Blow-up in reaction-diffusion equations with exponential and power-type nonlinearities

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Perustieteiden korkeakoulu | Doctoral thesis (article-based)
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Verkkokirja (451 KB, 51 s.)
Aalto University publication series DOCTORAL DISSERTATIONS, 45/2011
In this dissertation we study blow-up phenomena in semilinear parabolic equations with both exponential and power-type nonlinearities. We study the behavior of the solutions as the blow-up moment in time and the blow-up point in space are approached. Our focus is on the supercritical case; however, we also give some results on the subcritical case. We prove results concerning the blow-up rate of solutions, and we obtain the blow-up profile for limit L1-solutions both with respect to the similarity variables and at the blow-up moment. We use techniques that are applicable both for the exponential and power nonlinearities. We also consider immediate regularization for minimal L1-solutions and improve on some earlier results. We are also interested in the behavior of selfsimilar solutions and we prove the existence of regular selfsimilar solutions that intersect the singular one arbitrary number of times.
Supervising professor
Gripenberg, Gustaf, Prof.
Thesis advisor
Londen, Stig-Olof, Prof.
semilinear parabolic equation, supercritical case, exponential nonlinearity, power-type nonlinearity, blow-up, selfsimilar solutions, blow-up rate, blow-up profile, regularity, semigroup estimates
Other note
  • [Publication 1]: M. Fila, A. Pulkkinen, Backward selfsimilar solutions of supercritical parabolic equations, Applied Mathematics Letters 22 (2009), 897-901. © 2008 Elsevier Science. By permission.
  • [Publication 2]: M. Fila, A. Pulkkinen, Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation, Tohoku Mathematical Journal 60 (2008), 303-328. © 2008 Tohoku University, Mathematical Institute. By permission.
  • [Publication 3]: A. Pulkkinen, Blow-up profiles of solutions for the exponential reaction-diffusion equation, arXiv:1102.4158v1 [math.AP] (2011), 1-29, (accepted for publication in Mathematical Methods in the Applied Sciences). © 2011 by author.
  • [Publication 4]: A. Pulkkinen, Some comments concerning the blow-up of solutions of the exponential reaction-diffusion equation, arXiv:1102.4275v2 [math.AP] (2011), 1-18. © 2011 by author.