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A note on the parabolic differential and difference equations. (English) Zbl 1153.35353

Summary: The differential equation \(u'(t)+ Au(t)= f(t)\) \((-\infty< t<\infty)\) in a general Banach space \(E\) with the strongly positive operator \(A\) is ill-posed in the Banach space \(C(E)= C(\mathbb{R},E)\) with norm \(\|\varphi\|_{C(E)}= \sup_{-\infty< t< \infty}\|\varphi(t)\|_E\). In the present paper, the well-posedness of this equation in the Hölder space \(C^\alpha(E)= C^\alpha(\mathbb{R},E)\) with norm \[ \|\varphi\|_{C^\alpha(E)}= \sup_{-\infty< t<\infty}\|\varphi(t)\|_E+ \sup_{-\infty< t< t+ s< \infty}(\|\varphi(t+ s)- \varphi(t)\|_E/s^\alpha),\;0< \alpha< 1, \] is established. The almost coercivity inequality for solutions of the Rothe difference scheme in \(C(\mathbb{R}_\tau,E)\) spaces is proved. The well-posedness of this difference scheme in \(C^\alpha(\mathbb{R}_\tau,E)\) spaces is obtained.

MSC:

35K90 Abstract parabolic equations
34G10 Linear differential equations in abstract spaces
39A12 Discrete version of topics in analysis
47D06 One-parameter semigroups and linear evolution equations
47N20 Applications of operator theory to differential and integral equations
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References:

[1] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, “Nauka”, Moscow, Russia, 1967. · Zbl 0164.12302
[2] M. I. Vishik, A. D. Myshkis, and O. A. Oleinik, “Partial differential equations,” in Mathematics in USSR in the Last 40 Years, 1917-1957, Vol. 1, pp. 563-599, Fizmatgiz, Moscow, Russia, 1959.
[3] P. E. Sobolevskii, “Well-posedness of difference elliptic equation,” Discrete Dynamics in Nature and Society, vol. 1, no. 3, pp. 219-231, 1997. · Zbl 0928.39002 · doi:10.1155/S1026022697000228
[4] P. E. Sobolevskii, “Coerciveness inequalities for abstract parabolic equations,” Doklady Akademii Nauk SSSR, vol. 157, no. 1, pp. 52-55, 1964 (Russian). · Zbl 0149.36001
[5] P. E. Sobolevskii, “Some properties of the solutions of differential equations in fractional spaces,” Trudy Naucno-Issledovatel’skogi Instituta Matematiki VGU, vol. 14, pp. 68-74, 1975 (Russian), [RZh. Mat. 1975:7 B825].
[6] G. Da Prato and P. Grisvard, “Sommes d’opérateurs linéaires et équations différentielles opérationnelles,” Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 54, no. 3, pp. 305-387, 1975. · Zbl 0315.47009
[7] G. Da Prato and P. Grisvard, “Équations d’évolution abstraites non linéaires de type parabolique,” Comptes Rendus de l’Académie des Sciences Série A-B, vol. 283, no. 9, pp. A709-A711, 1976. · Zbl 0356.35048
[8] A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary value problem for parabolic differential equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53-61, 2001. · Zbl 0996.35027 · doi:10.1155/S1085337501000495
[9] A. Ashyralyev and P. E. Sobolevskii, Positive Operators and the Fractional Spaces. The Methodical Instructions for the Students of Engineering Grups, Offset Laboratory VSU, Voronezh, Russia, 1989.
[10] A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994. · Zbl 0803.65089
[11] A. Ashyralyev, I. Karatay, and P. E. Sobolevskii, “On well-posedness of the nonlocal boundary value problem for parabolic difference equations,” Discrete Dynamics in Nature and Society, vol. 2004, no. 2, pp. 273-286, 2004. · Zbl 1077.39015 · doi:10.1155/S1026022604403033
[12] A. E. Poli\vcka and P. E. Sobolevskii, “Correct solvability of parabolic difference equations in Bochner spaces,” Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 36, pp. 29-57, 294, 1978 (Russian).
[13] A. Ashyralyev, “Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces,” Journal of Evolution Equations, vol. 6, no. 1, pp. 1-28, 2006. · Zbl 1117.65077 · doi:10.1007/s00028-005-0194-y
[14] P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, no. 5, pp. 1063-1066, 1971 (Russian). · Zbl 0246.39002
[15] A. Ashyralyev and P. E. Sobolevskii, “The theory of interpolation of linear operators and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289-1291, 1984 (Russian). · Zbl 0598.65038
[16] A. Ashyralyev, S. Piskarev, and L. Weis, “On well-posedness of difference schemes for abstract parabolic equations in Lp([0,1];E) spaces,” Numerical Functional Analysis & Optimization, vol. 23, no. 7-8, pp. 669-693, 2002. · Zbl 1022.65095 · doi:10.1081/NFA-120016264
[17] D. Guidetti, B. Karasözen, and S. Piskarev, “Approximation of abstract differential equations,” Journal of Mathematical Sciences (New York), vol. 122, no. 2, pp. 3013-3054, 2004. · Zbl 1111.47063 · doi:10.1023/B:JOTH.0000029696.94590.94
[18] A. Ashyralyev and P. E. Sobolevskii, “Well-posed solvability of the Cauchy problem for difference equations of parabolic type,” Nonlinear Analysis, vol. 24, no. 2, pp. 257-264, 1995. · Zbl 0818.65047 · doi:10.1016/0362-546X(94)E0004-Z
[19] A. Ashiraliev and P. E. Sobolevskii, “Differential schemes of the high-order accuracy for parabolic equations with variable-coefficients,” Dopovidi Akademii Nauk Ukrainskoi RSR, Seriya À- Fiziko-Matematichni ta Technichni Nauki, vol. 6, pp. 3-7, 1988 (Russian).
[20] A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004. · Zbl 1060.65055
[21] H. Amann, “Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications,” Mathematische Nachrichten, vol. 186, pp. 5-56, 1997. · Zbl 0880.42007 · doi:10.1002/mana.3211860102
[22] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1978. · Zbl 0387.46032
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