< t < T where the solution of (1)- : T the maximal number such that the solution exists for 0 u(x, t)--+ oo if t--+ T. tn to t T) such that (xm, tm)· Such points (x T) are called blow-up points. (x) has at most two local maxima, see [14, 8]). f 4) that C > 0,

doi:10.1201/b16933-50

Chapter

ABSTRACT

Bernis and Friedman [4] have recently studied the nonlinear degenerate parabolic equation

{ u(x, 0) = g(x)

Friedman

where g, hE C4 and J( u) "' uP if u -+ oo , p > 1 ,

f(u) ~ -C if u-+ -oo. Assuming that the Cauchy problem {17), (18) does not have a solution for all t > 0, Caffarelli and Friedman [6) proved that if n = 1, then there exists a continuously differentiable curve r : t = T( x) ( x E R1) such that

- { A(1-a2) }1/(p-1) 1

They extended this result ton= 2, 3 (in [7]) provided f and g satisfy some (quite restrictive) conditions.