WELL-POSEDNESS, GLOBAL EXISTENCE AND DECAY ESTIMATES FOR THE HEAT EQUATION WITH GENERAL POWER-EXPONENTIAL NONLINEARITIES

摘要

In this paper we consider the problem: ?_tu - △u=f(u), u(0)= U_0 €exp L~p(R~N), where p> 1 and f : R → R having an exponential growth at infinity with f(0)= 0. We prove local well-posedness in exp Lp0 (R~N) for f(u) ~ e~(|u|q), 0 < q ≤ p, |u| →∞. However, if for some{formula} , then non-existence occurs in exp L~p (R~N). Under smallness condition on the initial data and for exponential nonlinearity f such that |f(u)| ~ |U|~m as u → 0, N(m-1)/2≥ p, we show that the solution is global. In particular, p - 1> 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.