Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term. (English) Zbl 1407.35122
Summary: In this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term
\[
u_t-\Delta u-\Delta u_t +\int^t_0 g(t-\tau)\Delta u(\tau)\mathrm{d}\tau=| u|^{p-2}u,\quad \text{in }\Omega\times (0,T),
\]
where \(\Omega\) is a bounded domain in \(\mathbb R^n(n\geq 1)\) with a Dirichlet boundary condition. Under suitable assumptions on the initial data \(u_0\) and the relaxation function \(g\), we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. \(J(u(0))\leq d(\infty)\)), by using the Galerkin method, the concavity method and an improved potential well method involving time \(t\). We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.
MSC:
| 35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B44 | Blow-up in context of PDEs |
References:
| [1] | Benjamin, Tb; Bona, Jl; Mahony, Jj, Model equations for long waves in nonlinear dispersive systems, Philos Trans R Soc London Ser A Math Phys Sci, 272, 1220, 47-78 (1972) · Zbl 0229.35013 |
| [2] | Padrón, Víctor, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans Amer Math Soc, 356, 7, 2739-2756 (2004) · Zbl 1056.35103 |
| [3] | Colton, David; Wimp, Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J Math Anal Appl, 69, 2, 411-418 (1979) · Zbl 0409.35050 |
| [4] | Korpusov, Mo; Sveshnikov, Ag, Three-dimensional nonlinear evolutionary pseudoparabolic equations in mathematical physics, Zh Vychisleni Mat Fizica, 43, 12, 1835-1869 (2003) · Zbl 1121.35329 |
| [5] | Showalter, Re; Ting, Tw, Pseudoparabolic partial differential equations, SIAM J Math Anal, 1, 1, 1-26 (1970) · Zbl 0199.42102 |
| [6] | Gopala Rao, Vr; Ting, Tw, Solutions of pseudo-heat equations in the whole space, Arch Ration Mech Anal, 49, 1, 57-78 (1972) · Zbl 0255.35049 |
| [7] | Brill, Heinz, A semilinear sobolev evolution equation in a Banach space, J Differ Equ, 24, 3, 412-425 (1977) · Zbl 0346.34046 |
| [8] | Dibenedetto, Emmanuele; Pierre, Michel, On the maximum principle for pseudoparabolic equations, Indiana Univ Math J, 30, 6, 821-854 (1981) · Zbl 0495.35047 |
| [9] | Yang, Chunxiao; Cao, Yang; Zheng, Sining, Second critical exponent and life span for pseudo-parabolic equation, J Differ Equ, 253, 12, 3286-3303 (2012) · Zbl 1278.35140 |
| [10] | Luo, Peng, Blow-up phenomena for a pseudo-parabolic equation, Math Meth Appl Sci, 38, 12, 2636-2641 (2015) · Zbl 1327.35216 |
| [11] | Chen, Hua; Tian, Shuying, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J Differ Equ, 258, 12, 4424-4442 (2015) · Zbl 1370.35190 |
| [12] | Zhu, Xiaoli; Li, Fuyi; Rong, Ting, Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun Pure Appl Anal, 14, 6, 2465-2485 (2015) · Zbl 1328.35118 |
| [13] | Zhu, Xiaoli; Li, Fuyi; Liang, Zhanping, A sufficient condition for blowup of solutions to a class of pseudo-parabolic equations with a nonlocal term, Math Meth Appl Sci, 39, 13, 3591-3606 (2016) · Zbl 1343.35141 |
| [14] | Di, Huafei; Shang, Yadong; Zheng, Xiaoxiao, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Disc Contin Dyn Syst Ser B, 21, 3, 781-801 (2016) · Zbl 1331.35204 |
| [15] | Ji, Shanming; Yin, Jingxue; Cao, Yang, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J Differ Equ, 261, 10, 5446-5464 (2016) · Zbl 1358.35063 |
| [16] | Xu, Runzhang; Su, Jia, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J Funct Anal, 264, 12, 2732-2763 (2013) · Zbl 1279.35065 |
| [17] | Li, Fushan; Zhao, Zengqin; Chen, Yanfu, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal Real World Appl, 12, 3, 1759-1773 (2011) · Zbl 1218.35040 |
| [18] | Li, Fushan; Li, Jinling, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions, J Math Anal Appl, 385, 2, 1005-1014 (2012) · Zbl 1227.35157 |
| [19] | Li, Fushan; Gao, Qingyong, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl Math Comput, 274, 383-392 (2016) · Zbl 1410.35085 |
| [20] | Xu, Fuyi; Zhang, Xinguang; Wu, Yonghong, Global existence and temporal decay for the 3D compressible Hall-magnetohydrodynamic system, J Math Anal Appl, 438, 1, 285-310 (2016) · Zbl 1334.35260 |
