Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension

Abstract

This paper concerns the grow-up rate of solutions to a semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the large-time behavior of a solution changes at the critical dimension $N=10$, and the grow-up phenomenon occurs for the case $N\ge 10$. For the inhomogeneous case with $N\ge 10$, the present author and a coauthor showed in [12] that the grow-up phenomenon disappears once $f$ exceeds a threshold. In this paper, we provide a quantitative characterization of the disappearance of the grow-up phenomenon by obtaining the grow-up rate in the critical dimension. Our result shows that a qualitatively different type of grow-up behavior occurs in the critical dimension compared with the case $N\ge 11$ studied in [12]. The difference is caused by a change in the outer behavior of the solution. Even in the case $f=0$, our result is new in that it provides a rigorous justification of the formal computation by Galaktionov and King [10].