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

TL;DR

This paper constructs and analyzes type I blowup solutions for a higher-order semilinear parabolic equation, extending known methods to cases with higher odd integer order and providing a rigorous foundation for previous formal results.

Contribution

It develops a spectral analysis-based method to construct non self-similar blowup solutions for higher-order equations, generalizing classical techniques to new settings.

Findings

Constructed type I blowup solutions for higher-order equations.

Provided a sharp asymptotic description of solutions.

Extended classical blowup analysis methods to higher-order cases.

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 $m \geq 1$ 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].