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:
[1]

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

[2]

P. Biler and J. Dolbeault, Long time behavior of solutions to Nernst-Planck and Debye-Hünkel drift-diffusion systems,, Ann. Henry Poincar\'e, 1 (2000), 461. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[33]

T. Suzuki, "Free Energy and Self-Interacting Particles,", in, 62 (2005). Google Scholar

[34]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. 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. 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. 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. Google Scholar

show all references

References:
[1]

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

[2]

P. Biler and J. Dolbeault, Long time behavior of solutions to Nernst-Planck and Debye-Hünkel drift-diffusion systems,, Ann. Henry Poincar\'e, 1 (2000), 461. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[33]

T. Suzuki, "Free Energy and Self-Interacting Particles,", in, 62 (2005). Google Scholar

[34]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. 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. 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. 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. Google Scholar

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