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

Abstract

We consider the higher-order semilinear parabolic equation $$ \\partial_t u =\n-(-\\Delta)^{m} u + u|u|^{p-1}, $$ in the whole space $\\mathbb{R}^N$, where $p >\n1$ and $m \\geq 1$ is an odd integer. We exhibit type I non self-similar blowup\nsolutions for this equation and obtain a sharp description of its asymptotic\nbehavior. The method of construction relies on the spectral analysis of a non\nself-adjoint linearized operator in an appropriate scaled variables setting. In\nview of known spectral and sectorial properties of the linearized operator\nobtained by [Galaktionov, rspa2011], we revisit the technique developed by\n[Merle-Zaag, duke1997] for the classical case $m = 1$, which consists in two\nsteps: the reduction of the problem to a finite dimensional one, then solving\nthe finite dimensional problem by a classical topological argument based on the\nindex theory. Our analysis provides a rigorous justification of a formal result\nin [Galaktionov, rspa2011].\n