Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

<p style='text-indent:20px;'>The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>

Long time existence of hyperbolic Ricci-Bourguignon flow on Riemannian Surfaces

We consider the hyperbolic Ricci-Bourguignon flow(HRBF) equation on Riemannian surfaces and we find a sufficient and necessary condition to this flow has global classical solution. Also, we show that the scalar curvature of the solution metric gij convergence to the flat curvature.

An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$ It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$ then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$ as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.