Blow-up in a degenerate parabolic equation with gradient nonlinearity

  • Blow-up in einer degeneriert parabolischen Gleichung mit nichtlinearem Gradiententerm

Stinner, Christian; Wiegner, Michael (Thesis advisor)

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

Aachen, Techn. Hochsch., Diss., 2008

Abstract

In this thesis we study positive classical solutions of the degenerate parabolic differential equation $u_t = u^p Delta u + u^q + kappa u^r|abla u|^2$ with zero Dirichlet boundary condition in smoothly bounded domains Omega. Here p>0, q>1, r>-1 and kappa are real parameters. The main aspect of this thesis is to show the influence of the parameters p, q, r and kappa with respect to blow-up of solutions and the size of the blow-up set. In this context blow-up of a solution means that the solution gets unbounded in finite time. During the past 20 years similar questions were considered for degenerate parabolic equations without gradient terms and the forced porous medium equation (see e.g. Galaktionov, Wiegner, Winkler). In particular, for the equation mentioned above in case of kappa = 0 it was proved that the exponent q=p+1 is critical with respect to blow-up. In this thesis we particularly show the influence of the gradient term on this behavior of the solutions for positive or negative values of kappa. In Chapter 1 we prove that the equation mentioned above has a maximal classical solution. This solution remains above of any positive classical solution and is unique. Moreover, we study the question whether positive classical solutions of this equation are unique. In Chapters 2 and 3 we present in detail which conditions ensure that the maximal solution of this equation is global in Omega x (0, infty) and which conditions ensure blow-up of the maximal solution. In this context - for fixed values of p, q, r and kappa – the following phenomena appear: the maximal solutions corresponding to any initial data exist global in time, or any maximal solution blows up independently of the initial data, or there are global solutions (for small initial data) as well as blow-up (for large initial data). In Chapter 2 we show that, if kappa is positive, the exponent r = 2p-q is a second critical exponent apart from q = p+1. In this case the gradient term is positive and enforces blow-up. In contrast to the case kappa > 0 we prove in Chapter 3 that r = q-2 is the second critical exponent apart from q = p+1, if kappa is negative. Here the negative gradient term can prevent blow-up. Furthermore, we study in case of the global existence of the maximal solution in Omega x (0, infty), whether this solution converges to 0 as t tends to infinity. We state that the exponents r = q-2 and q = p+1 are critical with respect to this question, too. In Chapter 4 we show the influence of the parameters with respect to the size of the blow-up set, if kappa is positive. Therefore we assume that the maximal solution of the equation is radially symmetric and gets unbounded after a finite and positive time T. Then, we prove for a class of initial data that the blow-up set is only a single point in case of q > max {p+1,r+2}, whereas it contains a ball, if q <= max {p+1,r+2}.

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