Decay estimate for subcritical semilinear damped wave equations with slowly decreasing data
Kazumasa Fujiwara, Vladimir Georgiev
TL;DR
This work analyzes pointwise decay for nonnegative solutions to the one-dimensional damped wave equation with a subcritical power nonlinearity by constructing a sharp supersolution strategy. The authors combine an explicit ODE-based supersolution with the linear solution operator $S(t)$ and a carefully chosen linear perturbation $u_L$ to obtain a pointwise decay bound of the form $0 \le u(t,x) \le C \left( \dfrac{u_L(t+T_0,x)^{p-1}}{(t+T_0)u_L(t+T_0,x)^{p-1} + 1} \right)^{1/(p-1)}$, valid for large times. They further derive a corollary showing that, for initial data with polynomial decay $\langle x \rangle^{-\rho}$ with $1<\rho<2/(p-1)$, the solution decays in $L^q$ like $t^{\frac{1}{q\rho(p-1)} - \frac{1}{p-1}}$, matching the heat-kernel rate and ensuring membership in $L^p(0,\infty; L^p)$. The results advance understanding of nonlinear effects in subcritical damped waves by connecting nonlinear decay to heat-kernel behavior via a robust supersolution framework.
Abstract
We study the decay properties of non-negative solutions to the one-dimensional defocusing damped wave equation in the Fujita subcritical case under a specific initial condition. Specifically, we assume that the initial data are positive, satisfy a condition ensuring the positiveness of solutions, and exhibit polynomial decay at infinity. To show the decay properties of the solution, we construct suitable supersolutions composed of an explicit function satisfying an ordinary differential inequality and the solution of the linear damped wave equation. Our estimates correspond to the optimal ones inferred from the analysis of the heat equation.