Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities

2016/07/10 by Majdoub, Mohamed, Tayachi, Slim
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.02723

Abstract

In this paper we consider the problem: ∂_t u- Δu=f(u), u(0)=u₀∈ 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 L^p₀(\R^N) for f(u)∼ e^|u|q, 00, \liminf_{s→ ∞}(f(s) \rme^-λsp)>0, 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)\over 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.

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