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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.

MSC:
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.
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