December  2009, 25(4): 1275-1286. doi: 10.3934/dcds.2009.25.1275

Strong traces for degenerate parabolic-hyperbolic equations

1. 

Department of Mathematics, Dong-A University, Busan 604-714, South Korea

Received  April 2008 Revised  July 2009 Published  September 2009

In this paper we consider bounded weak solutions $u$ of degenerate parabolic-hyperbolic equations defined in a subset $]0,T[\times\Omega\subset \R^{+}\times \R^d$. We define a strong notion of trace at the boundary $]0,T[\times\partial\Omega$ reached by $L^1$ convergence for a large class of functionals of $u$ and at $0 \times \Omega$ reached by $L^1$ convergence for solution $u$. This result develops the strong trace results of Kwon, Vasseur [13] and Panov [19, 20] for more general equations, namely, degenerate parabolic-hyperbolic equations.
Citation: Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275
[1]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203

[2]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205

[3]

Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477

[4]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226

[5]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

[6]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[7]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[8]

Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246

[9]

Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019

[10]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014

[11]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

[12]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[13]

Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154

[14]

Valerii Maltsev, Michael Pokojovy. On a parabolic-hyperbolic filter for multicolor image noise reduction. Evolution Equations & Control Theory, 2016, 5 (2) : 251-272. doi: 10.3934/eect.2016004

[15]

Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133

[16]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[17]

M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827

[18]

Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217

[19]

Tai Nguyen Phuoc, Laurent Véron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2033-2063. doi: 10.3934/dcds.2013.33.2033

[20]

Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks & Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]