## Abstract

We consider the existence and uniqueness of singular solutions for equations of the form u_{ 1}=div(|Du|^{p-2} Du)-φu), with initial data u(x, 0)=0 for x{upwards double arrow}0. The function φ{symbol} is a nondecreasing real function such that φ{symbol}(0)=0 and p>2. Under a growth condition on φ{symbol}(u) as u→∞, (H1), we prove that for every c>0 there exists a singular solution such that u(x, t)→cδ(x) as t→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→∞ as t→0. Finally, for functions φ{symbol} which behave like a power for large u we prove that the very singular solution is unique. This is our main result. In the case φ{symbol}(u)=u^{ q}, 1≤q, there are fundamental solutions for q<p*=p-1+(p/N) and very singular solutions for p-1<q<p*. These ranges are optimal.

Original language | English |
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Pages (from-to) | 51-74 |

Number of pages | 24 |

Journal | Journal d'Analyse Mathematique |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1992 |