Random data Cauchy theory for fully nonlocal telegraph equations

Abstract

We consider the random Cauchy problem for the fully nonlocal telegraph equation of power type with the general $(\mathcal{PC}^{\ast})$ type kernel $(a,b)$. This equation can effectively characterize high-frequency signal transmission in small-scale systems. We establish a new completely positive kernel induced by $b$ (see Appendix \refeq{app b}) and derive two novel solution operators by using the relaxation functions associated with the new kernel,which are closely related to the operators $\cos(\theta(-\Delta)^{\frac{\beta}{4}} )$ and $(-\Delta)^{-\frac{\beta}{4} }\sin(\theta(-\Delta)^{\frac{\beta}{4}} )$ for $\beta\in(1,2]$. These operators enable, for the first time, the derivation of mixed-norm $L_t^qL_x^{p'}$ estimates for the novel solution operators. Next, utilizing probabilistic randomization methods, we establish the average effects, the local existence and uniqueness for a large set of initial data $u^\omega \in L^{2}(\Omega, H^{s,p}(\mathbb R^3))$ ($p\in (1,2)$) while also obtaining probabilistic estimates for local existence under randomized initial conditions. The results reveal a critical phenomenon in the temporal regularity of the solution regarding the regularity index $s$ of the initial data $u^\omega$.