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May 2018, 17(3): 959-985. doi: 10.3934/cpaa.2018047

## Stability of traveling waves of models for image processing with non-convex nonlinearity

 Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Tong Li

Received  September 2017 Revised  September 2017 Published  January 2018

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.

Citation: Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047
##### References:
 [1] M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362. [2] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424. [3] J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698. [4] J. Goodman, A. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268. [5] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. [6] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. [7] S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. [8] J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370. [9] A. Kurganov, D. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473. [10] T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528. [11] T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270. [12] T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17. [13] T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. [14] T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. [15] T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106. [16] D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217. [17] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. [18] M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226. [19] M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068. [20] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978). [21] M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132. [22] J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358. [23] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639. [24] J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90. [25] A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021. [26] Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371. [27] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730.

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##### References:
 [1] M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362. [2] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424. [3] J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698. [4] J. Goodman, A. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268. [5] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. [6] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. [7] S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. [8] J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370. [9] A. Kurganov, D. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473. [10] T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528. [11] T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270. [12] T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17. [13] T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. [14] T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118. [15] T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106. [16] D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217. [17] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. [18] M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226. [19] M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068. [20] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978). [21] M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132. [22] J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358. [23] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639. [24] J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90. [25] A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021. [26] Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371. [27] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730.
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