Dispersive estimates for wave-type equations with time-dependent damping
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 $μ>0$, $f(u)=|u|^α$ with $α>1$, $θ\in\{0,1\}$, and the operator $L_{w^2}$ is defined on the Fourier transform by multiplication by $w(ξ)^2$. We prove the global (in time) existence of small data solutions for $α>α_{\mathrm{crit}}$, where the critical exponent $α_{\mathrm{crit}}$ depends on the choice of the operator $L_{w^2}$, the parameter $μ$, and the nonlinear term. In particular, we consider two model cases. For Boussinesq-type operators with $w(ξ)=\sqrt{|ξ|^2+|ξ|^4}$, combined with the derivative-type nonlinearity $Δ|u|^α$, we obtain a Strauss-type critical exponent. On the other hand, for plate-type operators with $w(ξ)=|ξ|^σ$, $σ\geq2$, and power-type nonlinearity $|u|^α$, the critical exponent is of Fujita type.