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Odd solutions to perturbed conservation laws. (English) Zbl 0896.35082
This paper treats the existence of odd solutions of the Cauchy problem for a perturbation of a conservation law. That is, we find a function \(u= u(t,x)\) satisfying the differential equation \[ u_t+ \text{div}(f(u))+ g(u)F(x)= 0\quad\text{for }t>0,\;x\in\mathbb{R}^n, \] and the initial condition \(u(0, x)=\phi(x)\) for \(x\in\mathbb{R}^n\). This paper was inspired by the paper of M. E. Schonbek [SIAM J. Math. Anal. 15, 1125-1139 (1984; Zbl 0567.35060)] who proved existence of solutions to singular scalar conservation laws of the form \[ u_t+ f(u)_x+ {g(u)\over | x|}= 0 \] by regularizing the equation and taking a singular limit using the theory of compensated compactness. We provide a semigroup approach to M. E. Schonbek’s work.

35L65 Hyperbolic conservation laws
35D05 Existence of generalized solutions of PDE (MSC2000)
35B25 Singular perturbations in context of PDEs
47H06 Nonlinear accretive operators, dissipative operators, etc.