Construction of type I blowup solutions for a higher order semilinear parabolic equation

Abstract

We consider the higher-order semilinear parabolic equation $${\partial }_{t}u=-(-\Delta {)}^{m}u+u|u{|}^{p-1},$$ in the whole space ${\mathbb{R}}^N$, where $p>1$ and $m\ge 1$ is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by [Galaktionov, rspa2011], we revisit the technique developed by [Merle-Zaag, duke1997] for the classical case $m=1$, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [Galaktionov, rspa2011].