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

Advances in Nonlinear Analysis

article

Tej-Eddine Ghoul¹  Van Tien Nguyen¹  Hatem Zaag² 

[1] New York University in Abu Dhabi, United Arab Emirates;Université Paris 13

关键词: Higher order parabolic equation;    Blowup solution;    Blowup profile;    Stability;   

DOI  :  10.1515/anona-2020-0006

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We consider the higher-order semilinear parabolic equation ∂tu=−(−Δ)mu+u|u|p−1, $$\begin{array}{} \displaystyle \partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1}, \end{array}$$ in the whole space ℝ N , 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].

【 授权许可】

CC BY