Convergence to self-similar solutions for a semilinear parabolic equation (Q934230)

The authors consider the following Cauchy problem \[ u_t=\Delta u+u^p,\quad x\in \mathbb{R}^N,\;t>0, \qquad u( x,0) =u_0( x) ,\quad x\in \mathbb{R}^N, \] where \(u_0\) is a nonnegative continuous function on \(\mathbb R^N\). Considering solutions which converge to self-similar solutions as \(t\to \infty\), the authors concern is the rate of convergence depending on the behavior of initial data. More precisely, given a specific decay rate of \(u_0\) as \(| x| \to \infty\), the authors determine the exact rate of the convergence. The article presented is a part of a research project on quantitative description of the behavior of solutions. So far, the authors have determined in previous articles the grow-up decay of solutions, the convergence rate to regular steady states and the decay rate to the trivial solution.