• Previous Article
    Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics
  • DCDS-B Home
  • This Issue
  • Next Article
    Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane
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

1. 

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

Received  April 2015 Revised  October 2015 Published  January 2016

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.
Citation: Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & 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, (2011).  doi: 10.1515/9783110255294.  Google Scholar

[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).   Google Scholar

[3]

B. K. Shivamoggi, A symmetric regularized long wave equation for shallow water waves,, Phys. Fluids, 29 (1986), 890.  doi: 10.1063/1.865895.  Google Scholar

[4]

P. Rosenau, Evolution and breaking of the ion-acoustic waves,, Phys. Fluids, 31 (1988), 1317.  doi: 10.1063/1.866723.  Google Scholar

[5]

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

[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.  doi: 10.1007/BF00281474.  Google Scholar

[7]

E. Milne, The diffusion of imprisoned radiation through a gas,, J. Lond. Math. Soc., 1 (1926), 40.  doi: 10.1112/jlms/s1-1.1.40.  Google Scholar

[8]

S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media,, Water Resour. Res., 29 (1993), 3389.  doi: 10.1029/93WR01495.  Google Scholar

[9]

L. Cueto-Felgueroso and R. Juanes, A phase-field model of unsaturated flow,, Water Resour. Res., 45 (2009).  doi: 10.1029/2009WR007945.  Google Scholar

[10]

L. I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media,, IZV. Akad. Nauk SSSR, 12 (1948), 27.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.11.007.  Google Scholar

[12]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, SIAM J. Math. Anal., 3 (1972), 527.  doi: 10.1137/0503051.  Google Scholar

[13]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type,, SIAM J. Math. Anal., 6 (1975), 25.  doi: 10.1137/0506004.  Google Scholar

[14]

E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications,, SIAM J. Math. Anal., 12 (1981), 731.  doi: 10.1137/0512062.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.11.022.  Google Scholar

[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.  doi: 10.1016/j.camwa.2015.02.009.  Google Scholar

[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.  doi: 10.1016/j.cam.2012.07.031.  Google Scholar

[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.  doi: 10.1016/j.cam.2008.05.001.  Google Scholar

[19]

B. Schweizer, Regularization of outflow problems in unsaturated porous media with dry regions,, J. Differ. Equations, 237 (2007), 278.  doi: 10.1016/j.jde.2007.03.011.  Google Scholar

[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.  doi: 10.1007/s10688-010-0022-1.  Google Scholar

[21]

E. I. Kaikina, Initial boundary value problems for nonlinear pseudo-parabolic equations in a critical case,, Electron. J. Differ. Equations, 109 (2007), 1.   Google Scholar

[22]

T. Matahashi and M. Tsutsumi, On a periodic problem for pseudo-parabolic equations of Sobolev-Galpen type,, Mathematica Japonica, 22 (1978), 535.   Google Scholar

[23]

T. Matahashi and M. Tsutsumi, Periodic solutions of semilinear pseudo-parabolic equations in Hilbert space,, Funkc. Ekvacioj, 22 (1979), 51.   Google Scholar

[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.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[25]

N. I. Bakiyevich and G. A. Shadrin, Cauchy problem for an equation in filtration theory,, Sb. Trudov Mosgospedinstituta, 7 (1978), 47.   Google Scholar

[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., 31 (2009), 119.   Google Scholar

[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.  doi: 10.1016/S0362-546X(95)00237-P.  Google Scholar

[28]

H. M. Yin, Weak and classical solutions of some nonlinear volterra intergro-differential equations,, Commun. Part. Diff. Eq., 17 (1992), 1369.  doi: 10.1080/03605309208820889.  Google Scholar

[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.  doi: 10.1007/3-7643-7385-7_19.  Google Scholar

[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.   Google Scholar

[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), 15 (2002), 40.   Google Scholar

[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.   Google Scholar

[33]

M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities,, Nonlinear Anal. Theory Methods Appl., 66 (2007), 2653.  doi: 10.1016/j.na.2006.03.046.  Google Scholar

[34]

R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems,, Academic Press, (1976).   Google Scholar

[35]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy,, M2AN Math. Model. Numer. Anal., 40 (2006), 765.  doi: 10.1051/m2an:2006029.  Google Scholar

[36]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, On a pseudoparabolic problem with constraint,, Differ. integral Equ., 19 (2006), 1391.   Google Scholar

[37]

G. Vallet, On a degenerated parabolic-hyperbolic problem arising from stratigraphy,, Numerical Mathematics and Advanced Applications, (2006), 412.  doi: 10.1007/978-3-540-34288-5_36.  Google Scholar

[38]

N. Igbida, Solutions auto-similaires pour une équation de Barenblatt,, Rev. Math. Appl., 17 (1996), 21.   Google Scholar

[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.   Google Scholar

[40]

M. Ptashnyk, Nonlinear Pseudoparabolic Equations and Variational Inequalities,, Master's Thesis, (2004).   Google Scholar

[41]

N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems,, Nonlinear Anal. Real World Appl., 12 (2011), 2625.  doi: 10.1016/j.nonrwa.2011.03.010.  Google Scholar

[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.  doi: 10.1016/j.jde.2004.04.011.  Google Scholar

[43]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIN: Control Optim. Calc. Var., 4 (1999), 419.  doi: 10.1051/cocv:1999116.  Google Scholar

[44]

J. L. Lions, Quelques Méthodes de Résolutions des Probléms aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[45]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar

[46]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differ. integral Equ., 6 (1933), 507.   Google Scholar

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, (2011).  doi: 10.1515/9783110255294.  Google Scholar

