2025/09/15 by Huang, Xi, Peng, Li, Pozo, Juan Carlos +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.11564
We consider the random Cauchy problem for the fully nonlocal telegraph equation of power type with the general (PC∗) 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 \refeqapp 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(θ(-Δ)^\fracβ4 ) and (-Δ)^-\fracβ4 sin(θ(-Δ)^\fracβ4 ) for β∈(1,2]. These operators enable, for the first time, the derivation of mixed-norm LtqLxp' 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 uω∈ L2(Ω, Hs,p(\mathbb R3)) (p∈ (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 uω.