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December  2021, 29(6): 3687-3720. doi: 10.3934/era.2021057

Semilinear pseudo-parabolic equations on manifolds with conical singularities

1. 

College of Mathematical Sciences, Harbin Engineering University, 150001, China

2. 

College of Intelligent Systems Science and Engineering, Harbin Engineering University, 150001, China

* Corresponding author

Received  March 2021 Revised  May 2021 Published  December 2021 Early access  August 2021

This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.

Citation: Yitian Wang, Xiaoping Liu, Yuxuan Chen. Semilinear pseudo-parabolic equations on manifolds with conical singularities. Electronic Research Archive, 2021, 29 (6) : 3687-3720. doi: 10.3934/era.2021057
References:
[1]

M. Alimohammady and M. K. Kalleji, Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration, J. Func. Anal., 265 (2013), 2331-2356.  doi: 10.1016/j.jfa.2013.07.013.  Google Scholar

[2]

D. Andreucci and A. F. Tedeev, The Cauchy Dirichlet problem for the porous media equation in cone-like domains, SIAM J. Math. Anal, 46 (2014), 1427-1455.  doi: 10.1137/130912177.  Google Scholar

[3]

A. BahrouniV. D. Rǎdulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.  doi: 10.1088/1361-6544/ab0b03.  Google Scholar

[4]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[5]

G. I. BarenblattI. P. Zeltov and I. N. Kockina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[6]

R. Bellman, The Stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643-647.   Google Scholar

[7]

G. BonannoG. Molica Bisci and V. D. Rǎdulescu, Weak solutions and energy estimates for a class of nonlinear elliptic Neumann problems, Adv. Nonlinear Stud., 13 (2013), 373-389.  doi: 10.1515/ans-2013-0207.  Google Scholar

[8]

Y. Cao and J. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 631-642.  doi: 10.3934/dcds.2016.36.631.  Google Scholar

[9]

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

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, Oxford, 1998.   Google Scholar
[11]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.  Google Scholar

[12]

H. ChenX. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar

[13]

H. ChenX. Liu and Y. Wei, Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents, J. Differential Equations, 252 (2012), 4200-4228.  doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[14]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[15]

Y. Chen and R. Z. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664.  Google Scholar

[16]

B. D. ColemanR. J. Duffin and V. J. Mizel, Instability, uniqueness and nonexistence theorems for the equation $u_t = u_xx-u_xtx$ on a strip, Arch. Ration. Mech. Anal., 19 (1965), 100-116.  doi: 10.1007/BF00282277.  Google Scholar

[17]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493–516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[18]

Y. V. Egorov and B. W. Schulze, Pseudo-differential operators, singularities, applications, In: Operator Theory: Advances and Applications, vol. 93. Birkhäser Verlag, Basel 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

[19]

A. El Hamidi and G. G. Laptev, Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains, Electron. J. Differential Equations, (2002), No. 97, 19 pp.  Google Scholar

[20]

R. FilippucciP. Pucci and V. Rǎdulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.  doi: 10.1080/03605300701518208.  Google Scholar

[21]

F. Gazzola and V. Rǎdulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbb{R}^n$, Differential Integral Equations, 13 (2000), 47-60.   Google Scholar

[22]

D. GoelV. D. Rǎdulescu and K. Sreenadh, Coron problem for nonlocal equations involving Choquard nonlinearity, Adv. Nonlinear Stud., 20 (2020), 141-161.  doi: 10.1515/ans-2019-2064.  Google Scholar

[23]

M. E. Gurtin and W. O. Williams, An axiomatic foundation for continuum thermodynamics, Arch. Ration. Mech. Anal., 26 (1967), 83-117.  doi: 10.1007/BF00285676.  Google Scholar

[24]

M. E. Gurtin and W. O. Williams, On the Clausius-Duhem inequality, Journal of Applied Mathematics and Physics (ZAMP), 17 (1966), 626-633.  doi: 10.1007/BF01597243.  Google Scholar

[25]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.   Google Scholar

[26]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Soviet Math., 10 (1978), 53-70.   Google Scholar

[27]

S. Khomrutai, Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in $\mathbb{R}^n$ with a sublinear source, Appl. Math. Lett., 48 (2015), 8-13.  doi: 10.1016/j.aml.2015.03.008.  Google Scholar

[28]

