Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms

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May 2016, 21(3): 781-801. doi: 10.3934/dcdsb.2016.21.781

Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms

Huafei Di ^1, , Yadong Shang ^1, and Xiaoxiao Zheng ^1,

1.	School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China, China, China

Received April 2015 Revised October 2015 Published January 2016

Abstract
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In this paper, we consider the initial boundary value problem for a fourth order pseudo-parabolic equation with memory and source terms. Under suitable assumptions on the function $g$, the initial data and the parameters in the equation, we not only prove the existence of global weak solutions by the combination of the Galerkin method and potential well theory, but also establish an explicit decay rate estimate of the energy adopting the ideas of Marcelo M. Cavalcanti et al. (J. Differ. Equations 203 (2004) 119-158) and Patrick Martinez (ESAIN: Control Optim. Calc. Var. 4 (1999) 419-444). Furthermore, the finite time blow up results for the solutions with both positive and negative initial energy are obtained under certain conditions.

Keywords: Galerkin method., blow up, Well-posedness, pseudo-parabolic, potential well, memory and source.

Mathematics Subject Classification: Primary: 35K35; Secondary: 35A01, 35B44, 35B4.

Citation: Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781

References:

[1]	A. B. Al'shin, M. O. Korpusov and A. G. Siveshnikov, Blow up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Aanlysis and Applicationss 15, Betlin, 2011. doi: 10.1515/9783110255294.
[2]	A. Y. Kolesov, E. F. Mishchenko and N. K. Rozov, Asymptotic methods of investiga-tion of periodic solutions of nonlinear hyperbolic equations, Trudy Mat. Inst. Steklova, 222 (1998), 192pp.
[3]	B. K. Shivamoggi, A symmetric regularized long wave equation for shallow water waves, Phys. Fluids, 29 (1986), 890-891. doi: 10.1063/1.865895.
[4]	P. Rosenau, Evolution and breaking of the ion-acoustic waves, Phys. Fluids, 31 (1988), 1317-1319. doi: 10.1063/1.866723.
[5]	R. E. Showalter and T. W. Ting, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001.
[6]	V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972), 57-78. doi: 10.1007/BF00281474.
[7]	E. Milne, The diffusion of imprisoned radiation through a gas, J. Lond. Math. Soc., 1 (1926), 40-51. doi: 10.1112/jlms/s1-1.1.40.
[8]	S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29 (1993), 3389-3405. doi: 10.1029/93WR01495.
[9]	L. Cueto-Felgueroso and R. Juanes, A phase-field model of unsaturated flow, Water Resour. Res., 45 (2009), W10409. doi: 10.1029/2009WR007945.
[10]	L. I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media, IZV. Akad. Nauk SSSR, Ser. Geogr., 12 (1948), 27-45.
[11]	B. C. Aslan, W. W. Hager and S. Moskow, A generalized eigenproblem for the Laplacian which arises in lightning, J. Math. Anal. Appl., 341 (2008), 1028-1041. doi: 10.1016/j.jmaa.2007.11.007.
[12]	R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543. doi: 10.1137/0503051.
[13]	R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42. doi: 10.1137/0506004.
[14]	E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.
[15]	A. Mikelić, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Differ. Equations, 248 (2010), 1561-1577. doi: 10.1016/j.jde.2009.11.022.
[16]	X. Cao and I. S. Pop, Two-phase porous media flows with dynamic capillary effects and hysteresis: Uniqueness of weak solutions, Comput. Math. Appl., 69 (2015), 688-695. doi: 10.1016/j.camwa.2015.02.009.
[17]	Y. Fan and I. S. Pop, Equivalent formulations and numerical schemes for a class of pseudo-parabolic equations, J. Comput. Appl. Math., 246 (2013), 86-93. doi: 10.1016/j.cam.2012.07.031.
[18]	C. M. Cuesta and I. S. Pop, Numerical schemes for a pseudo-parabolic Burgers equation: Discontinuous data and long-time behaviour, J. Comput. Appl. Math., 224 (2009), 269-283. doi: 10.1016/j.cam.2008.05.001.
[19]	B. Schweizer, Regularization of outflow problems in unsaturated porous media with dry regions, J. Differ. Equations, 237 (2007), 278-306. doi: 10.1016/j.jde.2007.03.011.
