doi: 10.3934/cpaa.2021190
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On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

1. 

University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam

2. 

Department of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City, Eastern International University, Nam Ky Khoi Nghia Str., Hoa Phu Ward, Thu Dau Mot City, Binh Duong Province, Vietnam

3. 

Nguyen Tat Thanh University, 300A Nguyen Tat Thanh Str., Dist. 4, Ho Chi Minh City, Vietnam

4. 

Department of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam

* Corresponding author

Received  April 2021 Revised  September 2021 Early access November 2021

Fund Project: This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2020-18-01

In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.

Citation: Le Thi Phuong Ngoc, Khong Thi Thao Uyen, Nguyen Huu Nhan, Nguyen Thanh Long. On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021190
References:
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Ch. J. AmickJ. L. Bona and M. E. Schonbeck, Decay of solutions of some nonlinear wave equations, J. Differ. Equ., 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

G. BarenblatI. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.   Google Scholar

[3]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

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A. Bouziani, Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition, Nonlinear Anal., 55 (2003), 883-904.  doi: 10.1016/j.na.2003.07.011.  Google Scholar

[5]

Y. CaoJ. Yin and C. Wang, Cauchy problems of semilinear pseudoparabolic equations, J. Differ. Equ., 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar

[6]

Y. CaoZ. Wang and J. Yin, A note on the lifespan of semilinear pseudo-parabolic equation, Appl. Math. Lett., 98 (2019), 406-410.  doi: 10.1016/j.aml.2019.06.039.  Google Scholar

[7]

S. Chen and J. Yu, Dynamics of a diffusive predator–prey system with anonlinear growth rate for the predator, J. Differ. Equ., 260 (2016), 7923-7939.  doi: 10.1016/j.jde.2016.02.007.  Google Scholar

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D. Q. Dai and Y. Huang, A moment problem for one-dimensional nonlinear pseudoparabolic equation, J. Math. Anal. Appl., 328 (2007), 1057-1067.  doi: 10.1016/j.jmaa.2006.06.010.  Google Scholar

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C. Goudjo, B. Lèye and M. Sy, Weak solution to a parabolic nonlinear system arising in biological dynamic in the soil, Int. J. Differ. Equ., 2011 (2011), 24 pp. doi: 10.1155/2011/831436.  Google Scholar

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T. HayatM. Khan and M. Ayub, Some analytical solutions for second grade fluid flows for cylindrical geometries, Math. Comp. Model., 43 (2006), 16-29.  doi: 10.1016/j.mcm.2005.04.009.  Google Scholar

[11]

T. HayatF. Shahzad and M. Ayub, Analytical solution for the steady flow of the third grade fluid in a porous half space, Appl. Math. Model., 31 (2007), 2424-2432.   Google Scholar

[12]

L. KongX. Wang and X. Zhao, Asymptotic analysis to a parabolic system with weighted localized sources and inner absorptions, Arch. Math., 99 (2012), 375-386.  doi: 10.1007/s00013-012-0433-8.  Google Scholar

[13]

B. LèyeN.N. DoanhO. MongaP. Garnier and N. Nunan, Simulating biological dynamics using partial differential equations: Application to decomposition of organic matter in 3D soil structure, Vietnam J. Math., 43 (2015), 801-817.  doi: 10.1007/s10013-015-0159-6.  Google Scholar

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J. L. Lions, Quelques méthodes de résolution des problémes aux limites non-linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Meth. Appl. Sci., 38 (2015), 2636-2641.  doi: 10.1002/mma.3253.  Google Scholar

[16]

A. Sh. Lyubanova, On some boundary value problems for systems of pseudoparabolic equations, Siberian Math. J., 56 (2015), 662-677.  doi: 10.1134/s0037446615040102.  Google Scholar

[17]

A Sh. Lyubanova, Nonlinear boundary value problem for pseudoparabolic equation, J. Math. Anal. Appl., 493 (2021), 124514.  doi: 10.1016/j.jmaa.2020.124514.  Google Scholar

[18]

S. A. Messaoudi and A. A. Talahmeh, Blow up in a semilinear pseudoparabolic equation with variable exponents, Annali Dell'Universita' Di Ferrara, 65 (2019), 311-326.  doi: 10.1007/s11565-019-00326-1.  Google Scholar

[19]

M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl., 352 (2009), 629-633.  doi: 10.1016/j.jmaa.2008.11.016.  Google Scholar

[20]

L. T. P. NgocN. H. Nhan and N. T. Long, General decay and blow-up results for a nonlinear pseudoparabolic equation with Robin-Dirichlet conditions, Math. Meth. Appl. Sci., 44 (2021), 8697-8725.  doi: 10.1002/mma.7299.  Google Scholar

[21]

N. T. Orumbayeva and A. B. Keldibekova, On one solution of a periodic boundary-value problem for a third-order pseudoparabolic equation, Lobachevskii J. Math., 41 (2020), 1864-1872.  doi: 10.1134/s1995080220090218.  Google Scholar

[22]

