Exploding solutions for a nonlocal quadratic evolution problem
The authors consider the following evolution equation on \(\mathbb{R}^{2}\): \[ \begin{cases} \partial _t u=-\nu(-\Delta)^{\alpha /2}u-\nabla\cdot (uB(u)),\,0<\alpha<2,\\u(0,x)=u_{0}(x), \end{cases}\tag{1} \] where \(\nu\geq 0\) is a viscosity coefficient, \((-\Delta)^{\alpha /2}\) is the fractional Laplacian defined via the Fourier transform, and \(B\) is a linear integral operator given by \[ B(u)(x)=\mu\int_{\mathbb{R}^{2}}\frac{x-y}{| x-y| ^{2}}u(y)\,dy, \] with \(\mu=\pm 1\). Equation (1) arises in physical processes involving diffusion and interaction of particles, see for instance \textit{M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel} and \textit{S. C. Venkataramani} [Nonlinearity 12, No. 4, 1071--1098 (1999; Zbl 0942.35018)]. The authors prove first a local existence and uniqueness result of solutions to (1). Namely, if \(\mu=\pm 1, \nu\geq 0\) and under the assumption that the initial data \(u_{0}\in H_{x}^{s}(\mathbb{R}^{2})\cap L_{x}^{q}(\mathbb{R}^{2})\) (for \(s>3, 1<q<2\)) it is proven that there exists a \(T>0\) (depending on the size of the initial data \(u_{0}\)) and a unique solution of (1) in \(C([0,T),H_{x}^{s}(\mathbb{R}^{2})\cap L_{x}^{q}(\mathbb{R}^{2}))\). Moreover, it is proved that either the solution is global (i.e., \(T=+\infty\)) or otherwise, if \(T<\infty\), one obtains the following blow-up result: \[ \lim_{t\to T}\int_{0}^{T}\| u(s) \|_{L_{x}^{\infty}(\mathbb{R}^{2})}\,ds=+\infty. \] In the repulsive case (corresponding to \(\mu=1\)) a global in time well-posedness result is obtained. Choosing \(\mu=-1,\nu>0\) and the initial data \(u_{0}\in H_{x}^{s}(\mathbb{R}^{2})\cap L_{x}^{1}(\mathbb{R}^{2})\) with \(s>3\) and \(u_{0}\geq 0\) the authors prove that there exists an open set of parameters \(\delta, a,b\) such that if \(u_{0}\) belongs to a set of admissible functions \(A_{\delta, a,b}\), then there exists a \(T<\infty\), such that \[ \lim_{t\to T}\| u(t)\| _{L_{x}^{\infty}(\mathbb{R}^{2})}=+\infty, \] i.e., the solution of equation (1) blows up in finite time.
scientific article; zbMATH DE number 5708823