[20]	E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Periodic boundary value problem for nonlinear Sobolev-type equation, Funct. Anal. Appl., 44 (2010), 171-181. doi: 10.1007/s10688-010-0022-1.
[21]	E. I. Kaikina, Initial boundary value problems for nonlinear pseudo-parabolic equations in a critical case, Electron. J. Differ. Equations, 109 (2007), 1-25.
[22]	T. Matahashi and M. Tsutsumi, On a periodic problem for pseudo-parabolic equations of Sobolev-Galpen type, Mathematica Japonica, 22 (1978), 535-553.
[23]	T. Matahashi and M. Tsutsumi, Periodic solutions of semilinear pseudo-parabolic equations in Hilbert space, Funkc. Ekvacioj, 22 (1979), 51-66.
[24]	R. Z. 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.
[25]	N. I. Bakiyevich and G. A. Shadrin, Cauchy problem for an equation in filtration theory, Sb. Trudov Mosgospedinstituta, 7 (1978), 47-63.
[26]	K. I. Khudaverdiyev and G. M. Farhadova, On global existence for generalized solution of one-dimensional non-self-adjoint mixed problem for a class of fourth order semilinear pseudo-parabolic equations, Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb., 31 (2009), 119-134.
[27]	H. J. Zhao and B. J. Xuan, Existence and convergence of solutions for the generalized BBM-Burgers equations, Nonlinear Anal. Theory Methods Appl., 28 (1997), 1835-1849. doi: 10.1016/S0362-546X(95)00237-P.
[28]	H. M. Yin, Weak and classical solutions of some nonlinear volterra intergro-differential equations, Commun. Part. Diff. Eq., 17 (1992), 1369-1385. doi: 10.1080/03605309208820889.
[29]	S. A. Messaoudi, Blow-up of solutions of a semilinear Heat Equation with a Visco-elastic Term, Progr. Nonlinear Differ. Equ. Appl., 64 (2005), 351-356. doi: 10.1007/3-7643-7385-7_19.
[30]	Y. D. Shang and B. L. Guo, On the problem of the existence of global solutions for a class of nonlinear convolutional integro-differential equations of pseudoparabolic type, Acta Math. Appl. Sin., 26 (2003), 511-524.
[31]	Y. D. Shang and B. L. Guo, Initial boundary value problem and initial value problem for the nonlinear integro-differential equations of pseudoparabolic type, Math. Appl.(Wuhan) 15 (2002), 40-45.
[32]	Y. D. Shang and B. L. Guo, Initial boundary value problem for a class of quasilinear integro-differential equations of pseudoparabolic type, Chin. J. Eng. Math., 20 (2003), 1-6.
[33]	M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal. Theory Methods Appl., 66 (2007), 2653-2675. doi: 10.1016/j.na.2006.03.046.
[34]	R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976.
[35]	S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy, M2AN Math. Model. Numer. Anal., 40 (2006), 765-784. doi: 10.1051/m2an:2006029.
[36]	S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, On a pseudoparabolic problem with constraint, Differ. integral Equ., 19 (2006), 1391-1412.
[37]	G. Vallet, On a degenerated parabolic-hyperbolic problem arising from stratigraphy, Numerical Mathematics and Advanced Applications, Springer, Berlin, (2006), 412-420. doi: 10.1007/978-3-540-34288-5_36.
[38]	N. Igbida, Solutions auto-similaires pour une équation de Barenblatt, Rev. Math. Appl., 17 (1996), 21-36.
[39]	C. Bauzet, J. Giacomoni and G. Vallet, On a class of quasilinear Barenblatt equations, Monografías de la Real Academia de Ciencias de Zaragoza, 38 (2012), 35-51.
[40]	M. Ptashnyk, Nonlinear Pseudoparabolic Equations and Variational Inequalities, Master's Thesis, Heidelberg, 2004. Available from: http://www.ub.uni-heidelberg.de/archiv/4802.
[41]	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.
[42]	M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differ. Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011.
[43]	P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIN: Control Optim. Calc. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116.
[44]	J. L. Lions, Quelques Méthodes de Résolutions des Probléms aux Limites Non Linéaires, Dunod, Paris, 1969.
[45]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[46]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. integral Equ., 6 (1933), 507-533.

