On The Cauchy Problem for The Equation u′ − Δu = u¦u¦γ−1 In The Supercritical Case

We prove global existence of classical solutions to the above problem, whenever the initial function is continuous and can be enclosed between a sub- and a supersolution of the corresponding stationary problem, where the latter are assumed to be smooth bounded functions lying in some L^p(?ⁿ) (Theorem 1). In addition we study the large-time behaviour of the solutions. As an example, it turns out from this that in case of the critical exponent γ = (n+2)/(n?2) the corresponding positive stationary solutions are unstable w.r.t. arbitrarily slight variations either from above or from below (Theorems 5 and 9).