Abstract
We consider the higher-order semilinear parabolic equation
$$\begin{array}{}
\displaystyle
\partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1},
\end{array}$$
in the whole space ℝN, where p > 1 and m ≥ 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 [15], we revisit the technique developed by Merle-Zaag [23] 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 [15].
Existence of solutions for a higher-order semilinear parabolic equation with singular initial data
