On a hyperbolic-parabolic mixed type equation

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

On a hyperbolic-parabolic mixed type equation

Huashui Zhan ^,

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.

Keywords: Hyperbolic-parabolic equation, homogeneous boundary condition, entropy solution, BV estimate.

Mathematics Subject Classification: Primary: 35L65, 35L85; Secondary: 35k35.

Citation: Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete and 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.
[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.
[3]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[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.
[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.
[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.
[7]	G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984.
[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.
[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.
[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.
[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.
[12]	L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004.
[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.
[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.
[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.
[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.
[17]	S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.
[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.
[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.
[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.
[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.
[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.
[23]	O. A. Oleinik, A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300.
[24]	O. A. Oleinik, Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140.
[25]	F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247.
[26]	G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450.
[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.
[28]	A. I. Volpert, BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302. doi: 10.1070/SM1967v002n02ABEH002340.
[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.
[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.
[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.
[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.
[33]	Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001. doi: 10.1142/9789812799791.
[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.
[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.
[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.
[37]	J. Zhao, Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165.
[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.

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.
[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.
[3]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[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.
[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.
[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.
[7]	G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984.
[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.
[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.
[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.
[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.
[12]	L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004.
[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.
[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.
[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.
[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.
[17]	S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.
[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.
[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.
[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.
[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.
[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.
[23]	O. A. Oleinik, A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300.
[24]	O. A. Oleinik, Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140.
[25]	F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247.
[26]	G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450.
[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.
[28]	A. I. Volpert, BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302. doi: 10.1070/SM1967v002n02ABEH002340.
[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.
[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.
[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.
[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.
[33]	Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001. doi: 10.1142/9789812799791.
[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.
[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.
[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.
[37]	J. Zhao, Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165.
[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.

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