局部任意阶增长的拟线性抛物方程解的存在性

Abstract

Abstract:

Existence theory for a kind of quasi-linear parabolic equations is established by the Schauder fixed point method. Actually a linearized map is defined by fixing the function variables in the coefficients and the right-hand term. Its domain is chosen to be bounded but a locally arbitrary growth condition is considered. Therefore its range is contained in a closed convex set through the maximum principle. This map is continuous since the solution smoothly depends on the data. Compactness is deduced from the embedding theorem, so there exists a fixed point. The coefficients are just required to be continuous with respect to the function variables, but can be only measurable with respect to the space and time variables. Moreover, memory terms are also considered.

Key words: Existence theory, Quasi-linear parabolic equations, Fixed point, Locally arbitrary growth condition

CLC Number:

O175.29

Ma Wenya, Zhang Qiaofu, Cui Junzhi. Global Existence for Some Quasi-Linear Parabolic Equations with Locally Arbitrary Growth and Memory[J].Acta mathematica scientia,Series A, 2015, 35(5): 833-844.

Trendmd

References

[1] Amadori A L. Nonlinear integro-differential evolution problems arising in option pricing:a viscosity solutions approach. Differ Integral Equ, 2003, 16:787-811
[2] Amann H. Non-local quasi-linear parabolic equations. Russian Math Surveys, 2005, 60:1021-1033
[3] Amann H. Quasilinear parabolic systems under nonlinear boundary conditions. Archive for Rational Mechanics and Analysis, 1986, 92:153-192
[4] Bear J. Dynamics of Fluids in Porous Media. New York:American Elsevier Publishing Company, 1972
[5] Chen S X. An Introduction to Mordern PDE(in Chinese). Beijing:China Science Press, 2005
[6] Chen Y Z. Second Order Parabolic Partial Differential Equations(in Chinese). Beijing:Peking University Press, 2003
[7] Chen Z M, Wu H J. Selected Topics in Finite Element Methods. Beijing:China Science Press, 2010
[8] Cioranescu D, Donato P. An Introduction to Homogenization. Oxford:Oxford University Press, 1999
[9] Garroni M G, Menaldi J L. Green Functions for Second Order Parabolic Integro-Differential Problems. Burnt Mill, Harlow, Essex, England and New York:Longman Scientific & Technical, 1992
[10] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin:Springer-Verlag, 2001
[11] Griepentrog J A. Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Advances in Differential Equations, 2007, 12:1031-1078
[12] Griepentrog J A. Sobolev-Morrey spaces associated with evolution equations. Advances in Differential Equations, 2007, 12:781-840
[13] Griepentrog J A, Recke L. Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems. Journal of Evolution Equations, 2010, 10:341-375
[14] Hale J K, Lunel S M V. Introduction to Functional Differential Equations. Applied Mathematical Sciences 99. New York:Springer-Verlag, 1993
[15] Ladyzhenskaya O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type. London:Academic Press, 1968
[16] Lieberman G M. Second Order Parabolic Differential Equations. River Edge:World Scientific Press, 2005
[17] Modest M F. Radiative Heat Transfer(2nd ed). San Diego:McGraw-Hill, 2003
[18] Wu Z Q, Yin J X, Wang C P. Elliptic and Parabolic Equations. Singapore:World Scientific Press, 2006
[19] Zhang Q F, Cui J Z. Existence theory for Rosseland equation and its homogenized equation. Applied Mathematics and Mechanics, 2012, 33:1595-1612
[20] Zhang Q F. Multi-Scale Analysis for Rosseland Equation with Small Periodic Oscillating Coefficients(in Chinese). Beijing:Chinese Academy of Sciences, 2012

Global Existence for Some Quasi-Linear Parabolic Equations with Locally Arbitrary Growth and Memory

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Piao Yongjie. Improvements of Fixed Point Theorems for Mappings with AContractions on Metric Spaces [J]. Acta mathematica scientia,Series A, 2018, 38(5): 855-863.
[2]	Zou Xia, Wu Shiliang. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Delay and Nonlinear Incidence [J]. Acta mathematica scientia,Series A, 2018, 38(3): 496-513.
[3]	Chang Shih-sen, Liu Zhenhai, Wen Ching-Feng, Tang Jinfang. Split Variational Inclusion Problem Involving Fixed Point for an Asymptotically Nonexpansive Semigroup with Application to Optimization Problem [J]. Acta mathematica scientia,Series A, 2018, 38(2): 231-243.
[4]	Li Haixia. Stability and Uniqueness of Positive Solutions for a Food-Chain Model [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1094-1104.
[5]	Guo Tingguang, Xu Zhiting. Existence of Traveling Waves in a Spatial Infectious Disease Model with Vaccination [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1129-1147.
[6]	Song Jiping, Wang Maofa. Common Fixed Point Results for Generalized Weak Contractions with a Rational Expression in Ordered Partial Metric Spaces [J]. Acta mathematica scientia,Series A, 2017, 37(3): 401-415.
[7]	Wang Xuewu. Convergence and Stability of Ishikawa Type Iterative Sequences for the φ-Hemi Pseudocontractive Operator Pair [J]. Acta mathematica scientia,Series A, 2017, 37(2): 248-256.
[8]	Hao Xinan, Liu Lishan. Positive Solutions for Nonlocal Boundary Value Problems of Fourth Order Differential Equation with Parameters [J]. Acta mathematica scientia,Series A, 2017, 37(2): 287-298.
[9]	Cai Gang, Yekini Shehu. Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(4): 623-638.
[10]	Deng Weiqi. A System of Split Equilibrium Problems for Sequences of Bifunctions [J]. Acta mathematica scientia,Series A, 2016, 36(4): 763-770.
[11]	Xia Zhinan. On Existence and Asymptotic Behavior of Solutions for Functional Integral Equation of Volterra-Stieltjes Type [J]. Acta mathematica scientia,Series A, 2016, 36(1): 130-143.
[12]	Luo Ting, Zhu Chuanxi. Tripled Coincidence and Tripled Fixed Point Theorems for Compatible Mappings Under Contractive Conditions in Partially Ordered Probabilistic Metric Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(1): 157-167.
[13]	Liu Yuanxing, Zhou Haiyun, Wang Peiyuan, Zhou Yu. Three Kinds of Iterative Algorithms for Solving the Constrained Split Common Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1115-1126.
[14]	Cai Gang. Strong Convergence Theorems by A Viscosity Iterative Method for Fixed Point Problems of Nonexpansive Mappings in Banach Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(3): 487-502.
[15]	Huang Huaping, Xu Shaoyuan, Xu Wangbin. T-Stability of Picard's Iteration in Cone Metric Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(2): 395-404.