[Paper Review] Construction of type I blowup solutions for a higher order semilinear parabolic equation

Tej‐Eddine Ghoul, Van Tien Nguyen+1 more|arXiv (Cornell University)|May 17, 2018

Stability and Controllability of Differential Equations19 references18 citations

TL;DR

This paper constructs type I non self-similar blowup solutions for a higher-order semilinear parabolic equation $\partial_t u = -(-\Delta)^m u + u|u|^{p-1}$ in $\mathbb{R}^N$, with $m \geq 1$ odd and $p > 1$. Using spectral analysis of a non-self-adjoint linearized operator in similarity variables, the authors reduce the infinite-dimensional problem to a finite-dimensional one and solve it via topological degree theory, rigorously justifying a formal result from Galaktionov (2001) and providing a sharp description of the blowup profile.

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].

Motivation & Objective

• To construct type I non self-similar blowup solutions for the higher-order semilinear parabolic equation $\partial_t u = -(-\Delta)^m u + u|u|^{p-1}$ in $\mathbb{R}^N$.
• To provide a sharp asymptotic description of the blowup behavior for such solutions.
• To rigorously justify a formal result on blowup dynamics previously proposed by Galaktionov (2001) using spectral and topological techniques.
• To extend the Merle-Zaag method—originally developed for the classical heat equation ($m=1$)—to higher-order operators with odd $m \geq 1$.
• To analyze the stability and structure of blowup solutions through linearization and scaling in similarity variables.

Proposed method

• Transform the original equation into similarity variables via the change $y = x/(T-t)^{1/(2m)}$, $s = -\log(T-t)$, leading to a time-dependent equation with a self-similar structure.
• Linearize the equation around a formally derived blowup profile $\varphi(y)$, resulting in a non-self-adjoint operator $\mathscr{L}_m$ whose spectral properties are analyzed.
• Decompose the solution error into components: low modes (associated with finitely many positive eigenvalues), high modes (controlled via weighted $L^\infty$ norms), and a localized error term.
• Use the spectral decomposition to reduce the infinite-dimensional problem to a finite-dimensional one by controlling only the components corresponding to the positive eigenvalues of $\mathscr{L}_m$.
• Apply topological degree theory (via index theory) to solve the finite-dimensional problem, ensuring existence of solutions satisfying the desired blowup behavior.
• Employ Gronwall-type estimates and semigroup estimates for $e^{s\mathscr{L}_m}$ to control error terms in weighted $L^\infty$ norms, particularly for high-order and localized components.

Experimental results

Research questions

• RQ1 Can type I non self-similar blowup solutions be rigorously constructed for higher-order semilinear parabolic equations with odd $m \geq 1$?
• RQ2 What is the precise asymptotic behavior of such blowup solutions, and how does it differ from self-similar profiles?
• RQ3 How can the Merle-Zaag method—previously used for $m=1$—be adapted to higher-order operators with non-self-adjoint linearizations?
• RQ4 What spectral properties of the linearized operator $\mathscr{L}_m$ enable the reduction to a finite-dimensional problem?
• RQ5 Is the formal blowup profile proposed by Galaktionov (2001) mathematically justified through rigorous analysis?

Key findings

• The authors construct type I non self-similar blowup solutions for the equation $\partial_t u = -(-\Delta)^m u + u|u|^{p-1}$ in $\mathbb{R}^N$ with $m \geq 1$ odd and $p > 1$.
• The blowup rate satisfies $\|u(t)\|_{L^\infty} \sim C(T-t)^{-1/(p-1)}$, confirming the Type I classification.
• The blowup profile is not self-similar, and its asymptotic structure is sharply described via the spectral decomposition in similarity variables.
• The solution is stable under small perturbations of initial data, as established by the topological argument based on index theory.
• The method provides a rigorous justification of a formal blowup dynamic previously conjectured by Galaktionov (2001), resolving a long-standing open problem.
• The analysis establishes uniform decay estimates for high-order and localized error components via weighted $L^\infty$ norms and Gronwall-type inequalities.