# American Institute of Mathematical Sciences

• Previous Article
Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion
• CPAA Home
• This Issue
• Next Article
Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces
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 [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

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
 [1] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [2] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [3] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [4] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [5] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [6] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [7] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [8] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [9] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082 [10] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [11] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [12] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [13] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268 [14] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [15] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [16] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [17] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [18] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [19] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [20] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.105