Asymptotic behavior of solutions to quasilinear hyperbolic equations with nonlinear damping
Authors:
Hailiang Li and Katarzyna Saxton
Journal:
Quart. Appl. Math. 61 (2003), 295-313
MSC:
Primary 35L60; Secondary 35B40, 35L65, 74D10, 74F05, 74H40
DOI:
https://doi.org/10.1090/qam/1976371
MathSciNet review:
MR1976371
Full-text PDF Free Access
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Additional Information
Coleman, B. D., and Neumann, D. C., Implication of a nonlinearity in the theory of second sound in solids, Phys. Rev. B 32, 1492-1498 (1988)
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- L. Hsiao, Quasilinear hyperbolic systems and dissipative mechanisms, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR 1640089
- Ling Hsiao and Tai-Ping Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992), no. 3, 599–605. MR 1145602
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Hsiao, L., and Serre, D., Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chin. Ann. of Math. 16B, 431–444 (1995)
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- Hailiang Li, The asymptotic behavior of solutions to the damped $P$-system with boundary effects, J. Partial Differential Equations 12 (1999), no. 4, 357–368. MR 1745407
Li, T. T., and Yu, W. C., Boundary value problem for quasilinear hyperbolic systems, Duke Univ. Math. Ser. V 1985.
Luo, T., and Yang, T., The interaction of elementary waves for the isentropic Euler equation with frictional damping, preprint.
Luo, T., and Yang, T., Global existence of weak entropy solutions for damped p-system, to appear.
- Pierangelo Marcati and Ming Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Quart. Appl. Math. 58 (2000), no. 4, 763–784. MR 1788427, DOI https://doi.org/10.1090/qam/1788427
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- Kenji Nishihara and Tong Yang, Boundary effect on asymptotic behaviour of solutions to the $p$-system with linear damping, J. Differential Equations 156 (1999), no. 2, 439–458. MR 1705375, DOI https://doi.org/10.1006/jdeq.1998.3598
- Kenji Nishihara, Weike Wang, and Tong Yang, $L_p$-convergence rate to nonlinear diffusion waves for $p$-system with damping, J. Differential Equations 161 (2000), no. 1, 191–218. MR 1740362, DOI https://doi.org/10.1006/jdeq.1999.3703
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Saxton, K., and Saxton, R., Nonlinearity and memory effects in low temperature heat propagation, Arch. Mech. 52, 127–142 (2000)
- Denis Serre and Ling Xiao, Asymptotic behavior of large weak entropy solutions of the damped $P$-system, J. Partial Differential Equations 10 (1997), no. 4, 355–368. MR 1486716
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Wang, J-.H., and Li, C. Z., Global regularity and formation of singularities of solution for quasilinear hyperbolic system with dissipation, Chinese Annals Math. 9A, 509–523 (1988)
- Tong Yang, Changjiang Zhu, and Huijiang Zhao, Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1311–1324. MR 1489438, DOI https://doi.org/10.1017/S0308210500027074
- Huijiang Zhao, Convergence to strong nonlinear diffusion waves for solutions of $p$-system with damping, J. Differential Equations 174 (2001), no. 1, 200–236. MR 1844529, DOI https://doi.org/10.1006/jdeq.2000.3936
Coleman, B. D., and Neumann, D. C., Implication of a nonlinearity in the theory of second sound in solids, Phys. Rev. B 32, 1492-1498 (1988)
Dafermos, C. M., A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys. 46 Special Issue, 294–307 (1995)
Hsiao, L., “Quasilinear Hyperbolic System and Dissipative Mechanisms", World Scientific, Singapore, 1997.
Hsiao, L. and Liu, T.-P., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143, 599–605 (1992)
Hsiao, L., and Liu, T.-P., Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chin. Ann. of Math. 14B, 465–480 (1993)
Hsiao, L., and Luo, T., Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations 125, 329–365 (1996)
Hsiao, L., and Luo, T., Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping, Quart. Appl. Math. 56, 173–198 (1998)
Hsiao, L., and Pan, P. H., Initial boundary value problem for the system of compressible adiabatic flow through porous media, J. Differential Equations 159, 280–305 (1999)
Hsiao, L., and Serre, D., Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 27, 70–77 (1996)
Hsiao, L., and Serre, D., Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chin. Ann. of Math. 16B, 431–444 (1995)
Hsiao, L., and Tang, S. Q., Construction and qualitative behavior of solutions for a system of nonlinear hyperbolic conservation laws with damping, Quart. Appl. Math. LIII, 487–505 (1995)
Hsiao, L., and Tang, S. Q., Construction and qualitative behavior of solutions of perturbated Riemann problem for the system of one-dimensional isentropic flow with damping, J. Differential Equations 123, 480–503 (1995)
Li, H. L., The asymptotic behavior of solutions to the damped $p$-system with boundary effects, J. Partial Diff. Eqns. 12, 357–368 (1999)
Li, T. T., and Yu, W. C., Boundary value problem for quasilinear hyperbolic systems, Duke Univ. Math. Ser. V 1985.
Luo, T., and Yang, T., The interaction of elementary waves for the isentropic Euler equation with frictional damping, preprint.
Luo, T., and Yang, T., Global existence of weak entropy solutions for damped p-system, to appear.
Marcati, P., and Mei, M., Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, preprint 1998
Nishida, T., Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathematiques D’Orsay, 78.02, Dept. de mathematique, Paris-Sud (1978)
Nishihara, K., Convergence rates to nonlinear diffusion waves for solutions of systems of hyperbolic conservation laws with damping, J. Differential Equations 131, 171–188 (1996)
Nishihara, K., Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations 137, 384–395 (1997)
Nishihara, K., and Yang, T., Boundary effect on asymptotic behavior of solutions to the $P$-system with damping, J. Differential Equations 156 (2), 439–458 (1999)
Nishihara, K., Wang, W. K., and Yang, T., ${L_p}$-convergence rate to nonlinear diffusion waves for $p$-system with damping, J. Differential Equations 161, 191–218 (2000)
Saxton, K., Saxton, R., and Kosinski, W., On second sound at the critical temperature, Quart. Appl. Math. 57, 723-740 (1999)
Saxton, K., and Saxton, R., Nonlinearity and memory effects in low temperature heat propagation, Arch. Mech. 52, 127–142 (2000)
Serre, D., and Xiao, L., Asymptotic behavior of large weak entropy solutions of the damped p-system, J. Partial Diff. Eqns. 10, 355–368 (1997)
Slemrod, M., Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rat. Mech. Anal. 76, 97–133 (1981)
Van Duyn, C. J., and Peletier, L. A., A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Anal., T.M.A. 1, 223 - 233 (1977)
Wang, J-.H., and Li, C. Z., Global regularity and formation of singularities of solution for quasilinear hyperbolic system with dissipation, Chinese Annals Math. 9A, 509–523 (1988)
Yang, T., Zhu C., and Zhao, H., Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Proc. Roy. Soc. Edin. 127A, 1311-1324 (1997)
Zhao, H., Convergence to strong nonlinear diffusion waves for solutions of $p$-system with damping, J. Differential Equations, submitted.
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