February  2022, 21(2): 585-623. doi: 10.3934/cpaa.2021190

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 Published  February 2022 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 and Applied Analysis, 2022, 21 (2) : 585-623. doi: 10.3934/cpaa.2021190
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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[39]

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

[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. 

[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.

[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.

[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.

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[39]

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

[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. 

[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.

[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.

[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.

[1]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121

[2]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

[3]

Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106

[4]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[5]

Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827

[6]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051

[7]

Kunio Hidano, Kazuyoshi Yokoyama. Global existence and blow up for systems of nonlinear wave equations related to the weak null condition. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4385-4414. doi: 10.3934/dcds.2022058

[8]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

[9]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

[10]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117

[11]

Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019

[12]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[13]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621

[14]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic and Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043

[15]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[16]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169

[17]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535

[18]

Quang-Minh Tran, Hong-Danh Pham. Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4521-4550. doi: 10.3934/dcdss.2021135

[19]

Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022111

[20]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (185)
  • HTML views (136)
  • Cited by (0)

[Back to Top]