一类非线性伪抛物型方程解的性质
Properties of Solutions for a Class of Nonlinear Pseudoparabolic Equations
DOI: 10.12677/AAM.2020.94066, PDF,  被引量    国家自然科学基金支持
作者: 程嘉卓, 金玲玉, 房少梅:华南农业大学,数学与信息学院,广东 广州
关键词: 先验估计Galerkin方法弱解A Priori Estimates Galerkin Method Weak Solution
摘要: 本文研究了一类分数阶非线性伪抛物方程的弱解问题。首先通过能量方法讨论得到方程解的先验估计,然后利用Galerkin方法构造近似解序列来证明方程弱解的存在唯一性。
Abstract: This paper studies the weak solutions of a class of fractional nonlinear pseudo-parabolic equations. First, a priori estimates of the solution of the equation are obtained through the energy method discussion, and then the Galerkin method is used to construct an approximate solution sequence to prove the existence and uniqueness of the weak solution of the equation.
文章引用:程嘉卓, 金玲玉, 房少梅. 一类非线性伪抛物型方程解的性质[J]. 应用数学进展, 2020, 9(4): 551-559. https://doi.org/10.12677/AAM.2020.94066

参考文献

[1] Coleman, B.D., Duffin, R.J. and Mizel, V.J. (1965) Instability, Uniqueness, and Nonexistence Theorems for the Equa-tion on a Strip. Archive for Rational Mechanics and Analysis, 19, 100-116. [Google Scholar] [CrossRef
[2] 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. [Google Scholar] [CrossRef
[3] 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 A. Mathematical, Physical and Engineering Sciences, 272, 47-78. [Google Scholar] [CrossRef
[4] Ting, T.W. (1963) Certain Non-Steady Flows of Second-Order Fluids. Archive for Rational Mechanics and Analysis, 14, 1-26. [Google Scholar] [CrossRef
[5] Padron, V. (2004) Effect of Aggregation on Population Recovery Modeled by a Forward-Backward Pseudoparabolic Equation. Transactions of the American Mathematical Society, 356, 2739-2756. [Google Scholar] [CrossRef
[6] Bona, J.L. and Dougalis, V.A. (1980) An Initial- and Boundary-Value Problem for a Model Equation for Propagation of Long Waves. Journal of Mathematical Analysis and Applications, 75, 503-522. [Google Scholar] [CrossRef
[7] Amick, C.J., Bona, J.L. and Schonbek, M.E. (1989) Decay of Solutions of Some Nonlinear Wave Equations (ENG). Journal of Differential Equations, 81, 1-49. [Google Scholar] [CrossRef
[8] Zhang, L. (1995) Decay of Solution of Generalized Benja-min-Bona-Mahony-Burgers Equations in N-Space Dimensions. Nonlinear Analysis Theory Methods & Applications, 25, 1343-1369. [Google Scholar] [CrossRef
[9] Medeiros, L.A. and Miranda, M.M. (1977) Weak Solutions for a Nonlinear Dispersive Equation. Journal of Mathematical Analysis and Applications, 59, 432-441. [Google Scholar] [CrossRef
[10] Bona, J.L., Pritchard, W.G. and Scott, L.R. (1980) Soli-tary-Wave Interaction. Physics of Fluids, 23, 438. [Google Scholar] [CrossRef
[11] Taha, A. and Mahomed, F.M. (2014) A Note on the Solutions of Some Nonlinear Equations Arising in Third-Grade Fluid Flows: An Exact Approach. The Scientific World Journal, 2014, Ar-ticle ID: 109128. [Google Scholar] [CrossRef] [PubMed]
[12] Hayat, T., Shahzad, F. and Ayub, M. (2007) Analytical Solution for the Steady Flow of the Third Grade Fluid in a Porous Half Space. Applied Mathematical Modelling, 31, 2424-2432. [Google Scholar] [CrossRef
[13] Sajid, M. and Hayat, T. (2008) Series Solution for Steady Flow of a Third Grade Fluid through Porous Space. Transport in Porous Media, 71, 173-183. [Google Scholar] [CrossRef
[14] Ngoc, L.T.P., Yen, D.T.H. and Long, N.T. (2018) Existence and Asymptotic Behavior of Solutions of the Dirichlet Problem for a Nonlinear Pseudoparabolic Equation. Electronic Journal of Differential Equations, 2018, 1-20.
[15] Kilbas, A.A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractinal Differential Equations. North-Holland Mathematics Studies, Vol. 204, Elsevier, Amster-dam.
[16] Pu, X., Guo, B. and Zhang, J. (2012) Global Weak Solutions to the 1-D Fractional Landau-Lifshitz Equation. Discrete and Continuous Dynamical Systems Series B (DCDS-B), 14, 199-207. [Google Scholar] [CrossRef
[17] Guo, B., Han, Y. and Xin, J. (2008) Existence of the Global Smooth Solution to the Period Boundary Value Problem of Fractional Nonlinear Schrodinger Equation. Applied Mathematics & Computation, 204, 468-477. [Google Scholar] [CrossRef
[18] Zhang, S. (2003) Existence of Positive Solution for Some Class of Nonlinear Fractional Differential Equations. Journal of Mathematical Analysis & Applications, 278, 136-148. [Google Scholar] [CrossRef