V. Padron, Effect of aggregation on population recovery modeled by a forward-backward, in pseudoparabolic equation,, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396.  doi: 10.1016/j.jmaa.2009.01.010.  Google Scholar

[25]

N. S. Popov, Solvability of a boundary value problem for a pseudoparabolic equation with nonlocal integral conditions, Differ. Equ., 51 (2015), 362-375.  doi: 10.1134/S0012266115030076.  Google Scholar

[26]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Higher Education, 1987.  Google Scholar

[27]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[28]

R. E. Showalter and T. W. Ting, Asymptotic behavior of solutions of pseudoparabolic partial differential equations, Annali Mat. Pura Appl., 90 (1971), 241-258.  doi: 10.1007/BF02415050.  Google Scholar

[29]

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.  Google Scholar

[30]

R. E. Showater, Hilbert space methods for partial differential equations, Electron. J. Differ. Equ., Monograph 01, 1994.  Google Scholar

[31]

S. L. Sobolev, A new problem in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50.   Google Scholar

[32]

F. SunL. Liu and Y. Wu, Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98 (2019), 735-755.  doi: 10.1080/00036811.2017.1400536.  Google Scholar

[33]

Y. Tian and Z. Xiang, Global solutions to a 3D chemotaxis-Stokes system with nonlinear cell diffusion and Robin signal boundary condition, J. Differ. Equ., 269 (2020), 2012-2056.  doi: 10.1016/j.jde.2020.01.031.  Google Scholar

[34]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.  Google Scholar

[35]

B. B. Tsegaw, Nonexistence of solutions to Cauchy problems for anisotropic pseudoparabolic equations, J. Ellip. Para. Equ., 6 (2020), 919-934.  doi: 10.1007/s41808-020-00087-5.  Google Scholar

[36]

E. Vitillaro, Global existence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171.  Google Scholar

[37]

R. 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.  Google Scholar

[38]

G. Xu and J. Zhou, Lifespan for a semilinear pseudo-parabolic equation, Math. Meth. Appl. Sci., 41 (2018), 705-713.  doi: 10.1002/mma.4639.  Google Scholar

[39]

E. V. Yushkov, Existence and blow-up of solutions of a pseudoparabolic equation, Differ. Equ., 47 (2011), 291-295.  doi: 10.1134/S0012266111020169.  Google Scholar

[40]

K. Zennir and T. Miyasita, Lifespan of solutions for a class of pseudoparabolic equation with weak memory, Alex. Engineer. J., 59 (2020), 957-964.   Google Scholar

[41]

L. Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Anal. TMA., 25 (1995), 1343-1369.  doi: 10.1016/0362-546X(94)00252-D.  Google Scholar

[42]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.  Google Scholar

[43]

X. ZhuF. Li and Y. Li, Global solutions and blow-up solutions to a class pseudoparabolic equations with nonlocal term, Appl. Math. Comp., 329 (2018), 38-51.  doi: 10.1016/j.amc.2018.02.003.  Google Scholar

show all references

References:
[1]

Ch. J. AmickJ. L. Bona and M. E. Schonbeck, Decay of solutions of some nonlinear wave equations, J. Differ. Equ., 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

G. BarenblatI. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.   Google Scholar

[3]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[4]

A. Bouziani, Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition, Nonlinear Anal., 55 (2003), 883-904.  doi: 10.1016/j.na.2003.07.011.  Google Scholar

[5]

Y. CaoJ. Yin and C. Wang, Cauchy problems of semilinear pseudoparabolic equations, J. Differ. Equ., 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar

[6]

Y. CaoZ. Wang and J. Yin, A note on the lifespan of semilinear pseudo-parabolic equation, Appl. Math. Lett., 98 (2019), 406-410.  doi: 10.1016/j.aml.2019.06.039.  Google Scholar

[7]

S. Chen and J. Yu, Dynamics of a diffusive predator–prey system with anonlinear growth rate for the predator, J. Differ. Equ., 260 (2016), 7923-7939.  doi: 10.1016/j.jde.2016.02.007.  Google Scholar

[8]

D. Q. Dai and Y. Huang, A moment problem for one-dimensional nonlinear pseudoparabolic equation, J. Math. Anal. Appl., 328 (2007), 1057-1067.  doi: 10.1016/j.jmaa.2006.06.010.  Google Scholar

[9]

C. Goudjo, B. Lèye and M. Sy, Weak solution to a parabolic nonlinear system arising in biological dynamic in the soil, Int. J. Differ. Equ., 2011 (2011), 24 pp. doi: 10.1155/2011/831436.  Google Scholar

[10]

T. HayatM. Khan and M. Ayub, Some analytical solutions for second grade fluid flows for cylindrical geometries, Math. Comp. Model., 43 (2006), 16-29.  doi: 10.1016/j.mcm.2005.04.009.  Google Scholar

[11]

T. HayatF. Shahzad and M. Ayub, Analytical solution for the steady flow of the third grade fluid in a porous half space, Appl. Math. Model., 31 (2007), 2424-2432.   Google Scholar

[12]

L. KongX. Wang and X. Zhao, Asymptotic analysis to a parabolic system with weighted localized sources and inner absorptions, Arch. Math., 99 (2012), 375-386.  doi: 10.1007/s00013-012-0433-8.  Google Scholar

