×

Well-posedness of parabolic differential and difference equations. (English) Zbl 1201.35064

Summary: We consider the abstract parabolic differential equation \(u'(t)+Au(t)=f(t)\), \(-\infty <t<\infty\) in a Banach space \(E\) with - \(A\) the infinitesimal generator of an analytic, exponentially decreasing semigroup \(\exp\{-tA\}\) \((t\geq 0)\). The main purpose of this paper is to establish the well-posedness of this equation in \(C^{\beta}(\mathbb{R}, E_{\alpha}\), \((\alpha, \beta\in [0,1])\), and the well-posedness of the corresponding Rothe difference scheme in \(C^{\beta}(\mathbb{R}_{\tau}, E_{\alpha}\), \((\alpha, \beta\in [0,1])\). Moreover, we apply our theoretical results to obtain new coercivity inequalities for the solution of parabolic difference equations.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ashyralyev, A.; Sobolevskii, P. E., (Well-Posedness of Parabolic Difference Equations. Well-Posedness of Parabolic Difference Equations, Operator Theory and Appl. (1994), Birkhäuser Verlag: Birkhäuser Verlag Basel, Boston, Berlin) · Zbl 1077.39015
[2] Ashyralyev, A.; Sözen, Y.; Sobolevskii, P. E., A note on the parabolic differential and difference equations, Abstr. Appl. Anal., 2007, 1-16 (2007) · Zbl 1153.35353
[3] Hino, Y.; Murakami, S.; Yoshizawa, T., Existence of almost periodic solutions of some functional differential equations with infinite delay in a Banach space, Tôhoku Math. J., 49, 133-147 (1997) · Zbl 0876.34077
[4] Kaplan, S., Abstract boundary value problems for linear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. Ser. \(3^e, 20, 2, 395-419 (1966)\) · Zbl 0163.12903
[5] Yamada, Y.; Nııkura, Y., Bifurcation of periodic solutions for nonlinear parabolic equations with infinite delay, Funkcial. Ekvac., 29, 309-333 (1986) · Zbl 0624.35007
[6] Zaidmann, S., Structure of bounded solutions for a class of abstract differential equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., XXII, 43-47 (1976)
[7] Zhang, B., Periodic solutions of nonlinear abstract differential equations with infinite delay, Funkcial. Ekvac., 36, 433-478 (1993) · Zbl 0794.34068
[8] H.C. Simpson, Periodic solutions of integrodifferential equations which arise in population dynamics, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1979.; H.C. Simpson, Periodic solutions of integrodifferential equations which arise in population dynamics, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1979.
[9] Muravnik, A. B., On Cauchy problem for parabolic differential-difference equations, Nonlinear Anal., 51, 215-238 (2002) · Zbl 1010.35111
[10] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’tseva, N. N., Linear and Quasilinear Equations of Parabolic Type (1967), Nauka: Nauka Moscow, (in Russian) · Zbl 0164.12302
[11] Sobolevskii, P. E., Well-posedness of difference elliptic equation, Discrete Dyn. Nat. Soc., 1, 3, 219-231 (1997) · Zbl 0928.39002
[12] Vishik, M. I.; Myshkis, A. D.; Oleinik, O. A., (Partial Differential Equations. Partial Differential Equations, Mathematics in USSR in the Last 40 Years, 1917-1957, vol. 1 (1959), Fizmatgiz: Fizmatgiz Moscow), 563-599, (in Russian)
[13] Alibekov, Kh. A.; Sobolevskii, P. E., Stability of difference schemes for parabolic equations, Dokl. Akad. Nauk, 232, 4, 737-740 (1977), (in Russian) · Zbl 0416.65063
[14] Amann, H., Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186, 5-55 (1997) · Zbl 0880.42007
[15] Ashyralyev, A. O.; Sobolevskii, P. E., The linear operator interpolation theory and the stability of difference—schemes, Dokl. Akad. Nauk, 275, 6, 1289-1291 (1984), (in Russian)
[16] Ashyralyev, A.; Sobolevskii, P. E., Difference schemes of the high-order accuracy for parabolic equations with variable-coefficients, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 6, 3-7 (1988), (in Russian) · Zbl 0652.65063
[17] A. Ashyralyev, A. Hanalyev, P.E. Sobolevskii, Coercive solvability of nonlocal boundary value problem for parabolic equations, 6 (1) (2001) 53-61.; A. Ashyralyev, A. Hanalyev, P.E. Sobolevskii, Coercive solvability of nonlocal boundary value problem for parabolic equations, 6 (1) (2001) 53-61. · Zbl 0996.35027
[18] Ashyralyev, A.; Piskarev, S.; Weis, L., On well-posedness of the difference schemes for abstract parabolic equations in \(L_p([0, 1], E)\) spaces, Numer. Funct. Anal. Optim., 23, 7-8, 669-693 (2002) · Zbl 1022.65095
[19] Ashyralyev, A.; Sobolevskii, P. E., (New Difference Schemes for Partial Differential Equations. New Difference Schemes for Partial Differential Equations, Operator Theory and Appl. (2004), Birkhäuser Verlag: Birkhäuser Verlag Basel, Boston, Berlin) · Zbl 1060.65055
[20] Ashyralyev, A., Nonlocal boundary value problems for abstract parabolic difference equations, well-posedness in Bochner spaces, J. Evol. Equ., 6, 1, 1-28 (2006) · Zbl 1117.65077
[21] Da Prato, G.; Grisvard, P., Sommes d’opérateus linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. (9), 54, 3, 305-387 (1975) · Zbl 0315.47009
[22] Da Prato, G.; Grisvard, P., Équations d’évolution abstraites non linéaires de type parabolique, C. R. Math. Acad. Sci. Paris A-B, 283, 9, A709-A711 (1976)
[23] Guidetti, D.; Karasozen, B.; Piskarev, S., Approximation of abstract differential equations, J. Math. Sci. (NY), 122, 2, 3013-3054 (2004) · Zbl 1111.47063
[24] Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhäuser Verlag: Birkhäuser Verlag Basel, Boston, Berlin · Zbl 0816.35001
[25] Polichka, A. E.; Sobolevskii, P. E., Correct solvability of parabolic difference equations in Bochner spaces, Trans. Moscow Math. Soc., 36, 29-57 (1978), (in Russian)
[26] 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
[27] Shakhmurov, V. B.; Agarwal, R. P., Linear and nonlinear degenerate boundary value problems in Besov spaces, Math. Comput. Modelling, 49, 5-6, 1244-1259 (2009) · Zbl 1165.35475
[28] Sobolevskii, P. E., Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk, 157, 1, 52-55 (1964), (in Russian)
[29] Sobolevskii, P. E., The coercive solvability of difference equations, Dokl. Akad. Nauk SSSR, 201, 5, 1063-1066 (1971), (in Russian) · Zbl 0246.39002
[30] Sobolevskii, P. E., Some properties of the solutions of differential equations in fractional spaces, Trudy Mat. Fak., Voronezh. Gos. Univ. (NS), 14, 68-74 (1974), (in Russian)
[31] Smirnitskii, Yu. A.; Sobolevskii, P. E., The positivity of difference operators, Vychisl. Sistemy, 87, 120-133 (1981), (in Russian) · Zbl 0516.47019
[32] Tribel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam, New York
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.