| Abstract: |
| In this talk we discuss the Cauchy problem for a class of semilinear evolution equations with scale-invariant time-dependent dissipation
\begin{equation} \label{Eq:Abstract}
\begin{cases}
u_{tt} + L_{w^2}u + \dfrac{\mu}{1+t}u_t = f(u), & t>0,\ x\in\mathbb{R}^n,\
u(0,x) = 0, \,\,\,\, u_t(0,x) = u_1(x), & x\in\mathbb{R}^n,
\end{cases}
\end{equation}
where $L_{w^2}$ is a Fourier multiplier operator defined by $L_{w^2}u = \mathcal{F}^{-1}(w(\xi)^2\hat{u})$, $f(u)=|u|^\alpha$ or $f(u)=\Delta|u|^\alpha$, and $\mu>0$, $\alpha>1$ are constants. We prove global (in time) existence of small data solutions for $\alpha>\alpha_{\mathrm{crit}}$, where the critical exponent $\alpha_{\mathrm{crit}}$ depends on the choice of the operator $L_{w^2}$. More precisely, it corresponds to a Strauss-type exponent in the case of Boussinesq-type operators $w(\xi)=\sqrt{|\xi|^2+ |\xi|^4}$, while it becomes a Fujita-type exponent for plate-type operators $w(\xi)=|\xi|^\sigma$, $\sigma\geq 2$. The main tools to prove this result are Duhamel`s principle, dispersive estimates of solutions of a family of
parameter-dependent linear Cauchy problems with vanishing right-hand side and Banach`s fixed point
theorem. |
|