A note on lifespan estimates for higher-order parabolic equations
Nurdaulet N. Tobakhanov, Berikbol T. Torebek
TL;DR
This paper analyzes the lifespan of solutions to a higher-order semilinear parabolic equation $u_t+(-\\Delta)^m u=|u|^p$ in $\\mathbb{R}^n$ with small initial data. It develops a two-pronged approach: lower bounds via contraction mapping in mixed $L^1$-$L^\\infty$ norms and an upper-bound strategy based on the test-function method and polyharmonic semigroup estimates (Ikeda–Sobajima style). The authors obtain sharp two-sided lifespan estimates depending on the Fujita exponent $p_{Fuj}=1+\\frac{2m}{n}$, namely $T_\\varepsilon \\simeq \\varepsilon^{-\\left(\\frac{1}{p-1}-\\frac{n}{2m}\\right)^{-1}}$ for $1<p<p_{Fuj}$ and $T_\\varepsilon \\simeq \\exp(\\varepsilon^{-(p-1)})$ at the critical, thereby extending previous upper-bound results. By proving results under the assumption $u_0\\in L^1\\cap L^{\\infty}$, the work sharpens prior decay-based conditions from Caristi–Mitidieri and Sun and strengthens the understanding of blow-up mechanisms in higher-order diffusion equations.
Abstract
We investigate the lifespan of solutions to the higher-order semilinear parabolic equation $$u_t+(-Δ)^m u=|u|^p, \quad x \in \mathbb{R}^n, t>0 $$ 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.