Blowup in infinite time of radial solutions for a parabolic-elliptic system in high-dimensional Euclidean spaces
Given two integers \(N\geq 11\) and \(J\geq 0\), define \[ \nu=\frac{-N-2 + \sqrt{(N-10)(N-2)}}{4} \;\;\;\text{ and }\;\;\; a_J = \frac{N+2J+6\nu+4}{-2\nu-2}\;. \] A global radially symmetric solution \((U,V)\) to the Smoluchowski-Poisson system \[ \partial_t U = \nabla\cdot\left( \nabla U - U \nabla V \right)\;, \qquad -\Delta V = U \;\;\;\text{ in }\;\;\; (0,\infty)\times{\mathbb R}^N \] is constructed with the following properties: \[ \sup_{(x,t)\in {\mathbb R}^N\times (0,\infty)}{\left\{ |x|^2 u(x,t) \right\}}<\infty\;, \qquad c t^{a_J} \leq \|U(t)\|_\infty \leq C t^{a_J} \] for \(t\geq T\), for some constants \(C>c>0\) and \(T>0\). Furthermore, if \(U_0\) is a radially symmetric, nonnegative, and continuous initial condition satisfying \(U_0(x)\leq 2(N-2)/|x|^2\) for \(x\in {\mathbb R}^N\), the corresponding solution \((U,V)\) to the Smoluchowski-Poisson system exists globally with \(\|U(t)\|_\infty \leq C t^{a_0}\) for \(t\) large enough. Similar results were already obtained for \(\partial_t u = \Delta u + u^p\) in [\textit{N.~Mizoguchi}, J. Differ. Equations 227, 652--669 (2006; Zbl 1102.35049)] and the same strategy of proof is used here.
scientific article; zbMATH DE number 5532901
scientific article