and boundedness of the solution, in case of an explosion I gave an estimation of the time when the explosion occurs.

Numerical part:

In the second part I compared three different numerical methods for calculating the solution of the equation, namely

• multi-grid iteration (with waveform relaxation for the smoothing part),
• the method of Runge-Kutta and
• the method of Crank-Nicolson combined with multi-grid iteration.

The third method was the core part of the evaluation. Due to this combination the instationary problem reduces to a sequence of stationary problems. For these problems I used the Picard iteration for the smoothing part. Unlike classical smoothing methods (like the Gau©¬-Seidel-Newton method or the Jakobi method) the Picard iteration has very good smoothing effects and very good convergence properties for these equations, which was shown in an example. I made a theoretical investigation of the smoothing effect and the convergence properties of the Picard iteration.

The results that I got from the three numerical methods were compared in an example where we know the exact time of explosion of the solution. The algorithms were implemented in Matlab.