The BV solution of the parabolic equation with degeneracy on the boundary. (English) Zbl 1346.35129
Summary: Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When \(1 \leq \alpha < p - 1\), the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35L85 | Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators |
| 35R35 | Free boundary problems for PDEs |
References:
| [1] | Yin J., Wang Ch., Properties of the boundary flux of a singular diffusion process., Chin. Ann. Math., 2004, 25B(2), 175-182.; Yin, J.; Wang, Ch, Properties of the boundary flux of a singular diffusion process, Chin. Ann. Math, 25B, 2, 175-182 (2004) · Zbl 1052.35105 |
| [2] | Zhan H., Yuan H., A diffusion convection equation with degeneracy on the boundary (in Chinese), Journal of Jilin University (Science Edition), 2015, 53(3), 353-358.; Zhan, H.; Yuan, H., A diffusion convection equation with degeneracy on the boundary (in Chinese), Journal of Jilin University (Science Edition, 53, 3, 353-358 (2015) |
| [3] | Zhan H., The boundary value condition of an evolutionary p.x/-Laplacian equation, Boundary Value Problems, 2015, 112, DOI: 10.1186/s13661-015-0377-6.; Zhan, H., The boundary value condition of an evolutionary p.x/-Laplacian equation, Boundary Value Problems, 112 (2015) · Zbl 1381.35098 · doi:10.1186/s13661-015-0377-6 |
| [4] | Zhan H. and Xie Q., The boundary degeneracy of a singular diffusion equation, Journal of inequalities and applications, 2014, 284, DOI: 10.1186/1029-242x-2014-284.; Zhan, H.; Xie, Q., The boundary degeneracy of a singular diffusion equation, Journal of inequalities and applications, 284 (2014) · Zbl 1332.35165 · doi:10.1186/1029-242x-2014-284 |
| [5] | Zhan H., The solutions of a hyperbolic-parabolic mixed type equation on half-space domain, J. Of Differential Equations, 2015, 259, 1449-1481.; Zhan, H., The solutions of a hyperbolic-parabolic mixed type equation on half-space domain, J. Of Differential Equations, 259, 1449-1481 (2015) · Zbl 1316.35213 |
| [6] | Zhan H., Homogeneous Dirichlet condition of an anisotropic degenerate parabolic equation, Boundary value problems, 2015, 22, DOI: 10.1186/s13661-015-0283-y.; Zhan, H., Homogeneous Dirichlet condition of an anisotropic degenerate parabolic equation, Boundary value problems, 22 (2015) · Zbl 1364.35171 · doi:10.1186/s13661-015-0283-y |
| [7] | Zhan H., The boundary degeneracy theory of a strongly degenerate parabolic equation, Boundary value problems, 2016, 2016, 2, DOI: 10.1186/s13661-015-0516-0.; Zhan, H., The boundary degeneracy theory of a strongly degenerate parabolic equation, Boundary value problems, 2016, 2 (2016) · Zbl 1335.35146 · doi:10.1186/s13661-015-0516-0 |
| [8] | Wu Z. and Zhao J., The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin. Ann. of Math., 1983, (1) 4B, 57-76.; Wu, Z.; Zhao, J., The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin. Ann. of Math, 4B, 1, 57-76 (1983) · Zbl 0508.35049 |
| [9] | Wu Z. and Zhao J., Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin. Ann. of Math., 1983, (3) 4B, 319-328.; Wu, Z.; Zhao, J., Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin. Ann. of Math, 4B, 3, 319-328 (1983) · Zbl 0516.35045 |
| [10] | Kobayasi K, Ohwa H., Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. of Diff. Equ.. 2012, 252, 137-167.; Kobayasi, K.; Ohwa, H., Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. of Diff. Equ, 252, 137-167 (2012) · Zbl 1237.35099 |
| [11] | Li Y., Wang Q., Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. of Diff. Equ., 2012. 252, 4719-4741.; Li, Y.; Wang, Q., Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. of Diff. Equ, 252, 4719-4741 (2012) · Zbl 1241.35139 |
| [12] | Wang J., Gao W., Su M., Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources, Applied Mathematics and Computation, 2010, 216, 1996-2009.; Wang, J.; Gao, W.; Su, M., Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources, Applied Mathematics and Computation, 216, 1996-2009 (2010) · Zbl 1425.35092 |
| [13] | Ye H., Yin J., Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection J. Math. Anal. Appl., 2015, 421, 1225-1237.; Ye, H.; Yin, J., Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection, J. Math. Anal. Appl, 421, 1225-1237 (2015) · Zbl 1408.35091 |
| [14] | Lee K., Petrosyan A. and Vazquez J. L., Large time geometric properties of solutions of the evolution p-Laplacian equation. J. Diff. Equ., 2006, 229, 389-411.; Lee, K.; Petrosyan, A.; Vazquez, J. L., Large time geometric properties of solutions of the evolution p-Laplacian equation, J. Diff. Equ, 229, 389-411 (2006) · Zbl 1108.35097 |
| [15] | Wang J., Wang Z. Yin J., A class of degenerate diffusion equations with mixed boundary conditions, J. Math. Anal. Appl., 2004, 298, 589-603.; Wang, J.; Wang, Z.; Yin, J., A class of degenerate diffusion equations with mixed boundary conditions, J. Math. Anal. Appl, 298, 589-603 (2004) · Zbl 1061.35031 |
| [16] | Yin J. and Wang Ch., Evolutionary weighted p-Laplacian with boundary degeneracy, J. Diff. Equ., 2007, 237, 421-445.; Yin, J.; Wang, Ch, Evolutionary weighted p-Laplacian with boundary degeneracy, J. Diff. Equ, 237, 421-445 (2007) · Zbl 1133.35065 |
| [17] | Gu L., Second order parabolic partial differential equations (in chinese), The Publishing Company of Xiamen University, 2002.; Gu, L., Second order parabolic partial differential equations (in chinese) (2002) |
| [18] | Taylor Michael E., Partial differential equations III, Springer-Verlag, 1999.; Taylor Michael, E., Partial differential equations III (1999) |
| [19] | Zhan H., The solution of convection-diffusion equation. Chinese Annals of Mathematics, 2013, 34(2), 235-256.; Zhan, H., The solution of convection-diffusion equation, Chinese Annals of Mathematics, 34, 2, 235-256 (2013) · Zbl 1299.35163 |
| [20] | Zhan H., A new kind of solutions of degenerate parabolic equations with unbounded convection term, Open Math., 2015, 13, 283-297.; Zhan, H., A new kind of solutions of degenerate parabolic equations with unbounded convection term, Open Math, 13, 283-297 (2015) · Zbl 1346.35083 |
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