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Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane
Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms
1. | School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China, China, China |
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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