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A note on the blow-up of a nonlinear evolution equation with nonlocal coefficients. (English) Zbl 0867.35011

We are interested in the study of blow-up present in solutions of the initial value problem \[ \partial_tu+\sum^n_{i=1}(-\Delta)^{-\beta_i\over 2}u\cdot\partial_{x_i}u=0,\;x\in\mathbb{R}^n,\;t\in\mathbb{R},\;\beta_i\in[0,n),\quad u(x,0)=u_0(x),\tag{1} \] with \[ (-\Delta)^{-\beta/2}f(x)=c_{\beta,n}\int_{\mathbb{R}^n} {f(y)\over|x-y|^{n-\beta}} dy.\tag{R} \] In [J. Math. Anal. Appl. 123, 104-113 (1987; Zbl 0622.35012)], the second author showed that for \(u_0\in C^\infty_0(\mathbb{R}^n)\) and \(\beta_i\in[1,n)\), this problem has a unique global solution.
In this note, we show that at least for \(n=1\) and \(\beta\in(0,1)\), the solutions of (1) with \(u_0\in C^\infty_0(\mathbb{R})\) may blow up in finite time: There exist \(u_0\in C^\infty_0(\mathbb{R})\) and \(T^*>0\) such that the solution \(u(x,t)\) of (1) with initial data \(u_0\) satisfies \[ \lim_{t\to T^*} |\partial_xu(\cdot,t)|_{L^\infty}=\infty. \]

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35S10 Initial value problems for PDEs with pseudodifferential operators
45K05 Integro-partial differential equations

Keywords:

blow-up

Citations:

Zbl 0622.35012
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