# American Institute of Mathematical Sciences

June  2017, 10(3): 605-624. doi: 10.3934/dcdss.2017030

## On a hyperbolic-parabolic mixed type equation

 School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China

* Corresponding author

Received  December 2015 Revised  October 2016 Published  February 2017

Fund Project: This work was partially supported by NSF of China (No. 11371297), NSF of Fujian Province in China (No. 2015J01592).

In this paper, the hyperbolic-parabolic mixed type equation
 $\frac{\partial u}{\partial t} = Δ A(u)+\text{div}(b(u)),\ \ (x,t)∈ Ω × (0,T),$
with the homogeneous boundary condition is considered. We find that only a part of the boundary condition is able to ensure the posedness of the solutions. By introducing a new kind of entropy solution matching the part boundary condition in a special way, we obtain the existence of the solution by the $BV$ estimate method, and establish the stability of the solutions by the Kruzkov bi-variables method.
Citation: Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030
##### References:
 [1] M. Bendahamane and K. H. Karlsen, Renarmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422. doi: 10.1137/S0036141003428937. Google Scholar [2] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial value problem for $u_{t}-Δ \varphi (u)=0$, J. Math.Pures et Appl., 58 (1979), 153-163. Google Scholar [3] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152. Google Scholar [4] G. Q. Chen and B. Perthame, Well-Posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. I. H. Poincare-AN, 20 (2003), 645-668. doi: 10.1016/S0294-1449(02)00014-8. Google Scholar [5] G. Q. Chen and E. DiBenedetto, Stability of entropy solutions to Cauchy problem for a class of nonlinear hyperbolic-parabolic equations, SIAM J.Math. Anal., 33 (2001), 751-762. doi: 10.1137/S0036141001363597. Google Scholar [6] B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equ., 151 (1999), 231-251. doi: 10.1006/jdeq.1998.3499. Google Scholar [7] G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984.Google Scholar [8] M. Escobedo, J. L. Vazquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.1090/S0002-9947-1994-1225573-2. Google Scholar [9] L. C. Evans, Weak convergence methods for nonlinear partial differential equations Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074. Google Scholar [10] G. Fichera, Sulle equazioni differenziatli lineari ellittico-paraboliche del secondo ordine, Atti Accd, Naz. Lincei. Mem, CI. Sci. Fis. Mat. Nat. Sez.1, 5 (1956), 1-30. Google Scholar [11] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in boundary problems, differential equations, Univ. of Wisconsin Press, Madison, Wis., 9 (1960), 97-120. Google Scholar [12] L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004.Google Scholar [13] F. R. Guarguaglini, V. Milišić and A. Terracina, A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions, J. Diff. Equ., 202 (2004), 183-207. doi: 10.1016/j.jde.2004.03.008. Google Scholar [14] K. H. Karlsen and N. H. Risebro, On the uniqueness of entropy solutions of nonlinear degenerate parabolic equations with rough coefficient, Discrete Contain. Dye. Sys., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081. Google Scholar [15] M. V. Keldyš, On certain cases of degeneration of elliptic type on the boundary of a domain, Dokl. Akad. Aauk SSSR, 77 (1951), 181-183. Google Scholar [16] K. Kobayasi and H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Diff. Equ., 252 (2012), 137-167. doi: 10.1016/j.jde.2011.09.008. Google Scholar [17] S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243. Google Scholar [18] Y. Li and Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. Diff. Equ., 252 (2012), 4719-4741. doi: 10.1016/j.jde.2012.01.027. Google Scholar [19] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formation of multidimensional conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [20] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184. Google Scholar [21] J. Málek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDES in Applied Mathematics and Mathematical Computation, 13. Chapman and HALL, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar [22] O. A. Oleinik and V. N. Samokhin, Mathematical Models in boundary, Layer Theorem Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 1999.Google Scholar [23] O. A. Oleinik, A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300. Google Scholar [24] O. A. Oleinik, Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140. Google Scholar [25] F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247. Google Scholar [26] G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450. Google Scholar [27] A. I. Vol'pert and S. I. Hudjaev, On the problem for quasilinear degenerate parabolic equations of second order (Russian), Mat. Sb., 78 (1969), 374-396. Google Scholar [28] A. I. Volpert, BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302. doi: 10.1070/SM1967v002n02ABEH002340. Google Scholar [29] A. I. Volpert and S. I. Hudjave, Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975.Google Scholar [30] Z. Wu and J. Yin, Some properties of functions in BV$_{x}$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 5 (1989), 395-422. Google Scholar [31] Z. Wu and J. Zhao, The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin.Ann. of Math., 4 (1983), 57-76. Google Scholar [32] Z. Wu and J. Zhao, Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin.Ann. of Math., 4 (1983), 319-328. Google Scholar [33] Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001. doi: 10.1142/9789812799791. Google Scholar [34] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. of Diff. Equ., 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012. Google Scholar [35] H. Zhan, The Study of the Cauchy Problem of a Second Order Quasilinear Degenerate Parabolic Equation and the Parallelism of a Riemannian Manifold Ph. D thesis, Xiamen University, 2004.Google Scholar [36] H. Zhan, The solution of a hyperbolic-parabolic mixed-type equation on half-space domain, J. Diff. Equ., 259 (2015), 1149-1181. doi: 10.1016/j.jde.2015.03.005. Google Scholar [37] J. Zhao, Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165. Google Scholar [38] J. Zhao and H. Zhan, Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations, Science in China Ser. A, 48 (2005), 583-593. doi: 10.1360/03ys0269. Google Scholar

