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Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces
Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source
1. | School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China, China, China |
References:
[1] |
C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411.
doi: 10.1007/s00526-008-0188-z. |
[2] |
H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425. |
[3] |
Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.
doi: 10.1016/j.jde.2009.03.021. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, |
[5] |
H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.
doi: 10.1016/j.jde.2015.01.038. |
[6] |
D. Colton, Pseudoparabolic equations in one space variable, J. Differential Equations, 12 (1972), 559-565. |
[7] |
D. Colton and J. Wimp, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418.
doi: 10.1016/0022-247X(79)90152-5. |
[8] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[9] |
E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.
doi: 10.1512/iumj.1981.30.30062. |
[10] |
F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
[11] |
M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type, J. Math. Sci. (N. Y.), 148 (2008), 1-142.
doi: 10.1007/s10958-007-0541-3. |
[12] |
H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386. |
[13] |
Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal., 68 (2008), 3332-3348.
doi: 10.1016/j.na.2007.03.029. |
[14] |
M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl., 352 (2009), 629-633.
doi: 10.1016/j.jmaa.2008.11.016. |
[15] |
J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana Univ. Math. J.}, 20 (): 1077.
|
[16] |
V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756 (electronic).
doi: 10.1090/S0002-9947-03-03340-3. |
[17] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. |
[18] |
M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system, Appl. Anal., 88 (2009), 1265-1282.
doi: 10.1080/00036810903277077. |
[19] |
W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains, J. Differential Equations, 27 (1978), 394-404. |
[20] |
D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172. |
[21] |
N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems, Nonlinear Anal. Real World Appl., 12 (2011), 2625-2639.
doi: 10.1016/j.nonrwa.2011.03.010. |
[22] |
R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543. |
[23] |
R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type, J. Differential Equations, 11 (1972), 252-265. |
[24] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. |
[25] |
T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26. |
[26] |
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. |
[27] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[28] |
R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
doi: 10.1016/j.jfa.2013.03.010. |
[29] |
C. Yang, Y. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.
doi: 10.1016/j.jde.2012.09.001. |
[30] |
X. Zhu, F. Li and Y. Li, Some sharp result about the global existence and blow up of solutions to a class of pseudo-parabolic equations,, preprint., ().
|
show all references
References:
[1] |
C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411.
doi: 10.1007/s00526-008-0188-z. |
[2] |
H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425. |
[3] |
Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.
doi: 10.1016/j.jde.2009.03.021. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998, |
[5] |
H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.
doi: 10.1016/j.jde.2015.01.038. |
[6] |
D. Colton, Pseudoparabolic equations in one space variable, J. Differential Equations, 12 (1972), 559-565. |
[7] |
D. Colton and J. Wimp, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418.
doi: 10.1016/0022-247X(79)90152-5. |
[8] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[9] |
E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.
doi: 10.1512/iumj.1981.30.30062. |
[10] |
F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
[11] |
M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type, J. Math. Sci. (N. Y.), 148 (2008), 1-142.
doi: 10.1007/s10958-007-0541-3. |
[12] |
H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386. |
[13] |
Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal., 68 (2008), 3332-3348.
doi: 10.1016/j.na.2007.03.029. |
[14] |
M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl., 352 (2009), 629-633.
doi: 10.1016/j.jmaa.2008.11.016. |
[15] |
J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana Univ. Math. J.}, 20 (): 1077.
|
[16] |
V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756 (electronic).
doi: 10.1090/S0002-9947-03-03340-3. |
[17] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. |
[18] |
M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system, Appl. Anal., 88 (2009), 1265-1282.
doi: 10.1080/00036810903277077. |
[19] |
W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains, J. Differential Equations, 27 (1978), 394-404. |
[20] |
D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172. |
[21] |
N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems, Nonlinear Anal. Real World Appl., 12 (2011), 2625-2639.
doi: 10.1016/j.nonrwa.2011.03.010. |
[22] |
R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543. |
[23] |
R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type, J. Differential Equations, 11 (1972), 252-265. |
[24] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. |
[25] |
T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26. |
[26] |
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. |
[27] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[28] |
R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
doi: 10.1016/j.jfa.2013.03.010. |
[29] |
C. Yang, Y. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.
doi: 10.1016/j.jde.2012.09.001. |
[30] |
X. Zhu, F. Li and Y. Li, Some sharp result about the global existence and blow up of solutions to a class of pseudo-parabolic equations,, preprint., ().
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