## Banach Journal of Mathematical Analysis

### Linear and nonlinear degenerate abstract differential equations with small parameter

Veli B. Shakhmurov

#### Abstract

The boundary value problems for linear and nonlinear regular degenerate abstract differential equations are studied. The equations have the principal variable coefficients and a small parameter. The linear problem is considered on a parameter-dependent domain (i.e., on a moving domain). The maximal regularity properties of linear problems and the optimal regularity of the nonlinear problem are obtained. In application, the well-posedness of the Cauchy problem for degenerate parabolic equations and boundary value problems for degenerate anisotropic differential equations are established.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 1 (2016), 147-168.

Dates
Received: 2 February 2015
Accepted: 6 May 2015
First available in Project Euclid: 11 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1447253826

Digital Object Identifier
doi:10.1215/17358787-3345071

Mathematical Reviews number (MathSciNet)
MR3453529

Zentralblatt MATH identifier
1335.35049

#### Citation

Shakhmurov, Veli B. Linear and nonlinear degenerate abstract differential equations with small parameter. Banach J. Math. Anal. 10 (2016), no. 1, 147--168. doi:10.1215/17358787-3345071. https://projecteuclid.org/euclid.bjma/1447253826

#### References

• [1] R. Agarwal, M. Bohner, and V. B. Shakhmurov, Maximal regular boundary value problems in Banach-valued weighted spaces, Bound. Value Probl. 1 (2005), no. 1, 9–42.
• [2] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147.
• [3] H. Amann, Linear and Quasi-linear Equations, Birkhäuser, Basel, 1995.
• [4] W. Arendt and M. Duelli, Maximal $L_{p}$-regularity for parabolic and elliptic equations on the line, J. Evol. Equ. 6 (2006), no 4, 773–790.
• [5] A. Ashyralyev, C. Cuevas, and S. Piskarev, On well-posedness of difference schemes for abstract elliptic problems in spaces, Numer. Func. Anal. Opt. 29 (2008), nos. 1–2, 43–65.
• [6] O. V. Besov, V. P. Ilin, and S. M. Nikolskii, Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975.
• [7] D. L. Burkholder, “A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions” in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, 270–286.
• [8] R. Denk, M. Hieber, and J. Prüss, $R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788.
• [9] C. Dore and S. Yakubov, Semigroup estimates and non coercive boundary value problems, Semigroup Forum 60 (2000), no. 1, 93–121.
• [10] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Dekker, New York, 1999.
• [11] A. Favini, V. Shakhmurov, and Y. Yakubov, Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum 79 (2009), no. 1, 22–54.
• [12] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.
• [13] S. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monogr., Amer. Math. Soc., Providence, 1971.
• [14] J.-L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems, Mir, Moscow, 1971.
• [15] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl. 16, Birkhäuser, Basel, 2003.
• [16] R. Shahmurov, Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annuals, J. Differential Equations 249 (2010), no. 3, 526–550.
• [17] R. Shahmurov, On strong solutions of a Robin problem modeling heat conduction in materials with corroded boundary, Nonlinear Anal. Real World Appl. 13 (2011), no. 1, 441–451.
• [18] V. B. Shakhmurov, Degenerate differential operators with parameters, Abstr. Appl. Anal. 2006 (2007), art. ID 51410.
• [19] V. B. Shakhmurov, Coercive boundary value problems for regular degenerate differential-operator equations, J. Math. Anal. Appl. 292 (2004), no. 2, 605–620.
• [20] V. B. Shakhmurov, Regular degenerate separable differential operators and applications, Potential Anal. 35 (2011), no. 3, 201–212.
• [21] P. E. Sobolevskii, Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk SSSR 157 (1964), no. 1, 27–40.
• [22] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library 18, North-Holland, Amsterdam, 1978.
• [23] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_{p}$ regularity, Math. Ann. 319 (2001), no. 4, 735–758.
• [24] S. Yakubov and Y. Yakubov, Differential-Operator Equations: Ordinary and Partial Differential Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 103, Chapman and Hall/CRC, Boca Raton, FL, 2000.