Asymptotic behavior of global solutions of the \(u_t=\Delta u + u^{p}\)

We study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem {u_t=\Delta u + u^{p}, x\in\mathbb{R}^{N}, t>0 u(0)=u_{0}, x\in\mathbb{R}^{N}, t=0. It is known that the nonnegative solution \(u(t)\) of this problem blows up in finite time for \(1 1+ 2/N\) and the norm of \(u_{0}\) is small enough, the problem admits global solution. In this work, we use the entropy method to obtain the decay rate of the global solution \(u(t)\).