Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source

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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

Xiaoli Zhu ^1, , Fuyi Li ^1, and Ting Rong ^1,

1.	School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China, China, China

Received June 2015 Revised August 2015 Published September 2015

Abstract
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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.

Keywords: potential wells, Pseudo-parabolic equation, blow up, global existence, sufficient condition.

Mathematics Subject Classification: 35K70, 35A01, 35B44.

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.
[2]	H. Brill, A semilinear Sobolev evolution equation in a Banach space,, \emph{J. Differential Equations}, 24 (1977), 412.
[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.
[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).
[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.
[6]	D. Colton, Pseudoparabolic equations in one space variable,, \emph{J. Differential Equations}, 12 (1972), 559.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2739. doi: 10.1090/S0002-9947-03-03340-3.
[17]	L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, \emph{Israel J. Math.}, 22 (1975), 273.
[18]	M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system,, \emph{Appl. Anal.}, 88 (2009), 1265. doi: 10.1080/00036810903277077.
[19]	W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains,, \emph{J. Differential Equations}, 27 (1978), 394.
[20]	D. H. Sattinger, On global solution of nonlinear hyperbolic equations,, \emph{Arch. Rational Mech. Anal.}, 30 (1968), 148.
[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.
[22]	R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, \emph{SIAM J. Math. Anal.}, 3 (1972), 527.
[23]	R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type,, \emph{J. Differential Equations}, 11 (1972), 252.
[24]	R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, \emph{SIAM J. Math. Anal.}, 1 (1970), 1.
[25]	T. W. Ting, Certain non-steady flows of second-order fluids,, \emph{Arch. Rational Mech. Anal.}, 14 (1963), 1.
[26]	N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, \emph{J. Math. Mech.}, 17 (1967), 473.
[27]	M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (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,, \emph{J. Funct. Anal.}, 264 (2013), 2732. 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,, \emph{J. Differential Equations}, 253 (2012), 3286. 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,, \emph{Calc. Var. Partial Differential Equations}, 34 (2009), 377. doi: 10.1007/s00526-008-0188-z.
[2]	H. Brill, A semilinear Sobolev evolution equation in a Banach space,, \emph{J. Differential Equations}, 24 (1977), 412.
[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.
[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).
[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.
[6]	D. Colton, Pseudoparabolic equations in one space variable,, \emph{J. Differential Equations}, 12 (1972), 559.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2739. doi: 10.1090/S0002-9947-03-03340-3.
[17]	L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, \emph{Israel J. Math.}, 22 (1975), 273.
[18]	M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system,, \emph{Appl. Anal.}, 88 (2009), 1265. doi: 10.1080/00036810903277077.
[19]	W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains,, \emph{J. Differential Equations}, 27 (1978), 394.
[20]	D. H. Sattinger, On global solution of nonlinear hyperbolic equations,, \emph{Arch. Rational Mech. Anal.}, 30 (1968), 148.
[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.
[22]	R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, \emph{SIAM J. Math. Anal.}, 3 (1972), 527.
[23]	R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type,, \emph{J. Differential Equations}, 11 (1972), 252.
[24]	R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, \emph{SIAM J. Math. Anal.}, 1 (1970), 1.
[25]	T. W. Ting, Certain non-steady flows of second-order fluids,, \emph{Arch. Rational Mech. Anal.}, 14 (1963), 1.
[26]	N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, \emph{J. Math. Mech.}, 17 (1967), 473.
[27]	M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (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,, \emph{J. Funct. Anal.}, 264 (2013), 2732. 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,, \emph{J. Differential Equations}, 253 (2012), 3286. 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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