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November  2015, 14(6): 2465-2485. doi: 10.3934/cpaa.2015.14.2465

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

Received  June 2015 Revised  August 2015 Published  September 2015

In this paper, a class of pseudo-parabolic equations with an exponential source is concerned. First, by the elliptic regularity theory, we establish the local existence and uniqueness of solutions. Sequently, the global existence and blow up of solutions with lower initial energy is considered via the potential wells method. Finally, we find a sufficient condition under which the solution blows up without any limit of initial energy by constructing a new functional which falls between the energy functional and Nehari functional. During this process, the properties of global solutions are also studied.
Citation: Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465
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,, \emph{Calc. Var. Partial Differential Equations}, 34 (2009), 377.  doi: 10.1007/s00526-008-0188-z.  Google Scholar

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Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations,, \emph{J. Differential Equations}, 246 (2009), 4568.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar

[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, (1998).   Google Scholar

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H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity,, \emph{J. Differential Equations}, 258 (2015), 4424.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

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D. Colton, Pseudoparabolic equations in one space variable,, \emph{J. Differential Equations}, 12 (1972), 559.   Google Scholar

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D. Colton and J. Wimp, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures,, \emph{J. Math. Anal. Appl.}, 69 (1979), 411.  doi: 10.1016/0022-247X(79)90152-5.  Google Scholar

[8]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range,, \emph{Calc. Var. Partial Differential Equations}, 3 (1995), 139.  doi: 10.1007/BF01205003.  Google Scholar

[9]

E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations,, \emph{Indiana Univ. Math. J.}, 30 (1981), 821.  doi: 10.1512/iumj.1981.30.30062.  Google Scholar

[10]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 185.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

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M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type,, \emph{J. Math. Sci. (N. Y.)}, 148 (2008), 1.  doi: 10.1007/s10958-007-0541-3.  Google Scholar

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H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$,, \emph{Arch. Rational Mech. Anal.}, 51 (1973), 371.   Google Scholar

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Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations,, \emph{Nonlinear Anal.}, 68 (2008), 3332.  doi: 10.1016/j.na.2007.03.029.  Google Scholar

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M. Meyvaci, Blow up of solutions of pseudoparabolic equations,, \emph{J. Math. Anal. Appl.}, 352 (2009), 629.  doi: 10.1016/j.jmaa.2008.11.016.  Google Scholar

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J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana Univ. Math. J.}, 20 (): 1077.   Google Scholar

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V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2739.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[17]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, \emph{Israel J. Math.}, 22 (1975), 273.   Google Scholar

[18]

M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system,, \emph{Appl. Anal.}, 88 (2009), 1265.  doi: 10.1080/00036810903277077.  Google Scholar

[19]

W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains,, \emph{J. Differential Equations}, 27 (1978), 394.   Google Scholar

[20]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations,, \emph{Arch. Rational Mech. Anal.}, 30 (1968), 148.   Google Scholar

[21]

N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 2625.  doi: 10.1016/j.nonrwa.2011.03.010.  Google Scholar

[22]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, \emph{SIAM J. Math. Anal.}, 3 (1972), 527.   Google Scholar

[23]

R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type,, \emph{J. Differential Equations}, 11 (1972), 252.   Google Scholar

[24]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, \emph{SIAM J. Math. Anal.}, 1 (1970), 1.   Google Scholar

[25]

T. W. Ting, Certain non-steady flows of second-order fluids,, \emph{Arch. Rational Mech. Anal.}, 14 (1963), 1.   Google Scholar

[26]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, \emph{J. Math. Mech.}, 17 (1967), 473.   Google Scholar

[27]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[28]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations,, \emph{J. Funct. Anal.}, 264 (2013), 2732.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[29]

C. Yang, Y. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation,, \emph{J. Differential Equations}, 253 (2012), 3286.  doi: 10.1016/j.jde.2012.09.001.  Google Scholar

[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., ().   Google Scholar

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,, \emph{Calc. Var. Partial Differential Equations}, 34 (2009), 377.  doi: 10.1007/s00526-008-0188-z.  Google Scholar

[2]

H. Brill, A semilinear Sobolev evolution equation in a Banach space,, \emph{J. Differential Equations}, 24 (1977), 412.   Google Scholar

[3]

Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations,, \emph{J. Differential Equations}, 246 (2009), 4568.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar

[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, (1998).   Google Scholar

[5]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity,, \emph{J. Differential Equations}, 258 (2015), 4424.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[6]

D. Colton, Pseudoparabolic equations in one space variable,, \emph{J. Differential Equations}, 12 (1972), 559.   Google Scholar

[7]

D. Colton and J. Wimp, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures,, \emph{J. Math. Anal. Appl.}, 69 (1979), 411.  doi: 10.1016/0022-247X(79)90152-5.  Google Scholar

[8]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range,, \emph{Calc. Var. Partial Differential Equations}, 3 (1995), 139.  doi: 10.1007/BF01205003.  Google Scholar

[9]

E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations,, \emph{Indiana Univ. Math. J.}, 30 (1981), 821.  doi: 10.1512/iumj.1981.30.30062.  Google Scholar

[10]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 185.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[11]

M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type,, \emph{J. Math. Sci. (N. Y.)}, 148 (2008), 1.  doi: 10.1007/s10958-007-0541-3.  Google Scholar

[12]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$,, \emph{Arch. Rational Mech. Anal.}, 51 (1973), 371.   Google Scholar

[13]

Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations,, \emph{Nonlinear Anal.}, 68 (2008), 3332.  doi: 10.1016/j.na.2007.03.029.  Google Scholar

[14]

M. Meyvaci, Blow up of solutions of pseudoparabolic equations,, \emph{J. Math. Anal. Appl.}, 352 (2009), 629.  doi: 10.1016/j.jmaa.2008.11.016.  Google Scholar

[15]

J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana Univ. Math. J.}, 20 (): 1077.   Google Scholar

[16]

V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2739.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[17]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, \emph{Israel J. Math.}, 22 (1975), 273.   Google Scholar

[18]

M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system,, \emph{Appl. Anal.}, 88 (2009), 1265.  doi: 10.1080/00036810903277077.  Google Scholar

[19]

W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains,, \emph{J. Differential Equations}, 27 (1978), 394.   Google Scholar

[20]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations,, \emph{Arch. Rational Mech. Anal.}, 30 (1968), 148.   Google Scholar

[21]

N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 2625.  doi: 10.1016/j.nonrwa.2011.03.010.  Google Scholar

[22]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, \emph{SIAM J. Math. Anal.}, 3 (1972), 527.   Google Scholar

[23]

R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type,, \emph{J. Differential Equations}, 11 (1972), 252.   Google Scholar

[24]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, \emph{SIAM J. Math. Anal.}, 1 (1970), 1.   Google Scholar

[25]

T. W. Ting, Certain non-steady flows of second-order fluids,, \emph{Arch. Rational Mech. Anal.}, 14 (1963), 1.   Google Scholar

[26]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, \emph{J. Math. Mech.}, 17 (1967), 473.   Google Scholar

[27]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[28]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations,, \emph{J. Funct. Anal.}, 264 (2013), 2732.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[29]

C. Yang, Y. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation,, \emph{J. Differential Equations}, 253 (2012), 3286.  doi: 10.1016/j.jde.2012.09.001.  Google Scholar

[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., ().   Google Scholar

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