一类具$p(\cdot,\cdot)$结构和$L^1$源的非线性抛物方程解的存在性

Acta mathematica scientia,Series A ›› 2024, Vol. 44 ›› Issue (1): 93-100.

Previous Articles Next Articles

Existence Result for a Class of Nonlinear Parabolic Equations with $p(\cdot,\cdot)$ Structure and an $L^1$ Source

Li Zhongqing()

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025

Received:2022-09-06 Revised:2023-09-11 Online:2024-02-26 Published:2024-01-10
Supported by:
NSFC(11901131);University-level Research Fund Project in Guizhou University of Finance and Economics(2022KYYB01)

Abstract

Abstract:

This paper is devoted to proving an existence result to a class of nonlinear parabolic equations with variable exponents. The zero order term produces a regularizing effect, which helps to get the $L^\infty$ estimate despite the poor summability of the source term. By choosing some appropriate test functions, we obtain the necessary a priori estimates for the approximate solution sequences denoted by $\{u^\epsilon\}_\varepsilon$ . Thanks to the Young measure method, the weak convergence of the nonlinear term is identified.

Key words: Variable exponents, Zero order term, Young measure

CLC Number:

O175.8

Li Zhongqing. Existence Result for a Class of Nonlinear Parabolic Equations with $p(\cdot,\cdot)$ Structure and an $L^1$ Source[J].Acta mathematica scientia,Series A, 2024, 44(1): 93-100.

Trendmd

References

[1]	Antontsev S, Shmarev S. Anisotropic parabolic equations with variable nonlinearity. Publ Mat, 2009, 53(2): 355-399
[2]	Arcoya D, Boccardo L. Regularizing effect of the interplay between coefficients in some elliptic equations. J Funct Anal, 2015, 268(5): 1153-1166
[3]	Bendahmane M, Wittbold P, Zimmermann A. Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$ -data. J Differential Equations, 2010, 249(6): 1483-1515
[4]	Blanchard D, Murat F, Redwane H. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J Differential Equations, 2001, 177(2): 331-374
[5]	Boccardo L, Croce G. Elliptic Partial Differential Equations:Existence and Regularity of Distributional Solutions. Berlin, Boston: DeGruyter, 2013
[6]	Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66(4): 1383-1406
[7]	Cirmi G R, Porzio M M. $L^\infty$ -solutions for some nonlinear degenerate elliptic and parabolic equations. Ann Mat Pura Appl, 1995, 169(4): 67-86
[8]	Diening L, Málek J, Steinhauer M. On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim Calc Var, 2008, 14(2): 211-232
[9]	Diening L, N?gele P, R??i?ka M. Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var Elliptic Equ, 2012, 57(11): 1209-1231
[10]	Dolzmann G, Hungerbühler N, Müller S. Non-linear elliptic systems with measure-valued right hand side. Math Z, 1997, 226(4): 545-574
[11]	Gwiazda P, ?wierczewska Gwiazda A. On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces. Topol Methods Nonlinear Anal, 2008, 32(1): 103-113
[12]	Hungerbühler N. A refinement of Ball's theorem on Young measures. New York J Math, 1997, 3: 48-53
[13]	Porzio M M. $L^\infty$ -regularity for degenerate and singular anisotropic parabolic equations. Boll Un Mat Ital A, 1997, 11(3): 697-707
[14]	R??i?ka M. Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer-Verlag, 2000
[15]	Simon J. Compact sets in the space $L^p(0,T;B)$ . Ann Mat Pura Appl, 1987, 146(4): 65-96
[16]	Zeidler E. Nonlinear Functional Analysis and Its Applications. II/B. New York: Springer-Verlag, 1990
[17]	Zhang C, Zhou S. Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$ data. J Differential Equations, 2010, 248(6): 1376-1400

Existence Result for a Class of Nonlinear Parabolic Equations with $p(\cdot,\cdot)$ Structure and an $L^1$ Source

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 0

Existence Result for a Class of Nonlinear Parabolic Equations withp(⋅,⋅)p(\cdot,\cdot)Structure and anL1L^1Source

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 0

Existence Result for a Class of Nonlinear Parabolic Equations with $p(\cdot,\cdot)$ Structure and an $L^1$ Source