Global solution and smoothing effect for a non-local regularization of a hyperbolic equation. (English) Zbl 1036.35123

Droniou, J.; Gallouët, T.; Vovelle, J.

J. Evol. Equ. 3, No. 3, 499-521 (2003).

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.

Reviewer: Marie Kopáčková (Praha)

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

Keywords:

Cauchy problem; parabolic regularization; weak solution