Abstract
The main equation of this paper is the special case of equation studied by Il’in and Oleinik for single model equation with convex nonlinearity [4], by considering \(f(u)=u^m\) and nonlinear diffusion, for \(m>0\). We are interested in the stability of the degenerate advection-diffusion equation by dealing with the singular term when \(u_+=0\). We first transform the original equation into the traveling waves by using the ansatz transformation. The weighted energy estimates of the transformed equation are then established, where the aim of this weighted function is to avoid the singular term when \(u_+ = 0\). At the final stage, the stability of traveling waves is shown based on the weighted energy estimates and appropriate perturbations.
Similar content being viewed by others
References
Buckmire, R., McMurtry, K., Mickens, R.E.: Numerical studies of a nonlinear heat equation with square root reaction term. Numerical Methods for Partial Differential Equations 25, 598–609 (2009)
Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhauser, Boston (1997)
Ghani, M., Li, J., Zhang, K.: Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B 26, 6253–6265 (2021)
Il’in, A.M., Oleinik, O.A.: Asymptotic behavior of solutions of the Cauchy problem for certain quasilinear equationsfor large time (in Russian). Mat. Sb. 51, 191–216 (1960)
Hu, Y.: Asymptotic nonlinear stability of traveling waves to a system of coupled Burgers equations. Journal of Mathematical Analysis and Applications. 397, 322–333 (2013)
Jin, H.Y., Li, J., Wang, Z.A.: Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity. J. Differential Equations. 255, 193–219 (2013)
Jordan, P.M.: A Note on the Lambert W-function: Applications in the mathematical and physical sciences. Contemporary Mathematics. 618, 247–263 (2014)
Li, J., Li, T., Wang, Z.A.: Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity. Math. Models Methods Appl. Sci. 24, 2819–2849 (2014)
Li, J., Wang, Z.A.: Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space. J. Differential Equations. 268, 6940–6970 (2020)
Li, T., Wang, Z.A.: Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis. SIAM J. Appl. Math. 70, 1522–1541 (2010)
Kawashima, S., Matsumura, A.: Stability of shock profiles in viscoelasticity with non-convex constitutive relations. Comm. Pure Appl. Math. 47, 1547–1569 (1994)
Li, T., Wang, Z.A.: Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J. Differential Equations 250, 1310–1333 (2011)
Martinez, V.R., Wang, Z.-A., Zhao, K.: Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology. Indiana Univ. Math. J. 67, 1383–1424 (2018)
Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan. J. Appl. Math. 2, 17–25 (1985)
Mickens, R.E.: Exact finite difference scheme for an advection equation having square-root dynamics. Journal of Difference Equations and Applications. 14, 1149–1157 (2008)
Mickens, R.E.: Wave front behavior of traveling waves solutions for a PDE having square-root dynamics. Mathematics and Computers in Simulation. 82, 1271–1277 (2012)
Mickens, R.E., Oyedeji, K.: Traveling wave solutions to modified Burgers and diffusionless Fisher PDE’s. Evolution Equations and Control Theory. 8, 139–147 (2019)
Nishida, T.: Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Math’ematiques d’Orsay 78–02. D’epartement de Math’ematique. Universit’e de ParisSud. Orsay, France (1978)
Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. Math. 22, 312–355 (1976)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1974)
Acknowledgements
Author would like to thank to the reviewer for the valuable comments and suggestions which helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ghani, M. Asymptotic Stability of Singular Traveling Waves to Degenerate Advection-Diffusion Equations Under Small Perturbation. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00602-1
Accepted:
Published:
DOI: https://doi.org/10.1007/s12591-022-00602-1