Advanced Search
Article Contents
Article Contents

The degenerate drift-diffusion system with the Sobolev critical exponent

Abstract Related Papers Cited by
  • We consider the drift-diffusion system of degenerated type. For $n\ge 3$,

    $\partial_t \rho -\Delta \rho^\alpha + \kappa\nabla\cdot (\rho \nabla \psi ) =0, t>0, x \in R^n,$

    $-\Delta \psi = \rho, t>0, x \in R^n,$

    $\rho(0,x) = \rho_0(x)\ge 0, x \in R^n,$

    where $\alpha>1$ and $\kappa=1$. There exists a critical exponent that classifies the global behavior of the weak solution. In particular, we consider the critical case $\alpha_*=\frac{2 n}{n+2}=(2^*)'$, where the Talenti function $U(x)$ solving $-2^*\Delta U^{\frac{n-2}{n+2}}=U$ in $R^n$ classifies the global existence of the weak solution and finite blow-up of the solution.

    Mathematics Subject Classification: Primary: 35K15, 35K55, 35Q60; Secondary: 78A35.


    \begin{equation} \\ \end{equation}
  • [1]

    P. Biler, Local and global solvability of some parabolic systems modeling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.


    P. Biler and J. Dolbeault, Long time behavior of solutions to Nernst-Planck and Debye-Hünkel drift-diffusion systems, Ann. Henry Poincaré, 1 (2000), 461-472.


    P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal. T.M.A., 23 (1994), 1189-1209.doi: doi:10.1016/0362-546X(94)90101-5.


    P. Biler, T. Nadzieja and R. Stanczy, Nonisothermal systems of self-attracting Fermi-Dirac particles, Banach Center Pulb., 66 (2004), 61-78.doi: doi:10.4064/bc66-0-5.


    A. Blanchet, J. Dobeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 2006 (2006), 32 pp. (electronic).


    J.I. Díaz, G. Galiano and A. Jüngel, On a quasilinear degenerate system arising in semiconductor theory, Part II, Nonlinear Anal., 36 (1999), 569-594.


    J. I. Díaz, G. Galiano and A. Jüngel, On a quasilinear degenerate system arising in semiconductor theory, Part I, Nonlinear Anal. Real World Appl., 2 (2001), 305-336.


    W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.doi: doi:10.1142/S0218202595000292.


    A. Jüngel, Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Model. Meth. Appl. Sci., 5 (1995), 497-518.


    E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.doi: doi:10.1016/0022-5193(70)90092-5.


    C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.doi: doi:10.1007/s00222-006-0011-4.


    C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.doi: doi:10.1007/s11511-008-0031-6.


    T. Kobayashi and T. OgawaFluid mechanical approximation to the degenerated drift-diffusion system from compressible Navier-Stokes-Poisson system, preprint.


    M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Comm. Pure Appl. Anal., 5 (2006), 97-106.


    M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations, 16 (2003), 427-452.


    M. Kurokiba and T. Ogawa, Wellposedness of the for the drift-diffusion system in $L^p$ arising from the semiconductor device simulation, J. Math. Anal. Appl., 342 (2008), 1052-1067.doi: doi:10.1016/j.jmaa.2007.11.017.


    M. S. Mock, An initial value problem from semiconductor devise theory, SIAM J. Math., 5 (1974), 597-612.doi: doi:10.1137/0505061.


    T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci.Appl., 5 (1995), 581-601.


    T. Nagai, Global existence of solutions to a parabolic system for chemotaxis in two space dimensions, Nonlinear Anal. T.M.A., 30 (1997), 5381-5388.doi: doi:10.1016/S0362-546X(97)00395-7.


    T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.doi: doi:10.1155/S1025583401000042.


    T. Nagai and T. OgawaGlobal existence of solutions to a parabolic-elliptic system of drift-diffusion type in $R^2$, preprint.


    T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.


    T. Nagai, T. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.


    T. Ogawa, Decay and asymptotic behavior of a solution of the Keller-Segel system of degenerated and non-degenerated type, Banach Center Publ., 74 (2006), 161-184.doi: doi:10.4064/bc74-0-10.


    T. Ogawa, Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.


    T. OgawaDecay and finite time blow-up of solutions to degenerate drift-diffusion system with the Sobolev critical exponent, preprint.


    F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. P.D.E., 26 (2001), 101-174.doi: doi:10.1081/PDE-100002243.


    D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30 (1968), 148-172.doi: doi:10.1007/BF00250942.


    T. Senba and T. Suzuki, Chemotaxis collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.


    T. Senba and T. Suzuki, Blow up behavior of solutions to the rescaled Jäger-Luckhaus system, Adv. Differential Equations, 8 (2003), 787-820.


    Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel system, Differential Integral Equations, 19 (2006), 841-876.


    Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.


    T. Suzuki, "Free Energy and Self-Interacting Particles," in "Progress in Nonlinear Differential Equations and Their Applications," 62, Birkhäuser Boston Inc., Boston, MA, 2005.


    G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.doi: doi:10.1007/BF02418013.


    M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RIMS Kyoto Univ., 8 (1972), 211-229.doi: doi:10.2977/prims/1195193108.


    G. Wolansky, Comparison between two models of self-gravitating clusters: Conditions for gravitational collapse, Nonlinear Anal. T.M.A., 24 (1995), 1119-1129.doi: doi:10.1016/0362-546X(94)E0028-F.


    A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265.

  • 加载中

Article Metrics

HTML views() PDF downloads(97) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint