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[J]. Electronic Research Archive, 2021, 29(6): 3687-3720. doi: 10.3934/era.2021057
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.
[1] | Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration. J. Func. Anal. (2013) 265: 2331-2356. |
[2] | The Cauchy Dirichlet problem for the porous media equation in cone-like domains. SIAM J. Math. Anal (2014) 46: 1427-1455. |
[3] | Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves. Nonlinearity (2019) 32: 2481-2495. |
[4] | Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford Ser. (1977) 28: 473-486. |
[5] | Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. (1960) 24: 1286-1303. |
[6] | The Stability of solutions of linear differential equations. Duke Math. J. (1943) 10: 643-647. |
[7] | Weak solutions and energy estimates for a class of nonlinear elliptic Neumann problems. Adv. Nonlinear Stud. (2013) 13: 373-389. |
[8] | Small perturbation of a semilinear pseudo-parabolic equation. Discrete Contin. Dyn. Syst. (2016) 36: 631-642. |
[9] | Cauchy problems of semilinear pseudo-parabolic equations. J. Differential Equations (2009) 246: 4568-4590. |
[10] | (1998) Perfect Incompressible Fluids. Oxford: Oxford University Press. |
[11] | Global existence and nonexistence for semilinear parabolic equations with conical degeneration. J. Pseudo-Differ. Oper. Appl. (2012) 3: 329-349. |
[12] | Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities. Calc. Var. Partial Differential Equations (2012) 43: 463-484. |
[13] | Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents. J. Differential Equations (2012) 252: 4200-4228. |
[14] | Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differential Equations (2015) 258: 4424-4442. |
[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 |
[16] | Instability, uniqueness and nonexistence theorems for the equation $u_t = u_xx-u_xtx$ on a strip. Arch. Ration. Mech. Anal. (1965) 19: 100-116. |
[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 |
[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 |
[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. |
[20] | Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions. Comm. Partial Differential Equations (2008) 33: 706-717. |
[21] | A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbb{R}^n$. Differential Integral Equations (2000) 13: 47-60. |
[22] | Coron problem for nonlocal equations involving Choquard nonlinearity. Adv. Nonlinear Stud. (2020) 20: 141-161. |
[23] | An axiomatic foundation for continuum thermodynamics. Arch. Ration. Mech. Anal. (1967) 26: 83-117. |
[24] | On the Clausius-Duhem inequality. Journal of Applied Mathematics and Physics (ZAMP) (1966) 17: 626-633. |
[25] | Three-manifolds with positive Ricci curvature. J. Differential Geom. (1982) 17: 255-306. |
[26] | The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. Soviet Math. (1978) 10: 53-70. |
[27] | Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in $\mathbb{R}^n$ with a sublinear source. Appl. Math. Lett. (2015) 48: 8-13. |
[28] | Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation. J. Differential Equations (2016) 260: 3598-3657. |
[29] | Boundary-value problems for partial differential equations in non-smooth domains. Uspekhi Mat. Nauk (1983) 38: 3-76. |
[30] | Non-existence of global solutions for higher-order evolution inequalities in unbounded cone-like domains. Mosc. Math. J. (2003) 3: 63-84. |
[31] | Deficiency indices for symmetric Dirac operators on manifolds with conical singularities. Topology (1993) 32: 611-623. |
[32] | Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. Z. Angew. Math. Phys. (2015) 66: 3181-3203. |
[33] | On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone. Arch. Math. (Basel) (2010) 94: 245-253. |
[34] | Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. |
[35] | J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, (French) Dunod; Gauthier-Villars, Paris, 1969. |
[36] | A note on blow-up of solution for a class ofsemilinear pseudo-parabolic equations. J. Funct. Anal. (2018) 274: 1276-1283. |
[37] | X. P. Liu, The Research on Well-Posedness of Solutions for two Classes of Nonlinear Parabolic Equation, Harbin Engneering University, Harbin, 2016. |
[38] | Ricci flow on surfaces with conic singularities. Anal. PDE (2015) 8: 839-882. |
[39] | R. Michele, "Footballs", Conical singularities and the Liouville equation, Physical Review D, 71 (2005), 044006, 16 pp. |
[40] | Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differential Equations (2014) 256: 2449-2479. |
[41] | Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. (2016) 16: 737-764. |
[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 |
[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 |
[44] | Existence and maximal $L_p$-regularity of solutions for the porous medium equation on manifolds with conical singularities. Comm. Partial Differential Equations (2016) 41: 1441-1471. |
[45] | B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester 1998. |
[46] | Compact sets in the space $L^p(0, T;B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96. |
[47] | Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal. (1963) 14: 1-26. |
[48] | Global well-posedness of coupled parabolic systems. Sci. China Math. (2020) 63: 321-356. |
[49] | Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations". J. Funct. Anal. (2016) 270: 4039-4041. |
[50] | Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763. |
[51] | Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial ernegy. Appl. Math. Lett. (2018) 83: 176-181. |
[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 |
[53] | The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete Contin. Dyn. Syst. (2017) 37: 5631-5649. |
[54] | Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up. Commun. Pure Appl. Anal. (2019) 18: 1351-1358. |