Random data Cauchy theory for fully nonlocal telegraph equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917054184947712 |
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| author | Huang, Xi Peng, Li Pozo, Juan Carlos Zhou, Yong |
| author_facet | Huang, Xi Peng, Li Pozo, Juan Carlos Zhou, Yong |
| contents | 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 $u^ω$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random data Cauchy theory for fully nonlocal telegraph equations Huang, Xi Peng, Li Pozo, Juan Carlos Zhou, Yong Analysis of PDEs 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 $u^ω$. |
| title | Random data Cauchy theory for fully nonlocal telegraph equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.11564 |