The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

AbstractBy constructing the comparison functions and the perturbed method, it is showed that any solution u∈C2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u|∂Ω=+∞ satisfies limd(x)→0u(x)Z(dμ(x))=[(2+σ)(2+ρ+σ)2c0(2+ρ)]1/ρ, where Ω is a bounded domain with smooth boundary in RN; limd(x)→0k(x)dσ(x)=c0, −2<σ, c0>0, μ=2+σ2; g∈C1[0,∞), g⩾0 and g(s)s is increasing on (0,∞), there exists ρ>0 such that lims→∞g′(sξ)g′(s)=ξρ, ∀ξ>0, ∫Z(s)∞dt2G(t)=s, G(t)=∫0tg(s)ds

The semi-discrete AKNS system: Conservation laws, reductions and continuum limits

In this paper, the semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy is shown in spirit composed by the Ablowitz-Ladik flows under certain combinations. Furthermore, we derive its explicit Lax pairs and infinitely many conservation laws, which are non-trivial in light of continuum limit. Reductions of the semi-discrete AKNS hierarchy are investigated to include the semi-discrete Korteweg-de Vries (KdV), the semi-discrete modified KdV, and the semi-discrete nonlinear Schr\"odinger hierarchies as its special cases. Finally, under the uniform continuum limit we introduce in the paper, the above results of the semi-discrete AKNS hierarchy, including Lax pairs, infinitely many conservation laws and reductions, recover their counterparts of the continuous AKNS hierarchy

Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space $B_{2,1}^{\frac{1}{2}}(\mathbb{T})$. Our results shows that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the $L^1$-integrability of the variable coefficients $\alpha(t)$ and $\gamma(t)$, but nothing to do with the shape or smoothness of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms.} Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters $\alpha(t)$ an $\gamma(t)$

UniBrain: Unify Image Reconstruction and Captioning All in One Diffusion Model from Human Brain Activity

Image reconstruction and captioning from brain activity evoked by visual stimuli allow researchers to further understand the connection between the human brain and the visual perception system. While deep generative models have recently been employed in this field, reconstructing realistic captions and images with both low-level details and high semantic fidelity is still a challenging problem. In this work, we propose UniBrain: Unify Image Reconstruction and Captioning All in One Diffusion Model from Human Brain Activity. For the first time, we unify image reconstruction and captioning from visual-evoked functional magnetic resonance imaging (fMRI) through a latent diffusion model termed Versatile Diffusion. Specifically, we transform fMRI voxels into text and image latent for low-level information and guide the backward diffusion process through fMRI-based image and text conditions derived from CLIP to generate realistic captions and images. UniBrain outperforms current methods both qualitatively and quantitatively in terms of image reconstruction and reports image captioning results for the first time on the Natural Scenes Dataset (NSD) dataset. Moreover, the ablation experiments and functional region-of-interest (ROI) analysis further exhibit the superiority of UniBrain and provide comprehensive insight for visual-evoked brain decoding

Research on Context-Awareness Mobile SNS Recommendation Algorithm

Although patterns of human activity show a large degree of freedom, they exhibit structural patterns subjected by geographic and social constraints. Aiming at various problems of personalized recommendation in mobile networks, a social network recommendation algorithm is proposed with a variety of context-aware information and combined with a series of social network analysis methods.Based on geographical location and temporal information, potential social relations among users are mined deeply to find the most similar set of users for the target user, then recommendations are carried out incorporating with social relations of the mobile users to effectively solve the problem of recommendation precision. The above study can not only help LBSN designers and developers to better understand their users and grasp their want, but also help to refine the design of their system to provide users with more appropriate applications and services.The experimental results on the real-world dataset verify the feasibility and effectiveness of the proposed algorithm, and it has higher prediction accuracy compared with existing recommendation algorithms

Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space B2,11/2(T). Our results show that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L1-integrability of the variable coefficients α(t) and γ(t), but nothing to do with the shape of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α(t) an γ(t)