S. Khomrutai, Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differential Equations, 260 (2016), 3598-3657.  doi: 10.1016/j.jde.2015.10.043.  Google Scholar

[29]

V. A. Kondrat'ev and O. A. Oleinik, Boundary-value problems for partial differential equations in non-smooth domains, Uspekhi Mat. Nauk, 38 (1983), 3-76.   Google Scholar

[30]

G. G. Laptev, Non-existence of global solutions for higher-order evolution inequalities in unbounded cone-like domains, Mosc. Math. J., 3 (2003), 63-84.  doi: 10.17323/1609-4514-2003-3-1-63-84.  Google Scholar

[31]

M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conical singularities, Topology, 32 (1993), 611-623.  doi: 10.1016/0040-9383(93)90012-K.  Google Scholar

[32]

Z. Li and W. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181-3203.  doi: 10.1007/s00033-015-0558-2.  Google Scholar

[33]

S. Lian and C. Liu, On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone, Arch. Math. (Basel), 94 (2010), 245-253.  doi: 10.1007/s00013-009-0081-9.  Google Scholar

[34]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[35]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[36]

W. Liu and J. Yu, A note on blow-up of solution for a class ofsemilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276-1283.  doi: 10.1016/j.jfa.2018.01.005.  Google Scholar

[37]

X. P. Liu, The Research on Well-Posedness of Solutions for two Classes of Nonlinear Parabolic Equation, Harbin Engneering University, Harbin, 2016. Google Scholar

[38]

R. MazzeoY. A. Rubinstein and N. Sesum, Ricci flow on surfaces with conic singularities, Anal. PDE, 8 (2015), 839-882.  doi: 10.2140/apde.2015.8.839.  Google Scholar

[39]

R. Michele, "Footballs", Conical singularities and the Liouville equation, Physical Review D, 71 (2005), 044006, 16 pp. Google Scholar

[40]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[41]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764.  doi: 10.1515/ans-2016-0023.  Google Scholar

[42]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear nonhomogeneous singular problems, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 9, 31 pp. doi: 10.1007/s00526-019-1667-0.  Google Scholar

[43]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[44]

N. Roidos and E. Schrohe, Existence and maximal $L_p$-regularity of solutions for the porous medium equation on manifolds with conical singularities, Comm. Partial Differential Equations, 41 (2016), 1441-1471.  doi: 10.1080/03605302.2016.1219745.  Google Scholar

[45]

B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester 1998.  Google Scholar

[46]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[47]

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

[48]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[49]

R. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations", J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[50]

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

[51]

R. XuX. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial ernegy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[52]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[53]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[54]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

show all references

References:
[1]

M. Alimohammady and M. K. Kalleji, Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration, J. Func. Anal., 265 (2013), 2331-2356.  doi: 10.1016/j.jfa.2013.07.013.  Google Scholar

[2]

D. Andreucci and A. F. Tedeev, The Cauchy Dirichlet problem for the porous media equation in cone-like domains, SIAM J. Math. Anal, 46 (2014), 1427-1455.  doi: 10.1137/130912177.  Google Scholar

[3]

A. BahrouniV. D. Rǎdulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.  doi: 10.1088/1361-6544/ab0b03.  Google Scholar

[4]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[5]

G. I. BarenblattI. P. Zeltov and I. N. Kockina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[6]

R. Bellman, The Stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643-647.   Google Scholar

[7]

G. BonannoG. Molica Bisci and V. D. Rǎdulescu, Weak solutions and energy estimates for a class of nonlinear elliptic Neumann problems, Adv. Nonlinear Stud., 13 (2013), 373-389.  doi: 10.1515/ans-2013-0207.  Google Scholar

[8]

Y. Cao and J. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 631-642.  doi: 10.3934/dcds.2016.36.631.  Google Scholar

[9]

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

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, Oxford, 1998.   Google Scholar
[11]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.  Google Scholar

[12]

H. ChenX. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar

[13]

H. ChenX. Liu and Y. Wei, Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents, J. Differential Equations, 252 (2012), 4200-4228.  doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[14]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[15]

Y. Chen and R. Z. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664.  Google Scholar

[16]

B. D. ColemanR. J. Duffin and V. J. Mizel, Instability, uniqueness and nonexistence theorems for the equation $u_t = u_xx-u_xtx$ on a strip, Arch. Ration. Mech. Anal., 19 (1965), 100-116.  doi: 10.1007/BF00282277.  Google Scholar

