Abstract:

The research mainly focuses on the Cauchy problem of the parabolic-parabolic Keller-Segel equation with a Logistic source. The equation is as follows

where the constants $c \in \mathbb{R}, \tau>0, d \geq 2$, initial values $u_{0} \in \mathcal{P} \mathcal{M}^{d-2}\left(\mathbb{R}^{d}\right), \varphi_{0} \in \mathcal{S}\left(\mathbb{R}^{d}\right)$. When $c=0$, Biler-Boritchev-Brandolese proved that for any initial value equation of arbitrary size, a global solution exists in the case of the diffusion parameter $\tau \gg 1$. Using the fixed point lemma and the method of scale invariance to obtain the existence and uniqueness of mild solutions for the parabolic-parabolic Keller-Segel equation with a square Logistic source, under initial conditions $u_{0}$ and $\varphi_{0}$ have certain restrictive conditions (which depend on the parameter $\tau$).

Key words: Keller-Segel equations, parabolic equations, inviscid limit