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Some results on blow up for semilinear parabolic problems Herrero, Miguel A. Velázquez, J.J. L. Ni, Wei-Ming Peletier, L. A. Vázquez, Juan Luis Proceedings of the IMA Workshop held at the University of Minnesota, Minneapolis, Minnesota, May 13–18, 1991 The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed. 2023-06-20T21:07:51Z 2023-06-20T21:07:51Z 1993 book part 0-387-94068-5 10.1007/978-1-4612-0885-3_7 https://hdl.handle.net/20.500.14352/60763 http://link.springer.com/chapter/10.1007%2F978-1-4612-0885-3_7 http://www.springer.com eng IMA Volumes in Mathematics and its Applications open access Springer