Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis
The authors study a nonlinear singular chemotaxis parabolic system (with a logarithmic sensitivity function) modeling the onset of tumor angiogenesis in two and three dimensional space. By a Cole-Hopf type transformation, the singular chemotaxis system is converted into a nonsingular hyperbolic system \[ \begin{aligned} u_t -\Delta u & =\nabla\cdot(u\boldsymbol{v}), \\ \boldsymbol{v}_t-\varepsilon\Delta \boldsymbol{v} & =\nabla(-\varepsilon|\boldsymbol{v}|^2+u). \end{aligned} \] The global existence, asymptotic decay rates and diffusion convergence rate of solutions (when \(\varepsilon\to 0\)) with the initial data which are small perturbations of constant states are shown by the method of energy estimates. An interesting feature of those results is revealing a hidden interactive dissipation structure in the system by which the energy dissipation is established.
scientific article; zbMATH DE number 6517070
scientific article