Cauchy problems for Keller-Segel type time-space fractional diffusion equation

Abstract

This paper investigates Cauchy problems for nonlinear fractional time-space generalized Keller-Segel equation ${}_0^c{D}_t^{\beta }\rho +(-△{)}^{\frac{\alpha }{2}}\rho +\nabla \cdot (\rho B(\rho ))=0$, where Caputo derivative ${}_0^c{D}_t^{\beta }\rho$ models memory effects in time, fractional Laplacian $(-△{)}^{\frac{\alpha }{2}}\rho$ represents Lévy diffusion and $B(\rho )=-s_{n,\gamma }{\int }_{R^n}\frac{x-y}{|x-y{|}^{n-\gamma +2}}\rho (y)dy$ is the general potential with a singular kernel which takes into account the long rang interaction. We first establish $L^r-L^q$ estimates and weighted estimates of the fundamental solutions $(P(x,t),Y(x,t))$ (or equivalently, the solution operators $(S_{\alpha }^{\beta }(t),T_{\alpha }^{\beta }(t))$). Then, we prove the existence and uniqueness of the mild solutions when initial data are in $L^p$ spaces, or the weighted spaces. Similar to Keller-Segel equations, if the initial data are small in critical space $L^{p_c}({\mathbb{R}}^n)$ ($p_c=\frac{n}{\alpha +\gamma -2}$), we construct the global existence. Furthermore, we prove the $L^1$ integrability and integral preservation when the initial data are in $L^1({\mathbb{R}}^n)\cap L^p({\mathbb{R}}^n)$ or $L^1({\mathbb{R}}^n)\cap L^{p_c}({\mathbb{R}}^n)$. Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.

References in corpus (3)

• Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations
• A generalized definition of Caputo derivatives and its application to fractional ODEs
• Some compactness criteria for weak solutions of time fractional PDEs