show all references

References:

[1]	A. B. Al'shin, M. O. Korpusov and A. G. Siveshnikov, Blow up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Aanlysis and Applicationss 15, Betlin, 2011. doi: 10.1515/9783110255294.
[2]	A. Y. Kolesov, E. F. Mishchenko and N. K. Rozov, Asymptotic methods of investiga-tion of periodic solutions of nonlinear hyperbolic equations, Trudy Mat. Inst. Steklova, 222 (1998), 192pp.
[3]	B. K. Shivamoggi, A symmetric regularized long wave equation for shallow water waves, Phys. Fluids, 29 (1986), 890-891. doi: 10.1063/1.865895.
[4]	P. Rosenau, Evolution and breaking of the ion-acoustic waves, Phys. Fluids, 31 (1988), 1317-1319. doi: 10.1063/1.866723.
[5]	R. E. Showalter and T. W. Ting, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001.
[6]	V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972), 57-78. doi: 10.1007/BF00281474.
[7]	E. Milne, The diffusion of imprisoned radiation through a gas, J. Lond. Math. Soc., 1 (1926), 40-51. doi: 10.1112/jlms/s1-1.1.40.
[8]	S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29 (1993), 3389-3405. doi: 10.1029/93WR01495.
[9]	L. Cueto-Felgueroso and R. Juanes, A phase-field model of unsaturated flow, Water Resour. Res., 45 (2009), W10409. doi: 10.1029/2009WR007945.
[10]	L. I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media, IZV. Akad. Nauk SSSR, Ser. Geogr., 12 (1948), 27-45.
[11]	B. C. Aslan, W. W. Hager and S. Moskow, A generalized eigenproblem for the Laplacian which arises in lightning, J. Math. Anal. Appl., 341 (2008), 1028-1041. doi: 10.1016/j.jmaa.2007.11.007.
[12]	R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543. doi: 10.1137/0503051.
[13]	R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42. doi: 10.1137/0506004.
[14]	E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.
[15]	A. Mikelić, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Differ. Equations, 248 (2010), 1561-1577. doi: 10.1016/j.jde.2009.11.022.
[16]	X. Cao and I. S. Pop, Two-phase porous media flows with dynamic capillary effects and hysteresis: Uniqueness of weak solutions, Comput. Math. Appl., 69 (2015), 688-695. doi: 10.1016/j.camwa.2015.02.009.
[17]	Y. Fan and I. S. Pop, Equivalent formulations and numerical schemes for a class of pseudo-parabolic equations, J. Comput. Appl. Math., 246 (2013), 86-93. doi: 10.1016/j.cam.2012.07.031.
[18]	C. M. Cuesta and I. S. Pop, Numerical schemes for a pseudo-parabolic Burgers equation: Discontinuous data and long-time behaviour, J. Comput. Appl. Math., 224 (2009), 269-283. doi: 10.1016/j.cam.2008.05.001.
[19]	B. Schweizer, Regularization of outflow problems in unsaturated porous media with dry regions, J. Differ. Equations, 237 (2007), 278-306. doi: 10.1016/j.jde.2007.03.011.
[20]	E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Periodic boundary value problem for nonlinear Sobolev-type equation, Funct. Anal. Appl., 44 (2010), 171-181. doi: 10.1007/s10688-010-0022-1.
[21]	E. I. Kaikina, Initial boundary value problems for nonlinear pseudo-parabolic equations in a critical case, Electron. J. Differ. Equations, 109 (2007), 1-25.
[22]	T. Matahashi and M. Tsutsumi, On a periodic problem for pseudo-parabolic equations of Sobolev-Galpen type, Mathematica Japonica, 22 (1978), 535-553.
[23]	T. Matahashi and M. Tsutsumi, Periodic solutions of semilinear pseudo-parabolic equations in Hilbert space, Funkc. Ekvacioj, 22 (1979), 51-66.
[24]	R. Z. 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.