[13]

B. LèyeN.N. DoanhO. MongaP. Garnier and N. Nunan, Simulating biological dynamics using partial differential equations: Application to decomposition of organic matter in 3D soil structure, Vietnam J. Math., 43 (2015), 801-817.  doi: 10.1007/s10013-015-0159-6.  Google Scholar

[14]

J. L. Lions, Quelques méthodes de résolution des problémes aux limites non-linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Meth. Appl. Sci., 38 (2015), 2636-2641.  doi: 10.1002/mma.3253.  Google Scholar

[16]

A. Sh. Lyubanova, On some boundary value problems for systems of pseudoparabolic equations, Siberian Math. J., 56 (2015), 662-677.  doi: 10.1134/s0037446615040102.  Google Scholar

[17]

A Sh. Lyubanova, Nonlinear boundary value problem for pseudoparabolic equation, J. Math. Anal. Appl., 493 (2021), 124514.  doi: 10.1016/j.jmaa.2020.124514.  Google Scholar

[18]

S. A. Messaoudi and A. A. Talahmeh, Blow up in a semilinear pseudoparabolic equation with variable exponents, Annali Dell'Universita' Di Ferrara, 65 (2019), 311-326.  doi: 10.1007/s11565-019-00326-1.  Google Scholar

[19]

M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl., 352 (2009), 629-633.  doi: 10.1016/j.jmaa.2008.11.016.  Google Scholar

[20]

L. T. P. NgocN. H. Nhan and N. T. Long, General decay and blow-up results for a nonlinear pseudoparabolic equation with Robin-Dirichlet conditions, Math. Meth. Appl. Sci., 44 (2021), 8697-8725.  doi: 10.1002/mma.7299.  Google Scholar

[21]

N. T. Orumbayeva and A. B. Keldibekova, On one solution of a periodic boundary-value problem for a third-order pseudoparabolic equation, Lobachevskii J. Math., 41 (2020), 1864-1872.  doi: 10.1134/s1995080220090218.  Google Scholar

[22]

V. Padron, Effect of aggregation on population recovery modeled by a forward-backward, in pseudoparabolic equation,, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396.  doi: 10.1016/j.jmaa.2009.01.010.  Google Scholar

[25]

N. S. Popov, Solvability of a boundary value problem for a pseudoparabolic equation with nonlocal integral conditions, Differ. Equ., 51 (2015), 362-375.  doi: 10.1134/S0012266115030076.  Google Scholar

[26]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Higher Education, 1987.  Google Scholar

[27]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[28]

R. E. Showalter and T. W. Ting, Asymptotic behavior of solutions of pseudoparabolic partial differential equations, Annali Mat. Pura Appl., 90 (1971), 241-258.  doi: 10.1007/BF02415050.  Google Scholar

[29]

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.  Google Scholar

[30]

R. E. Showater, Hilbert space methods for partial differential equations, Electron. J. Differ. Equ., Monograph 01, 1994.  Google Scholar

[31]

S. L. Sobolev, A new problem in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50.   Google Scholar

[32]

F. SunL. Liu and Y. Wu, Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98 (2019), 735-755.  doi: 10.1080/00036811.2017.1400536.  Google Scholar

[33]

Y. Tian and Z. Xiang, Global solutions to a 3D chemotaxis-Stokes system with nonlinear cell diffusion and Robin signal boundary condition, J. Differ. Equ., 269 (2020), 2012-2056.  doi: 10.1016/j.jde.2020.01.031.  Google Scholar

[34]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.  Google Scholar

[35]

B. B. Tsegaw, Nonexistence of solutions to Cauchy problems for anisotropic pseudoparabolic equations, J. Ellip. Para. Equ., 6 (2020), 919-934.  doi: 10.1007/s41808-020-00087-5.  Google Scholar

[36]

E. Vitillaro, Global existence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171.  Google Scholar

[37]

R. 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.  Google Scholar

[38]

G. Xu and J. Zhou, Lifespan for a semilinear pseudo-parabolic equation, Math. Meth. Appl. Sci., 41 (2018), 705-713.  doi: 10.1002/mma.4639.  Google Scholar

[39]

E. V. Yushkov, Existence and blow-up of solutions of a pseudoparabolic equation, Differ. Equ., 47 (2011), 291-295.  doi: 10.1134/S0012266111020169.  Google Scholar

[40]

K. Zennir and T. Miyasita, Lifespan of solutions for a class of pseudoparabolic equation with weak memory, Alex. Engineer. J., 59 (2020), 957-964.   Google Scholar

[41]

L. Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Anal. TMA., 25 (1995), 1343-1369.  doi: 10.1016/0362-546X(94)00252-D.  Google Scholar

[42]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.  Google Scholar

[43]

X. ZhuF. Li and Y. Li, Global solutions and blow-up solutions to a class pseudoparabolic equations with nonlocal term, Appl. Math. Comp., 329 (2018), 38-51.  doi: 10.1016/j.amc.2018.02.003.  Google Scholar

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