[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).   Google Scholar

[3]

B. K. Shivamoggi, A symmetric regularized long wave equation for shallow water waves,, Phys. Fluids, 29 (1986), 890.  doi: 10.1063/1.865895.  Google Scholar

[4]

P. Rosenau, Evolution and breaking of the ion-acoustic waves,, Phys. Fluids, 31 (1988), 1317.  doi: 10.1063/1.866723.  Google Scholar

[5]

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

[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.  doi: 10.1007/BF00281474.  Google Scholar

[7]

E. Milne, The diffusion of imprisoned radiation through a gas,, J. Lond. Math. Soc., 1 (1926), 40.  doi: 10.1112/jlms/s1-1.1.40.  Google Scholar

[8]

S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media,, Water Resour. Res., 29 (1993), 3389.  doi: 10.1029/93WR01495.  Google Scholar

[9]

L. Cueto-Felgueroso and R. Juanes, A phase-field model of unsaturated flow,, Water Resour. Res., 45 (2009).  doi: 10.1029/2009WR007945.  Google Scholar

[10]

L. I. Rubinstein, On the problem of the process of propagation of heat in heterogeneous media,, IZV. Akad. Nauk SSSR, 12 (1948), 27.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.11.007.  Google Scholar

[12]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space,, SIAM J. Math. Anal., 3 (1972), 527.  doi: 10.1137/0503051.  Google Scholar

[13]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type,, SIAM J. Math. Anal., 6 (1975), 25.  doi: 10.1137/0506004.  Google Scholar

[14]

E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications,, SIAM J. Math. Anal., 12 (1981), 731.  doi: 10.1137/0512062.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.11.022.  Google Scholar

[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.  doi: 10.1016/j.camwa.2015.02.009.  Google Scholar

[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.  doi: 10.1016/j.cam.2012.07.031.  Google Scholar

[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.  doi: 10.1016/j.cam.2008.05.001.  Google Scholar

[19]

B. Schweizer, Regularization of outflow problems in unsaturated porous media with dry regions,, J. Differ. Equations, 237 (2007), 278.  doi: 10.1016/j.jde.2007.03.011.  Google Scholar

[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.  doi: 10.1007/s10688-010-0022-1.  Google Scholar

[21]

E. I. Kaikina, Initial boundary value problems for nonlinear pseudo-parabolic equations in a critical case,, Electron. J. Differ. Equations, 109 (2007), 1.   Google Scholar

[22]

T. Matahashi and M. Tsutsumi, On a periodic problem for pseudo-parabolic equations of Sobolev-Galpen type,, Mathematica Japonica, 22 (1978), 535.   Google Scholar

[23]

T. Matahashi and M. Tsutsumi, Periodic solutions of semilinear pseudo-parabolic equations in Hilbert space,, Funkc. Ekvacioj, 22 (1979), 51.   Google Scholar

[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.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[25]

N. I. Bakiyevich and G. A. Shadrin, Cauchy problem for an equation in filtration theory,, Sb. Trudov Mosgospedinstituta, 7 (1978), 47.   Google Scholar

[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., 31 (2009), 119.   Google Scholar

[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.  doi: 10.1016/S0362-546X(95)00237-P.  Google Scholar

[28]

H. M. Yin, Weak and classical solutions of some nonlinear volterra intergro-differential equations,, Commun. Part. Diff. Eq., 17 (1992), 1369.  doi: 10.1080/03605309208820889.  Google Scholar

[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.  doi: 10.1007/3-7643-7385-7_19.  Google Scholar

[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.   Google Scholar

[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), 15 (2002), 40.   Google Scholar

[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.   Google Scholar

[33]

M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities,, Nonlinear Anal. Theory Methods Appl., 66 (2007), 2653.  doi: 10.1016/j.na.2006.03.046.  Google Scholar

[34]

R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems,, Academic Press, (1976).   Google Scholar

[35]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy,, M2AN Math. Model. Numer. Anal., 40 (2006), 765.  doi: 10.1051/m2an:2006029.  Google Scholar

[36]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, On a pseudoparabolic problem with constraint,, Differ. integral Equ., 19 (2006), 1391.   Google Scholar

[37]

G. Vallet, On a degenerated parabolic-hyperbolic problem arising from stratigraphy,, Numerical Mathematics and Advanced Applications, (2006), 412.  doi: 10.1007/978-3-540-34288-5_36.  Google Scholar

[38]

N. Igbida, Solutions auto-similaires pour une équation de Barenblatt,, Rev. Math. Appl., 17 (1996), 21.   Google Scholar

[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.   Google Scholar

[40]

M. Ptashnyk, Nonlinear Pseudoparabolic Equations and Variational Inequalities,, Master's Thesis, (2004).   Google Scholar

[41]

N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems,, Nonlinear Anal. Real World Appl., 12 (2011), 2625.  doi: 10.1016/j.nonrwa.2011.03.010.  Google Scholar

[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.  doi: 10.1016/j.jde.2004.04.011.  Google Scholar

[43]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIN: Control Optim. Calc. Var., 4 (1999), 419.  doi: 10.1051/cocv:1999116.  Google Scholar

[44]

J. L. Lions, Quelques Méthodes de Résolutions des Probléms aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar

[45]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar

[46]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differ. integral Equ., 6 (1933), 507.   Google Scholar

[1]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[2]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[3]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[4]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[5]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[6]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[7]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[8]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[9]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282

[10]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109

[11]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[12]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[13]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[14]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[15]

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, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[16]

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, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[17]

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

[18]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[19]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

[20]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (99)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]