| Contents |
| The goal of the lecture is to study the Cauchy problem for structurally damped $\sigma$-evolution models
u_{tt} + (-\Delta)^\sigma u + (-\Delta}^\delta u_t=f(u,u_t,|D|^\alpha u),\,\,u(0,x)=u_0(x),\,\,u_t(0,x)=u_1(x) ,
where $\sigma \geq 1$, $\delta \in (0,\sigma]$ and $\alpha \in [0,\sigma]$. Our main issue is the global existence (in time) of small data solutions. It is related to determine the critical exponent (of Fujita type).
The considerations base on a very precise linear theory. The main tool are Strichartz decay estimates not necessarily on the conjugate line. |
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