Qualitative behavior of conservation laws with reaction term and nonconvex flux
Author:
Corrado Mascia
Journal:
Quart. Appl. Math. 58 (2000), 739-761
MSC:
Primary 35L65; Secondary 35L60, 74J30
DOI:
https://doi.org/10.1090/qam/1788426
MathSciNet review:
MR1788426
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Abstract: The aim of the paper is to study qualitative behavior of solutions to the equation \[ \frac {{\partial u}}{{\partial t}} + \frac {{\partial f\left ( u \right )}}{{\partial x}} = g\left ( u \right ) ,\] where $\left ( x, t \right ) \in \mathbb {R} \times {\mathbb {R}_ + }, u = u\left ( x, t \right ) \in \mathbb {R}$. The main new feature with respect to previous works is that the flux function $f$ may have finitely many inflection points, intervals in which it is affine, and corner points. The function $g$ is supposed to be zero at 0 and 1, and positive in between.
- C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behaviour of solutions, Nonlinear analysis and mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, Pitman, London, 1977, pp. 1–58. Res. Notes in Math., No. 17. MR 0481581
- C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI https://doi.org/10.1512/iumj.1977.26.26088
- C. M. Dafermos, Large time behavior of solutions of hyperbolic balance laws, Bull. Soc. Math. Grèce (N.S.) 25 (1984), 15–29. MR 815565
- C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), no. 3-4, 201–239. MR 785530, DOI https://doi.org/10.1017/S0308210500014256
- Hai Tao Fan and Jack K. Hale, Attractors in inhomogeneous conservation laws and parabolic regularizations, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1239–1254. MR 1270661, DOI https://doi.org/10.1090/S0002-9947-1995-1270661-9
- Hai Tao Fan and Jack K. Hale, Large time behavior in inhomogeneous conservation laws, Arch. Rational Mech. Anal. 125 (1993), no. 3, 201–216. MR 1245070, DOI https://doi.org/10.1007/BF00383219
- Brian T. Hayes, Stability of solutions to a destabilized Hopf equation, Comm. Pure Appl. Math. 48 (1995), no. 2, 157–166. MR 1319699, DOI https://doi.org/10.1002/cpa.3160480204
A. Kolmogorov, I. Petrovsky, and N. Piscounov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à, un problème biologique, Boll. Univ. Moscow Ser. Internat. Sec. A 1, 1–25 (1937)
S. N. Kruz̆kov, First order quasilinear equations in several independent variables, Mat. Sb. 51, 99–128 (1960) (Russian); English transl. in Amer. Math. Soc. Transl. (2) 42, 199–231 (1964)
S. N. Kruz̆kov and N. S. Petrosjan, Asymptotic behaviour of the solutions of the Cauchy problem for nonlinear first order equations, Uspekhi Mat. Nauk 42, 3–40 (1987) (Russian); English transl. in Russian Math. Surveys 42, 1–47 (1987)
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI https://doi.org/10.1002/cpa.3160100406
- Tai Ping Liu, Invariants and asymptotic behavior of solutions of a conservation law, Proc. Amer. Math. Soc. 71 (1978), no. 2, 227–231. MR 500495, DOI https://doi.org/10.1090/S0002-9939-1978-0500495-7
- Athanasios N. Lyberopoulos, A Poincaré-Bendixson theorem for scalar balance laws, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 3, 589–607. MR 1286920, DOI https://doi.org/10.1017/S0308210500028791
- Athanasios N. Lyberopoulos, Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation, Quart. Appl. Math. 48 (1990), no. 4, 755–765. MR 1079918, DOI https://doi.org/10.1090/qam/1079918
- Athanasios N. Lyberopoulos, Large-time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field, J. Differential Equations 99 (1992), no. 2, 342–380. MR 1184059, DOI https://doi.org/10.1016/0022-0396%2892%2990026-J
- Corrado Mascia, Continuity in finite time of entropy solutions for nonconvex conservation laws with reaction term, Comm. Partial Differential Equations 23 (1998), no. 5-6, 913–931. MR 1632756, DOI https://doi.org/10.1080/03605309808821372
- C. Mascia, Travelling wave solutions for a balance law, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 3, 567–593. MR 1453282, DOI https://doi.org/10.1017/S0308210500029917
- Corrado Mascia and Carlo Sinestrari, The perturbed Riemann problem for a balance law, Adv. Differential Equations 2 (1997), no. 5, 779–810. MR 1751427
- R. Natalini and A. Tesei, On a class of perturbed conservation laws, Adv. in Appl. Math. 13 (1992), no. 4, 429–453. MR 1190121, DOI https://doi.org/10.1016/0196-8858%2892%2990020-W
O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, Amer. Math. Soc. Transl. Ser. 2 33, 285–290 (1964)
