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Heteroclinic orbits between rotating waves in hyperbolic balance laws. (English) Zbl 0951.35079

Summary: We deal with the large-time behavior of scalar hyperbolic conservation laws with source terms \[ u_t+ f(u)_x= g(u), \] which are often called hyperbolic balance laws. H. Fan and J. K. Hale [Trans. Am. Math. Soc. 347, 1239-1254 (1995; Zbl 0831.35103)] proved the existence of a global attractor \({\mathcal A}_0\) for this equation with \(x\in S^1\). \({\mathcal A}_0\) consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.

MSC:

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0831.35103
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References:

[1] DOI: 10.2307/2154808 · Zbl 0831.35103 · doi:10.2307/2154808
[2] DOI: 10.1007/BF00383219 · Zbl 0807.35085 · doi:10.1007/BF00383219
[3] DOI: 10.1512/iumj.1977.26.26088 · Zbl 0377.35051 · doi:10.1512/iumj.1977.26.26088
[4] DOI: 10.1090/S0002-9947-1988-0940217-X · doi:10.1090/S0002-9947-1988-0940217-X
[5] DOI: 10.1006/jdeq.1996.3223 · Zbl 0879.35105 · doi:10.1006/jdeq.1996.3223
[6] DOI: 10.1007/BF01194015 · Zbl 0814.35073 · doi:10.1007/BF01194015
[7] Härterich, Equilibrium solutions of viscous scalar balance laws with a convex flux (1997)
[8] Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves (1973) · Zbl 0268.35062 · doi:10.1137/1.9781611970562
[9] Lyberopoulos, Proc. R. Soc. Edinb. A 124 pp 589– (1994) · Zbl 0806.35111 · doi:10.1017/S0308210500028791
[10] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803 · doi:10.1002/cpa.3160100406
[11] DOI: 10.1070/SM1970v010n02ABEH002156 · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[12] DOI: 10.1006/jdeq.1997.3342 · Zbl 0897.35041 · doi:10.1006/jdeq.1997.3342
[13] Matano, Discr. Cont. Dyn. Systems 3 pp 1– (1997) · Zbl 0995.37015 · doi:10.1007/BF02471759
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