[25]	N. I. Bakiyevich and G. A. Shadrin, Cauchy problem for an equation in filtration theory, Sb. Trudov Mosgospedinstituta, 7 (1978), 47-63.
[26]	K. I. Khudaverdiyev and G. M. Farhadova, On global existence for generalized solution of one-dimensional non-self-adjoint mixed problem for a class of fourth order semilinear pseudo-parabolic equations, Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb., 31 (2009), 119-134.
[27]	H. J. Zhao and B. J. Xuan, Existence and convergence of solutions for the generalized BBM-Burgers equations, Nonlinear Anal. Theory Methods Appl., 28 (1997), 1835-1849. doi: 10.1016/S0362-546X(95)00237-P.
[28]	H. M. Yin, Weak and classical solutions of some nonlinear volterra intergro-differential equations, Commun. Part. Diff. Eq., 17 (1992), 1369-1385. doi: 10.1080/03605309208820889.
[29]	S. A. Messaoudi, Blow-up of solutions of a semilinear Heat Equation with a Visco-elastic Term, Progr. Nonlinear Differ. Equ. Appl., 64 (2005), 351-356. doi: 10.1007/3-7643-7385-7_19.
[30]	Y. D. Shang and B. L. Guo, On the problem of the existence of global solutions for a class of nonlinear convolutional integro-differential equations of pseudoparabolic type, Acta Math. Appl. Sin., 26 (2003), 511-524.
[31]	Y. D. Shang and B. L. Guo, Initial boundary value problem and initial value problem for the nonlinear integro-differential equations of pseudoparabolic type, Math. Appl.(Wuhan) 15 (2002), 40-45.
[32]	Y. D. Shang and B. L. Guo, Initial boundary value problem for a class of quasilinear integro-differential equations of pseudoparabolic type, Chin. J. Eng. Math., 20 (2003), 1-6.
[33]	M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal. Theory Methods Appl., 66 (2007), 2653-2675. doi: 10.1016/j.na.2006.03.046.
[34]	R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976.
[35]	S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy, M2AN Math. Model. Numer. Anal., 40 (2006), 765-784. doi: 10.1051/m2an:2006029.
[36]	S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, On a pseudoparabolic problem with constraint, Differ. integral Equ., 19 (2006), 1391-1412.
[37]	G. Vallet, On a degenerated parabolic-hyperbolic problem arising from stratigraphy, Numerical Mathematics and Advanced Applications, Springer, Berlin, (2006), 412-420. doi: 10.1007/978-3-540-34288-5_36.
[38]	N. Igbida, Solutions auto-similaires pour une équation de Barenblatt, Rev. Math. Appl., 17 (1996), 21-36.
[39]	C. Bauzet, J. Giacomoni and G. Vallet, On a class of quasilinear Barenblatt equations, Monografías de la Real Academia de Ciencias de Zaragoza, 38 (2012), 35-51.
[40]	M. Ptashnyk, Nonlinear Pseudoparabolic Equations and Variational Inequalities, Master's Thesis, Heidelberg, 2004. Available from: http://www.ub.uni-heidelberg.de/archiv/4802.
[41]	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.
[42]	M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differ. Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011.
[43]	P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIN: Control Optim. Calc. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116.
[44]	J. L. Lions, Quelques Méthodes de Résolutions des Probléms aux Limites Non Linéaires, Dunod, Paris, 1969.
[45]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[46]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. integral Equ., 6 (1933), 507-533.

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