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Global existence, singularities and ill-posedness for a nonlocal flux. (English) Zbl 1186.35002
The authors consider the following nonlocal equations:
\[ \partial_t u+ (u{\mathcal H}u)_x=0 \quad\text{on }\mathbb{R}\times\mathbb{R}^+\tag{1} \]
\[ u(x,0)= u_0(t),\tag{2} \] where \({\mathcal H}u\) is the Hilbert transform of the function \(u\), which is defined by the expression
\[ ({\mathcal H}u)(x)=\frac 1\pi \text{ p.v }\int\frac{u(y)}{x-y}\,dy. \tag{3} \] The authors present global existence, local existence, blow-up in finite time and ill-posedness depending on the sign of the initial data for classical solutions of (1)–(2).

MSC:
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35R25 Ill-posed problems for PDEs
35A20 Analyticity in context of PDEs
35B44 Blow-up in context of PDEs
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