[17]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493–516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[18]

Y. V. Egorov and B. W. Schulze, Pseudo-differential operators, singularities, applications, In: Operator Theory: Advances and Applications, vol. 93. Birkhäser Verlag, Basel 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

[19]

A. El Hamidi and G. G. Laptev, Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains, Electron. J. Differential Equations, (2002), No. 97, 19 pp.  Google Scholar

[20]

R. FilippucciP. Pucci and V. Rǎdulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.  doi: 10.1080/03605300701518208.  Google Scholar

[21]

F. Gazzola and V. Rǎdulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbb{R}^n$, Differential Integral Equations, 13 (2000), 47-60.   Google Scholar

[22]

D. GoelV. D. Rǎdulescu and K. Sreenadh, Coron problem for nonlocal equations involving Choquard nonlinearity, Adv. Nonlinear Stud., 20 (2020), 141-161.  doi: 10.1515/ans-2019-2064.  Google Scholar

[23]

M. E. Gurtin and W. O. Williams, An axiomatic foundation for continuum thermodynamics, Arch. Ration. Mech. Anal., 26 (1967), 83-117.  doi: 10.1007/BF00285676.  Google Scholar

[24]

M. E. Gurtin and W. O. Williams, On the Clausius-Duhem inequality, Journal of Applied Mathematics and Physics (ZAMP), 17 (1966), 626-633.  doi: 10.1007/BF01597243.  Google Scholar

[25]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.   Google Scholar

[26]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Soviet Math., 10 (1978), 53-70.   Google Scholar

[27]

S. Khomrutai, Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in $\mathbb{R}^n$ with a sublinear source, Appl. Math. Lett., 48 (2015), 8-13.  doi: 10.1016/j.aml.2015.03.008.  Google Scholar

[28]

S. Khomrutai, Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differential Equations, 260 (2016), 3598-3657.  doi: 10.1016/j.jde.2015.10.043.  Google Scholar

[29]

V. A. Kondrat'ev and O. A. Oleinik, Boundary-value problems for partial differential equations in non-smooth domains, Uspekhi Mat. Nauk, 38 (1983), 3-76.   Google Scholar

[30]

G. G. Laptev, Non-existence of global solutions for higher-order evolution inequalities in unbounded cone-like domains, Mosc. Math. J., 3 (2003), 63-84.  doi: 10.17323/1609-4514-2003-3-1-63-84.  Google Scholar

[31]

M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conical singularities, Topology, 32 (1993), 611-623.  doi: 10.1016/0040-9383(93)90012-K.  Google Scholar

[32]

Z. Li and W. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181-3203.  doi: 10.1007/s00033-015-0558-2.  Google Scholar

[33]

S. Lian and C. Liu, On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone, Arch. Math. (Basel), 94 (2010), 245-253.  doi: 10.1007/s00013-009-0081-9.  Google Scholar

[34]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[35]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[36]

W. Liu and J. Yu, A note on blow-up of solution for a class ofsemilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276-1283.  doi: 10.1016/j.jfa.2018.01.005.  Google Scholar

[37]

X. P. Liu, The Research on Well-Posedness of Solutions for two Classes of Nonlinear Parabolic Equation, Harbin Engneering University, Harbin, 2016. Google Scholar

[38]

R. MazzeoY. A. Rubinstein and N. Sesum, Ricci flow on surfaces with conic singularities, Anal. PDE, 8 (2015), 839-882.  doi: 10.2140/apde.2015.8.839.  Google Scholar

[39]

R. Michele, "Footballs", Conical singularities and the Liouville equation, Physical Review D, 71 (2005), 044006, 16 pp. Google Scholar

[40]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[41]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764.  doi: 10.1515/ans-2016-0023.  Google Scholar

[42]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear nonhomogeneous singular problems, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 9, 31 pp. doi: 10.1007/s00526-019-1667-0.  Google Scholar

[43]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[44]

N. Roidos and E. Schrohe, Existence and maximal $L_p$-regularity of solutions for the porous medium equation on manifolds with conical singularities, Comm. Partial Differential Equations, 41 (2016), 1441-1471.  doi: 10.1080/03605302.2016.1219745.  Google Scholar

[45]

B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester 1998.  Google Scholar

[46]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[47]

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

[48]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[49]

R. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations", J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[50]

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

[51]

R. XuX. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial ernegy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[52]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[53]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[54]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

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