Singular solutions of some nonlinear parabolic equations

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.