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具对数源的p-Laplacian型抛物方程解的爆破准则
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Abstract:
文章旨在研究一类具对数非线性源的p-Laplacian型伪抛物方程，该方程曾于[Comput.Math. Appl.73（2017）2076-2091]中被探讨。原文献应用位势井方法，针对亚临界和临界初始能量情 形，给出了解全局存在或有限时间爆破的阈值结果。本文构造了一个新的不变集，在超临界初始 能量情形下，确立了新的有限时间爆破准则，并进一步借助喷泉定理，阐明该问题在任意高初始 能量下总存在有限时间爆破解。此外，本文还从上方估计了爆破解的生命跨度。这部分结果拓展 了[Comput.Math.Appl.73(2017)2076-2091]中获得的爆破结论。
In this paper the authors investigate a p-Laplacian type pseudo-parabolic equation with logarithmic nonlinearity which was considered in [Comput. Math. Appl. 73(2017) 2076-2091], where threshold results for the solutions to exist globally or to blow up in finite time were given for subcritical and critical initial energy. A new finite time blow-up criterion for supercritical initial energy is established in this paper, which in particular implies that the problem admits finite time blow-up solutions at arbitrarily high initial energy level. Moreover, the lifespan of the blow-up solutions is estimated from above. This partially extends the blow-up results obtained in [Comput. Math. Appl. 73(2017) 2076-2091].
| [1] | Barenblatt, G.I., Zheltov, I.P. and Kochina, I.N. (1960) Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks [Strata]. Journal of Applied Mathematics and Mechanics, 24, 1286-1303. https://doi.org/10.1016/0021-8928(60)90107-6 |
| [2] | Benjamin, T.B., Bona, J.L. and Mahony, J.J. (1972) Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272, 47-78. https://doi.org/10.1098/rsta.1972.0032 |
| [3] | Padrón, V. (2003) Effect of Aggregation on Population Recovery Modeled by a Forward- Backward Pseudoparabolic Equation. Transactions of the American Mathematical Society, 356, 2739-2756. https://doi.org/10.1090/s0002-9947-03-03340-3 |
| [4] | Chen, H. and Tian, S. (2015) Initial Boundary Value Problem for a Class of Semilinear PseudoParabolic Equations with Logarithmic Nonlinearity. Journal of Differential Equations, 258, 4424-4442. https://doi.org/10.1016/j.jde.2015.01.038 |
| [5] | Nhan, L.C. and Truong, L.X. (2017) Global Solution and Blow-Up for a Class of Pseudo p-Laplacian Evolution Equations with Logarithmic Nonlinearity. Computers ？ Mathematics with Applications, 73, 2076-2091. https://doi.org/10.1016/j.camwa.2017.02.030 |
| [6] | Sattinger, D.H. (1968) On Global Solution of Nonlinear Hyperbolic Equations. Archive for Rational Mechanics and Analysis, 30, 148-172. https://doi.org/10.1007/bf00250942 |
| [7] | Han, Y., Cao, C. and Sun, P. (2018) A p-Laplace Equation with Logarithmic Nonlinearity at High Initial Energy Level. Acta Applicandae Mathematicae, 164, 155-164. https://doi.org/10.1007/s10440-018-00230-4 |
| [8] | Gazzola, F. and Weth, T. (2005) Finite Time Blow-Up and Global Solutions for Semilinear Parabolic Equations with Initial Data at High Energy Level. Differential and Integral Equations, 18, 961-990. https://doi.org/10.57262/die/1356060117 |
| [9] | Cao, Y. and Liu, C. (2018) Initial Boundary Value Problem for a Mixed Pseudo-Parabolic p-Laplacian Type Equation with Logarithmic Nonlinearity. Electronic Journal of Differential Equations, 2018, 1-19. |
| [10] | He, Y., Gao, H. and Wang, H. (2018) Blow-Up and Decay for a Class of Pseudo-Parabolic p-Laplacian Equation with Logarithmic Nonlinearity. Computers e Mathematics with Applications, 75, 459-469. https://doi.org/10.1016/j.camwa.2017.09.027 |
| [11] | Dai, P., Mu, C. and Xu, G. (2020) Blow-Up Phenomena for a Pseudo-Parabolic Equation with p-Laplacian and Logarithmic Nonlinearity Terms. Journal of Mathematical Analysis and Applications, 481, Article 123439. https://doi.org/10.1016/j.jmaa.2019.123439 |
| [12] | Ding, H. and Zhou, J. (2019) Global Existence and Blow-Up for a Mixed Pseudo-Parabolic p-Laplacian Type Equation with Logarithmic Nonlinearity. Journal of Mathematical Analysis and Applications, 478, 393-420. https://doi.org/10.1016/j.jmaa.2019.05.018 |
| [13] | Di, H., Shang, Y. and Song, Z. (2020) Initial Boundary Value Problem for a Class of Strongly Damped Semilinear Wave Equations with Logarithmic Nonlinearity. Nonlinear Analysis: Real World Applications, 51, Article 102968. https://doi.org/10.1016/j.nonrwa.2019.102968 |
| [14] | DiBenedetto, E. (1993) Degenerate Parabolic Equations. Springer-Verlag. |
| [15] | Ladyzhenskaia, O., Solonnikov, V. and Ural'tseva, N. (1988) Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society. |
| [16] | Levine, H.A. (1973) Some Nonexistence and Instability Theorems for Solutions of Formally Parabolic Equations of the Form Put = –Au + F(u). Archive for Rational Mechanics and Analysis, 51, 371-386. https://doi.org/10.1007/bf00263041 |
| [17] | Levine, H. (1973) Remarks on the Growth and Nonexistence of Solutions to Nonlinear Wave Equations. A Seminar on PDEs, 59-70. |
| [18] | Sun, F., Liu, L. and Wu, Y. (2017) Infinitely Many Sign-Changing Solutions for a Class of Biharmonic Equation with p-Laplacian and Neumann Boundary Condition. Applied Mathematics Letters, 73, 128-135. https://doi.org/10.1016/j.aml.2017.05.001 |
| [19] | Willem, M. (1997) Minimax Theorems. Vol. 24, Springer Science & Business Media. |
| [20] | Fan, X. and Zhang, Q. (2003) Existence of Solutions for p(x)-Laplacian Dirichlet Problem. Nonlinear Analysis: Theory, Methods e Applications, 52, 1843-1852. https://doi.org/10.1016/s0362-546x(02)00150-5 |
| [21] | Kichenassamy, S. and Veron, L. (1986) Singular Solutions of Thep-Laplace Equation. Mathematische Annalen, 275, 599-615. https://doi.org/10.1007/bf01459140 |
| [22] | Simsen, J., Nascimento, M.J.D. and Simsen, M.S. (2014) Existence and Upper Semicontinuity of Pullback Attractors for Non-Autonomous p-Laplacian Parabolic Problems. Journal of Mathematical Analysis and Applications, 413, 685-699. https://doi.org/10.1016/j.jmaa.2013.12.019 |
