A new gap in the critical exponent for semi-linear structurally damped evolution equations

Our aim in this paper is to discuss the critical exponent in semi-linear structurally damped wave and beam equations with additional dispersion term. The special model we have in mind is $$u_{tt}(t,x)+(-\Delta)^{\sigma}u(t,x)+(-\Delta)^{2\delta}u(t,x)+2(-\Delta)^{\delta}u_{t}(t,x)=\left|u(t,x)\right| ^{p}$$ where the initial displacement $u(0,x)=u_{0}(x)$, the initial velocity $u_{t}(0,x)=u_{1}(x)$ and the parameters $t\in [0,\infty)$, $x\in \mathbb{R}^{n}$, $\sigma\geq 1$, $\delta\in(0,\frac{\sigma}{2})$, $p>1$. The solution to the linear equation at low frequency region involves an interplay of diffusion and oscillation phenomena represented by a real-complex Fourier multiplier of the form $$m(t,\xi)=\frac{e^{-|\xi|^{2\delta}t\pm i|\xi|^{\sigma}t}}{2i|\xi|^{\sigma}}, \ \ \xi\in \mathbb{R}^{n}, \ \ i=\sqrt{-1}.$$ The scaling argument shows that the diffusive part leads to faster decay rates compared to the oscillatory one. This interplay creates a new gap in the critical exponent between the blow up (in finite time) result when $1<p<1+\frac{4\delta}{n-2\delta}$ (sub-critical case) and the global (in time) existence result when $p>1+\frac{\sigma+2\delta}{n-\sigma}$ (super-critical case). We leave an open to show if this gap will be closed at least in low or high space dimensions because, to the best of authors knowledge, the necessary Fourier multiplier that leads to the sub-critical case does not explicitly appear in $m(t,\xi)$.