Dispersive estimates for wave-type equations with time-dependent damping

Abstract

In this paper, we study the Cauchy problem for a class of semilinear evolution equations with scale-invariant time-dependent dissipation \begin{equation*} \begin{cases} u_{tt} + L_{w^2}u + \dfracμ{1+t}u_t = Δ^θ f(u), & t>0,\ x\in\mathbb{R}^n,\\ u(0,x) = 0,\qquad u_t(0,x) = u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{equation*} where $\hat{I}¼>0$, $f(u)=|u{|}^{\hat{I}}\pm$ with $\hat{I}\pm >1$, $\hat{I}¸\in \{0,1\}$, and the operator $L_{w^2}$ is defined on the Fourier transform by multiplication by $w(\hat{I}¾{)}^2$. We prove the global (in time) existence of small data solutions for $\hat{I}\pm >\hat{I}{\pm }_{\mathrm{crit}}$, where the critical exponent $\hat{I}{\pm }_{\mathrm{crit}}$ depends on the choice of the operator $L_{w^2}$, the parameter $\hat{I}¼$, and the nonlinear term. In particular, we consider two model cases. For Boussinesq-type operators with $w(\hat{I}¾)=\sqrt{|\hat{I}¾{|}^2+|\hat{I}¾{|}^4}$, combined with the derivative-type nonlinearity $\hat{I}”|u{|}^{\hat{I}}\pm$, we obtain a Strauss-type critical exponent. On the other hand, for plate-type operators with $w(\hat{I}¾)=|\hat{I}¾{|}^{\ddot{I}}ƒ$, $\ddot{I}ƒ\ge 2$, and power-type nonlinearity $|u{|}^{\hat{I}}\pm$, the critical exponent is of Fujita type.

26 pages, 1 figure