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Global solution and smoothing effect for a non-local regularization of a hyperbolic equation. (English) Zbl 1036.35123
The authors study the Cauchy problem for the equation \(u_t+f(u)_x+g(u)=0\), where \(f\in C^{\infty} (\mathbb{R})\), \(g\in L^{\infty }(\mathbb{R})\), \(g\) is a non-local operator defined through the Fourier transform \(F(g(u))(p)=| p| ^{\lambda }F(u)(p)\) with \(\lambda \in (1,2\rangle \). The existence and uniqueness of a weak solution, the regularizing effect and the maximum principle is proved.

MSC:
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35L45 Initial value problems for first-order hyperbolic systems
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