caa a22 4500 469066784 CHVBK 20180323132910.0 cr unu---uuuuu 170328e19921201xx s 000 0 eng 10.1007/BF02790217 doi (NATIONALLICENCE)springer-10.1007/BF02790217 Singular solutions of some nonlinear parabolic equations [Elektronische Daten] [Shoshana Kamin, Juan Vazquez] We consider the existence and uniqueness of singular solutions for equations of the formu 1=div(|Du|p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. The Magnes Press, The Hebrew University, 1992 Kamin Shoshana School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel aut Vazquez Juan División de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain aut Journal d'Analyse Mathématique Springer-Verlag 59/1(1992-12-01), 51-74 0021-7670 59:1<51 1992 59 11854 https://doi.org/10.1007/BF02790217 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02790217 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kamin Shoshana School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel aut NATIONALLICENCE 700 1- Vazquez Juan División de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain aut NATIONALLICENCE 773 0- Journal d'Analyse Mathématique Springer-Verlag 59/1(1992-12-01), 51-74 0021-7670 59:1<51 1992 59 11854 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer