A note on lifespan estimates for higher-order parabolic equations

TL;DR

This paper analyzes the lifespan of solutions to higher-order semilinear parabolic equations, providing precise asymptotic estimates that refine previous bounds and extend understanding of solution behavior near critical exponents.

Contribution

The paper derives sharp upper and lower bounds for solution lifespans, improving previous results by replacing initial data assumptions with $L^1 igcap L^ abla$ conditions.

Findings

Lifespan estimates depend on the exponent p relative to the Fujita critical exponent.

Derived explicit formulas for lifespan bounds in subcritical and critical cases.

Extended previous upper bound results to include lower bounds under sharper initial data conditions.

Abstract

We investigate the lifespan of solutions to the higher-order semilinear parabolic equation $$u_t+(-\Delta)^m u=|u|^p, \quad x \in \mathbb{R}^n, t>0 $$ with initial data. We focus on the precise asymptotic behavior of the lifespan of nontrivial solutions. By combining the test function method and semigroup estimates, we derive both upper and lower bounds for the lifespan of solutions $$T_{\varepsilon} \simeq \left\{\begin{array}{l}\varepsilon^{-\left(\frac{1}{p-1}-\frac{n}{2m}\right)^{-1}}, \,\, 1<p<p_{\text {Fuj}}, \\ \exp\left(\varepsilon^{-(p-1)}\right), \,\, p=p_{\text {Fuj}},\end{array}\right.$$ where $p_{Fuj}=1+\frac{2m}{n}$ is the critical exponent of Fujita. These estimates refine and extend the earlier results of Caristi-Mitidieri [J. Math. Anal. Appl., 279:2 (2003), 710-722] and Sun [Electron. J. Differential Equations, 17 (2010)], who obtained only upper bounds under slowly decaying initial data assumptions. In our setting, the above condition on the initial data is replaced by the assumption $L^1\cap L^\infty$, which sharpens the results of the aforementioned works.