| [21] | Zhu, Bo; Liu, Lishan; Wu, Yonghong, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl Math Lett, 61, 73-79 (2016) · Zbl 1355.35196 |
| [22] | Zhu, Bo; Liu, Lishan; Wu, Yonghong, Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Comput Math Appl (2016) · Zbl 1355.35196 · doi:10.1016/j.camwa.2016.01.028 |
| [23] | Quittner, Pavol; Souplet, Philippe, Superlinear parabolic problems blow-up, global existence and steady states. Birkhäuser advanced texts (2007), Basel: Birkhäuser Verlag, Basel · Zbl 1128.35003 |
| [24] | Sattinger, Dh, On global solution of nonlinear hyperbolic equations, Arch Ration Mech Anal, 30, 2, 148-172 (1968) · Zbl 0159.39102 |
| [25] | Payne, Le; Sattinger, Dh, Saddle points and instability of nonlinear hyperbolic equations, Israel J Math, 22, 3, 273-303 (1975) · Zbl 0317.35059 |
| [26] | Ikehata, Ryo; Suzuki, Takashi, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math J, 26, 3, 475-491 (1996) · Zbl 0873.35010 |
| [27] | Matsuyama, Tokio; Ikehata, Ryo, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J Math Anal Appl, 204, 3, 729-753 (1996) · Zbl 0962.35025 |
| [28] | Ono, Kosuke, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J Math Anal Appl, 216, 1, 321-342 (1997) · Zbl 0893.35078 |
| [29] | Liu, Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, J Differ Equ, 192, 1, 155-169 (2003) · Zbl 1024.35078 |
| [30] | Gazzola, Filippo; Weth, Tobias, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ Integral Equ, 18, 9, 961-990 (2005) · Zbl 1212.35248 |
| [31] | Gazzola, Filippo; Squassina, Marco, Global solutions and finite time blow up for damped semilinear wave equations, Ann l’Inst Henri Poincare (C) Non Linear Anal, 23, 2, 185-207 (2006) · Zbl 1094.35082 |
| [32] | Zhou, Jun, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl Math Comput, 265, 807-818 (2015) · Zbl 1410.35237 |
| [33] | Liu, Yacheng; Xu, Runzhang, A class of fourth order wave equations with dissipative and nonlinear strain terms, J Differ Equ, 244, 1, 200-228 (2008) · Zbl 1138.35066 |
| [34] | Liu, Yacheng; Xu, Runzhang; Yu, Tao, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal Theory Meth Appl, 68, 11, 3332-3348 (2008) · Zbl 1149.35367 |
| [35] | Chen, Hua; Liu, Gongwei, Global existence, uniform decay and exponential growth for a class of semi-linear wave equation with strong damping, Acta Math Sci, 33, 1, 41-58 (2013) · Zbl 1289.35202 |
| [36] | Liu, Gongwei; Xia, Suxia, Global existence and finite time blow up for a class of semilinear wave equations on ℝ\(^N\), Comput Math Appl, 70, 6, 1345-1356 (2015) · Zbl 1443.35104 |
| [37] | Li, Qingwei; Gao, Wenjie; Han, Yuzhu, Global existence blow up and extinction for a class of thin-film equation, Nonlinear Anal Theory Meth Appl, 147, 96-109 (2016) · Zbl 1488.35289 |
| [38] | Messaoudi, Salim A.; Tatar, Nasser-Eddine, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math Meth Appl Sci, 30, 6, 665-680 (2007) · Zbl 1121.35015 |
| [39] | Xu, Runzhang; Yang, Yanbing; Liu, Yacheng, Global well-posedness for strongly damped viscoelastic wave equation, Appl Anal, 92, 1, 138-157 (2013) · Zbl 1282.35226 |
| [40] | Di, Huafei; Shang, Yadong, Global existence and nonexistence of solutions for the nonlinear pseudo-parabolic equation with a memory term, Math Meth Appl Sci, 38, 17, 3923-3936 (2015) · Zbl 1334.35162 |
| [41] | Levine, Howard A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(P_{u_t} \) = F(u), Arch Ration Mech Anal, 51, 5, 371-386 (1973) · Zbl 0278.35052 |
| [42] | Song, Haitao; Xue, Desheng, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal Theory Meth Appl, 109, 245-251 (2014) · Zbl 1297.35056 |
| [43] | Song, Haitao, Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation, Nonlinear Anal Real World Appl, 26, 306-314 (2015) · Zbl 1331.35238 |
| [44] | Guo, Yanqiu; Rammaha, Mohammad A.; Sakuntasathien, Sawanya, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J Differ Equ, 257, 10, 3778-3812 (2014) · Zbl 1302.35259 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
