Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term
Global existence of non-negative solutions \((u,v)\) to the degenerate chemotaxis system \[ \begin{alignedat}{2} \partial_t u & = \nabla \cdot (\nabla u^m - u^{q-1} \nabla v), &&\quad (t,x)\in (0,\infty)\times{\mathbb R}^N,\\ \tau \partial_t v & = \Delta v + u - v, &&\quad (t,x)\in (0,\infty)\times{\mathbb R}^N,\end{alignedat} \] is investigated according to the values of \(m\geq 1\), \(q\geq 2\), and \(\tau\in\{0,1\}\). When \(\tau=1\), \(q\in [2,m]\) or \(\tau=0\), \(q\in [2,m+(2/N))\) (with the additional constraint \(q\in [2,m+1)\) if \(N=1\)), global existence and boundedness of a non-negative weak solution \((u,v)\) are established without any size restriction on the initial data. In addition, if \(\tau=0\) and \(q\geq m+(2/N)\), there is a global non-negative weak solution \((u,v)\) if the \(L^{N(q-m)/2}\)-norm of \(u(0)\) is sufficiently small. In that case, the \(L^p\)-norm of \(u(t)\) decays at the same temporal rate as the solution \(z\) to the porous medium equation \(\partial_t z = \Delta z^m\) for \(p\in (1,\infty)\). The critical exponent \(m+(2/N)\) also arises in the study of the blow-up of non-negative solutions to \(\partial_t z = \Delta z^m + z^q\). For \(\tau=0\), these results have been improved in [\textit{S.~Luckhaus} and \textit{Y.~Sugiyama}, M2AN Math. Model. Numer. Anal. 40, 597--621 (2006)].
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