On very singular similarity solutions of a higher-order semilinear parabolic equation

Nonlinearity ◽  
2004 ◽  
Vol 17 (3) ◽  
pp. 1075-1099 ◽  
Author(s):  
V A Galaktionov ◽  
J F Williams
2019 ◽  
Vol 9 (1) ◽  
pp. 388-412 ◽  
Author(s):  
Tej-Eddine Ghoul ◽  
Van Tien Nguyen ◽  
Hatem Zaag

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].


2018 ◽  
Vol 56 (6) ◽  
pp. 4434-4460
Author(s):  
Eduardo Casas ◽  
Mariano Mateos ◽  
Fredi Tröltzsch

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