A fast diffusion equation which generates a monotone local semiflow. I. Local existence and uniqueness
Peter Takáč · Differential and Integral Equations · 1991
The Cauchy problem for the fast diffusion equation [Jtn = d•8x(n-1 •8xn), (x, t) E I. x 1.+, with the boundary conditions lim n-1 • 8xn = c and lim n = b is investigated. x-+-oo x-+ooHere b, c, d E (0, oo) are given constants.It is proved that, when viewed as an abstract evolution equation in a suitable Sobolev space X, this problem has a unique mild solution which exists locally in time and is C 00 in I. x (0, r) for some T > 0, whenever n(x, 0) E X is given.These solutions form a monotone local semiflow in X.All results are stated in terms of an equivalent singular and degenerate parabolic problem in (0, 1) x 1.+ with fast transport.