show all references

##### References:
 [1] M. Bendahamane and K. H. Karlsen, Renarmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422. doi: 10.1137/S0036141003428937. Google Scholar [2] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial value problem for $u_{t}-Δ \varphi (u)=0$, J. Math.Pures et Appl., 58 (1979), 153-163. Google Scholar [3] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152. Google Scholar [4] G. Q. Chen and B. Perthame, Well-Posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. I. H. Poincare-AN, 20 (2003), 645-668. doi: 10.1016/S0294-1449(02)00014-8. Google Scholar [5] G. Q. Chen and E. DiBenedetto, Stability of entropy solutions to Cauchy problem for a class of nonlinear hyperbolic-parabolic equations, SIAM J.Math. Anal., 33 (2001), 751-762. doi: 10.1137/S0036141001363597. Google Scholar [6] B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equ., 151 (1999), 231-251. doi: 10.1006/jdeq.1998.3499. Google Scholar [7] G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984.Google Scholar [8] M. Escobedo, J. L. Vazquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.1090/S0002-9947-1994-1225573-2. Google Scholar [9] L. C. Evans, Weak convergence methods for nonlinear partial differential equations Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. doi: 10.1090/cbms/074. Google Scholar [10] G. Fichera, Sulle equazioni differenziatli lineari ellittico-paraboliche del secondo ordine, Atti Accd, Naz. Lincei. Mem, CI. Sci. Fis. Mat. Nat. Sez.1, 5 (1956), 1-30. Google Scholar [11] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in boundary problems, differential equations, Univ. of Wisconsin Press, Madison, Wis., 9 (1960), 97-120. Google Scholar [12] L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004.Google Scholar [13] F. R. Guarguaglini, V. Milišić and A. Terracina, A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions, J. Diff. Equ., 202 (2004), 183-207. doi: 10.1016/j.jde.2004.03.008. Google Scholar [14] K. H. Karlsen and N. H. Risebro, On the uniqueness of entropy solutions of nonlinear degenerate parabolic equations with rough coefficient, Discrete Contain. Dye. Sys., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081. Google Scholar [15] M. V. Keldyš, On certain cases of degeneration of elliptic type on the boundary of a domain, Dokl. Akad. Aauk SSSR, 77 (1951), 181-183. Google Scholar [16] K. Kobayasi and H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Diff. Equ., 252 (2012), 137-167. doi: 10.1016/j.jde.2011.09.008. Google Scholar [17] S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243. Google Scholar [18] Y. Li and Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. Diff. Equ., 252 (2012), 4719-4741. doi: 10.1016/j.jde.2012.01.027. Google Scholar [19] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formation of multidimensional conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [20] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184. Google Scholar [21] J. Málek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDES in Applied Mathematics and Mathematical Computation, 13. Chapman and HALL, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar [22] O. A. Oleinik and V. N. Samokhin, Mathematical Models in boundary, Layer Theorem Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 1999.Google Scholar [23] O. A. Oleinik, A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300. Google Scholar [24] O. A. Oleinik, Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140. Google Scholar [25] F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247. Google Scholar [26] G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450. Google Scholar [27] A. I. Vol'pert and S. I. Hudjaev, On the problem for quasilinear degenerate parabolic equations of second order (Russian), Mat. Sb., 78 (1969), 374-396. Google Scholar [28] A. I. Volpert, BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302. doi: 10.1070/SM1967v002n02ABEH002340. Google Scholar [29] A. I. Volpert and S. I. Hudjave, Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975.Google Scholar [30] Z. Wu and J. Yin, Some properties of functions in BV$_{x}$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 5 (1989), 395-422. Google Scholar [31] Z. Wu and J. Zhao, The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin.Ann. of Math., 4 (1983), 57-76. Google Scholar [32] Z. Wu and J. Zhao, Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin.Ann. of Math., 4 (1983), 319-328. Google Scholar [33] Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001. doi: 10.1142/9789812799791. Google Scholar [34] J. Yin and C. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. of Diff. Equ., 237 (2007), 421-445. doi: 10.1016/j.jde.2007.03.012. Google Scholar [35] H. Zhan, The Study of the Cauchy Problem of a Second Order Quasilinear Degenerate Parabolic Equation and the Parallelism of a Riemannian Manifold Ph. D thesis, Xiamen University, 2004.Google Scholar [36] H. Zhan, The solution of a hyperbolic-parabolic mixed-type equation on half-space domain, J. Diff. Equ., 259 (2015), 1149-1181. doi: 10.1016/j.jde.2015.03.005. Google Scholar [37] J. Zhao, Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165. Google Scholar [38] J. Zhao and H. Zhan, Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations, Science in China Ser. A, 48 (2005), 583-593. doi: 10.1360/03ys0269. Google Scholar
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