×

Linear and nonlinear abstract equations with parameters. (English) Zbl 1215.34067

The linear abstract equation
\[ -tu^{(2)}(x)+ Au(x)+ t^{1/2} B_1(x)u^{(1)}(x)+ B_2(x) u(x)= f(x) \]
with a parameter \(t\) is considered. Here, \(A\) and \(B_1(x)\), \(B_2(x)\) for \(x\in (0,1)\) are linear operators in a Banach space. The nonlocal boundary conditions contain the parameter \(t\) as well.
Under some assumptions, the existence of the unique solution in a Sobolev space and a coercive uniform estimation is established. Also, the behavior of the solution for \(t\to 0\) and the smoothness properties of the solution with respect to the parameter \(t\) are investigated and the discreteness of the corresponding differential operator is proved.
For the nonlinear problem with right side \(f(x,u, u^{(1)})\), the existence and uniqueness of maximal regular solution is obtained.
An application to the equation
\[ -t_1 D^2_x u(x,y)- t_2 D^2_y u(x,y)+ du(x,y)+ t^{1/2}_1 D_x u(x,y)+ t^{1/2}_2 D_y u(x,y)= f(x,y) \]
on the region \((0,a)\times (0,b)\) is given.
Reviewer: S. Burys (Kraków)

MSC:

34G10 Linear differential equations in abstract spaces
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
34G20 Nonlinear differential equations in abstract spaces
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Krein, S. G., Linear Differential Equations in Banach Space (1971), American Mathematical Society: American Mathematical Society Providence · Zbl 0636.34056
[2] Fattorini, H. O., The Cauchy Problems (1983), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0493.34005
[3] Yakubov, S.; Yakubov, Ya., Differential-Operator Equations. Ordinary and Partial Differential Equations (2000), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton · Zbl 0936.35002
[4] Goldstain, J. A., Semigroups of Linear Operators and Applications (1985), Oxford University Press: Oxford University Press Oxford
[5] Amann, H., Linear and Quasi-Linear Equations, 1 (1995), Birkhauser: Birkhauser Basel
[6] Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhauser, Verlag: Birkhauser, Verlag Basel · Zbl 0816.35001
[7] Shklyar, A. Ya., Complete Second Order Linear Differential Equations in Hilbert Spaces (1997), Birkhauser Verlag: Birkhauser Verlag Basel · Zbl 0873.34049
[8] Dore, C.; Yakubov, S., Semigroup estimates and non coercive boundary value problems, Semigroup Forum, 60, 93-121 (2000) · Zbl 0947.47033
[9] Sobolevskii, P. E., Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk SSSR, 57, 1, 27-40 (1964)
[10] Ashyralyev, A.; Claudio, Cuevas; Piskarev, S., On well-posedness of difference schemes for abstract elliptic problems in spaces, Numer. Funct. Anal. Optim., 29, 1-2, 43-65 (2008) · Zbl 1140.65073
[11] Shakhmurov, V. B., Theorems about of compact embedding and applications, Dokl. Akad. Nauk SSSR, 241, 6, 1285-1288 (1978)
[12] Shakhmurov, V. B., Nonlinear abstract boundary value problems in vector-valued function spaces and applications, Nonlinear Anal. Series A: TMA, 67, 3, 745-762 (2006) · Zbl 1169.35383
[13] Shakhmurov, V. B., Imbedding theorems and their applications to degenerate equations, Differ. Equ., 24, 4, 475-482 (1988) · Zbl 0672.46014
[14] Shakhmurov, V. B., Coercive boundary value problems for regular degenerate differential-operator equations, J. Math. Anal. Appl., 292, 2, 605-620 (2004) · Zbl 1060.35045
[15] Shakhmurov, V. B., Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces, J. Inequal. Appl., 2, 4, 329-345 (2005) · Zbl 1119.46034
[16] Shakhmurov, V. B., Degenerate differential operators with parameters, Abstr. Appl. Anal., 2006, 1-27 (2007) · Zbl 1181.35112
[17] Favini, A.; Shakhmurov, V.; Yakubov, Y., Regular boundary value problems for complete second order elliptic differential operator equations in \(U M D\) Banach spaces, Semigroup Forum, 79, 22-54 (2009) · Zbl 1177.47089
[18] D.L. Burkholder, A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions, in: Proc. conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, Wads Worth, Belmont, 1983, pp. 270-286.; D.L. Burkholder, A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions, in: Proc. conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, Wads Worth, Belmont, 1983, pp. 270-286.
[19] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam · Zbl 0387.46032
[20] Weis, L., Operator-valued Fourier multiplier theorems and maximal \(L_p\) regularity, Math. Ann., 319, 735-758 (2001) · Zbl 0989.47025
[21] Denk, R.; Hieber, M.; Prüss, J., \(R\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166, 788 (2003) · Zbl 1274.35002
[22] Haller, R.; Heck, H.; Noll, A., Mikhlin’s theorem for operator-valued Fourier multipliers in \(n\) variables, Math. Nachr., 244, 110-130 (2002) · Zbl 1054.47013
[23] Besov, O. V.; Ilin, V. P.; Nikolskii, S. M., Integral Representations of Functions and Embedding Theorems (1975), Nauka: Nauka Moscow
[24] Komatsu, H., Fractional powers of operators, Pas. J. Math., 19, 285-346 (1966) · Zbl 0154.16104
[25] Grisvard, P., Commutativité de deux foncteurs d’interpolation et applications, J. Math. Pures Appl. 9, 45, 143-290 (1966) · Zbl 0173.15803
[26] Lions, J. L.; Peetre, J., Sur une classe d’espaces d’interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19, 5-68 (1964) · Zbl 0148.11403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.