- Carlo Sinestrari, Asymptotic profile of solutions of conservation laws with source, Differential Integral Equations 9 (1996), no. 3, 499–525. MR 1371704
- Carlo Sinestrari, Large time behaviour of solutions of balance laws with periodic initial data, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 1, 111–131. MR 1322204, DOI https://doi.org/10.1007/BF01194015
- Carlo Sinestrari, The Riemann problem for an inhomogeneous conservation law without convexity, SIAM J. Math. Anal. 28 (1997), no. 1, 109–135. MR 1427730, DOI https://doi.org/10.1137/S003614109427446X
- Carlo Sinestrari, Instability of discontinuous traveling waves for hyperbolic balance laws, J. Differential Equations 134 (1997), no. 2, 269–285. MR 1432096, DOI https://doi.org/10.1006/jdeq.1996.3223
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR 1297766
- H. F. Weinberger, Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 5, 407–425 (English, with French summary). MR 1138530, DOI https://doi.org/10.1016/S0294-1449%2816%2930284-0
C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behavior of solutions, in Nonlinear Analysis and Mechanics, Vol. 1 (R. J. Knops, ed.), Research Notes in Mathematics 17, Pitman, 1977, pp. 1–58
C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. 26, 1097–1119 (1977)
C. M. Dafermos, Large time behavior of solutions of hyperbolic balance laws, Bull. Soc. Math. Grèce (N.S.) 25, 15–29 (1984)
C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99, 201–239 (1985)
H. Fan and J. K. Hale, Attractors in inhomogeneous conservation laws and parabolic regularizations, Trans. Amer. Math. Soc. 347, 1239–1254 (1995)
H. Fan and J. K. Hale, Large-time behavior in inhomogeneous conservation laws, Arch. Rational Mech. Anal. 125, 201–216 (1993)
B. T. Hayes, Stability of solutions to a destabilized Hopf equation, Comm. Pure Appl. Math. 48, 157–166 (1995)
A. Kolmogorov, I. Petrovsky, and N. Piscounov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à, un problème biologique, Boll. Univ. Moscow Ser. Internat. Sec. A 1, 1–25 (1937)
S. N. Kruz̆kov, First order quasilinear equations in several independent variables, Mat. Sb. 51, 99–128 (1960) (Russian); English transl. in Amer. Math. Soc. Transl. (2) 42, 199–231 (1964)
S. N. Kruz̆kov and N. S. Petrosjan, Asymptotic behaviour of the solutions of the Cauchy problem for nonlinear first order equations, Uspekhi Mat. Nauk 42, 3–40 (1987) (Russian); English transl. in Russian Math. Surveys 42, 1–47 (1987)
P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, 537–566 (1957)
T. P. Liu, Invariants and asymptotic behavior of solutions of a conservation law, Proc. Amer. Math. Soc. 71, 227–231 (1978)
A. N. Lyberopoulos, A Poincaré-Bendixson theorem for scalar balance laws, Proc. Roy Soc. Edinburgh Sect. A 124, 589–607 (1994)
A. N. Lyberopoulos, Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation, Quart. Appl. Math. 48, 755–765 (1990)
A. N. Lyberopoulos, Large time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field, J. Differential Equations 99, 342–380 (1992)
C. Mascia, Continuity in finite time of entropy solutions for nonconvex conservation laws with reaction term, Comm. Partial Differential Equations 23, 913–931 (1998)
C. Mascia, Travelling wave solutions for a balance law, Proc. Roy. Soc. Edinburgh Sect. A 127, 567–593 (1997)
C. Mascia and C. Sinestrari, The perturbed Riemann problem for a balance law, Advances in Differential Equations 2, 779–810 (1997)
R. Natalini and A. Tesei, On a class of perturbed conservation laws, Adv. in Appl. Math. 13, 429–453 (1992)
O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, Amer. Math. Soc. Transl. Ser. 2 33, 285–290 (1964)
C. Sinestrari, Asymptotic profile of solutions of conservation laws with source, Differential Integral Equations 9, 499–525 (1966)
C. Sinestrari, Large time behaviour of solutions of balance laws with periodic initial data, NoDEA Nonlinear Differential Equations Appl. 2, 111–131 (1995)
C. Sinestrari, The Riemann problem for an inhomogeneous conservation law without convexity, SIAM J. Math. Anal. 28, 109–135 (1997)
C. Sinestrari, Instability of discontinuous traveling waves for hyperbolic balance laws, J. Differential Equations 134, 269–285 (1997)
J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983
A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs 140, American Mathematical Society, Providence, RI, 1994
H. F. Weinberger, Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity, Ann. Inst. H. Poincaré (Anal. Non Linéaire) 7, 407–425 (1990)
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