Random data Cauchy theory for fully nonlocal telegraph equations

TL;DR

The paper addresses the random Cauchy problem for a fully nonlocal telegraph equation with kernels in the $(\mathcal{PC}^{\ast})$ class, coupling temporal memory with spatial fractional diffusion. It develops a novel completely positive kernel framework and constructs two solution operators, $\mathcal{C}(t)$ and $\mathcal{S}(t)$, enabling a mild formulation and explicit mixed-norm space estimates in $L^q_tL^{p'}_x$. Through probabilistic randomization of initial data, the authors establish averaging effects and almost-sure local existence/uniqueness results, with a sharp delineation between subcritical and (super)critical temporal regularity regimes governed by the critical index $s_{\mathrm{crit}}=\tfrac{3(\kappa-1)}{\kappa+1}$. The work integrates Volterra kernel theory, complete positivity, and stochastic analysis to advance the understanding of nonlocal hyperbolic equations with memory and fractional diffusion, including enhanced temporal regularity for data above the critical threshold. Overall, it provides a rigorous framework for random data Cauchy problems in fully nonlocal telegraph models and highlights the delicate balance between memory, diffusion, nonlinearities, and randomness.

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 $\cos(θ(-Δ)^{\fracβ{4}} )$ and $(-Δ)^{-\fracβ{4} }\sin(θ(-Δ)^{\fracβ{4}} )$ for $β\in(1,2]$. 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 $u^ω\in L^{2}(Ω, H^{s,p}(\mathbb R^3))$ ($p\in (1,2)$) 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 $u^ω$.

Theorems & Definitions (39)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.1
• proof
• Lemma 4.1
• proof
• Remark 4.1
• Lemma 4.2
• proof