In this article, we investigate a high-order quasilinear hyperbolic equation that involves Kirchhoff damping and logarithmic source:

$$ \begin{align*}u_{tt}-{\operatorname*{div}}(|\nabla u|^{p-2}\nabla u)+\sigma(\|\nabla u\|_{2}^{2} )u_t+\Delta^{2}u=|u|^{q-2}u\log|u|,\end{align*} $$

$\Omega \times (0,T_{\mathrm {max}})$

$\Omega \subset \mathbb {R}^{n}$

$p,q>2$

$\sigma (\|\nabla u\|_{2}^{2} )$

$u_t$

$q\geq p$

$q>p$

$q<p$

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.

In this paper, the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabola

\begin{equation*}T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t}\end{equation*}

$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$

$\frac{1}{2}\lt r\lt\infty$

$L^\infty\times L^2\to L^2$

We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$ for $\alpha \in (0,3/2)$.

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.