# American Institute of Mathematical Sciences

August  2011, 4(4): 875-886. doi: 10.3934/dcdss.2011.4.875

## The degenerate drift-diffusion system with the Sobolev critical exponent

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578

Received  September 2009 Revised  December 2009 Published  November 2010

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.

Citation: T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
##### References:
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Differential Equations, 8 (2003), 787-820.  Google Scholar [31] 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.  Google Scholar [32] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.  Google Scholar [33] T. Suzuki, "Free Energy and Self-Interacting Particles," in "Progress in Nonlinear Differential Equations and Their Applications," 62, Birkhäuser Boston Inc., Boston, MA, 2005.  Google Scholar [34] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: doi:10.1007/BF02418013.  Google Scholar [35] 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.  Google Scholar [36] 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.  Google Scholar [37] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265.  Google Scholar

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##### References:
 [1] P. Biler, Local and global solvability of some parabolic systems modeling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.  Google Scholar [2] 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.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [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).  Google Scholar [6] 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.  Google Scholar [7] 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.  Google Scholar [8] 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.  Google Scholar [9] A. Jüngel, Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Model. Meth. Appl. Sci., 5 (1995), 497-518.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] T. Kobayashi and T. Ogawa, Fluid mechanical approximation to the degenerated drift-diffusion system from compressible Navier-Stokes-Poisson system,, preprint., ().   Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] M. S. Mock, An initial value problem from semiconductor devise theory, SIAM J. Math., 5 (1974), 597-612. doi: doi:10.1137/0505061.  Google Scholar [18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci.Appl., 5 (1995), 581-601.  Google Scholar [19] 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.  Google Scholar [20] 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.  Google Scholar [21] T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $R^2$,, preprint., ().   Google Scholar [22] 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.  Google Scholar [23] T. Nagai, T. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.  Google Scholar [24] 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.  Google Scholar [25] T. Ogawa, Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type, Differential Integral Equations, 21 (2008), 1113-1154.  Google Scholar [26] T. Ogawa, Decay and finite time blow-up of solutions to degenerate drift-diffusion system with the Sobolev critical exponent,, preprint., ().   Google Scholar [27] 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.  Google Scholar [28] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30 (1968), 148-172. doi: doi:10.1007/BF00250942.  Google Scholar [29] T. Senba and T. Suzuki, Chemotaxis collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.  Google Scholar [30] T. Senba and T. Suzuki, Blow up behavior of solutions to the rescaled Jäger-Luckhaus system, Adv. Differential Equations, 8 (2003), 787-820.  Google Scholar [31] 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.  Google Scholar [32] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.  Google Scholar [33] T. Suzuki, "Free Energy and Self-Interacting Particles," in "Progress in Nonlinear Differential Equations and Their Applications," 62, Birkhäuser Boston Inc., Boston, MA, 2005.  Google Scholar [34] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: doi:10.1007/BF02418013.  Google Scholar [35] 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.  Google Scholar [36] 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.  Google Scholar [37] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265.  Google Scholar
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