The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions

TL;DR

This paper derives a precise upper bound for the lifespan of solutions to high-dimensional semilinear damped wave equations with Fujita critical power, using heat kernel methods to improve understanding of solution blow-up times.

Contribution

It establishes the sharp upper bound of the lifespan for these equations in high dimensions, advancing the theoretical understanding of solution behavior at critical exponents.

Findings

Lifespan T(ε) is bounded above by an exponential function of ε.

The bound is sharp and matches known lower bounds.

Heat kernel is effectively used as a test function in the analysis.

Abstract

This paper is concerned about the lifespan estimate to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent in high dimensions$(n\geq 4)$. We establish the sharp upper bound of the lifespan in the following form \begin{equation}\nonumber\\ \begin{aligned} T(\varepsilon)\leq \exp(C\varepsilon^{-\frac 2n}), \end{aligned} \end{equation} by using the heat kernel as the test function.