Well-posedness of delay parabolic equations with unbounded operators acting on delay terms

In the present paper, the well-posedness of the initial value problem for the delay differential equation dv(t)dt+Av(t)=B(t)v(t−ω)+f(t), t≥0; v(t)=g(t) (−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spaces Eα are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.MSC:35G15.

[1]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[2]  Gabriella Di Blasio,et al.  Delay differential equations with unbounded operators acting on delay terms , 2003 .

[3]  Ali Fuat Yeniçerioğlu.  The behavior of solutions of second order delay differential equations , 2007 .

[4]  Ali Fuat Yeniçerioglu.  Stability properties of second order delay integro-differential equations , 2008, Comput. Math. Appl..

[5]  L. Torelli,et al.  Stability of numerical methods for delay differential equations , 1989 .

[6]  Allaberen Ashyralyev,et al.  Finite Difference Method for Delay Parabolic Equations , 2011 .

[7]  A. N. Al-Mutib.  Stability properties of numerical methods for solving delay differential equations , 1984 .

[8]  Allaberen Ashyralyev,et al.  On convergence of difference schemes for delay parabolic equations , 2013, Comput. Math. Appl..

[9]  Marino Zennaro,et al.  Stability analysis of one-step methods for neutral delay-differential equations , 1988 .

[10]  Allaberen Ashyralyev,et al.  Fractional spaces generated by the positive differential and difference operators in a Banach space , 2007 .

[11]  Pengzhen Dong,et al.  Sufficient conditions for inverse anticipating synchronization of unidirectional coupled chaotic systems with multiple time delays , 2010, 2010 Chinese Control and Decision Conference.

[12]  Alfredo Bellen,et al.  One-step collocation for delay differential equations , 1984 .

[13]  Allaberen Ashyralyev,et al.  Well-posedness of delay parabolic difference equations , 2014 .

[14]  H. Stewart.  Generation of analytic semigroups by strongly elliptic operators , 1974 .

[15]  Deniz Agirseven,et al.  Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition , 2012 .

[16]  Allaberen Ashyralyev,et al.  On the stability of the linear delay differential and difference equations , 2001 .

[17]  A. Ashyralyev,et al.  Positivity of two-dimensional elliptic differential operators in Hölder spaces , 2012 .

[18]  Allaberen Ashyralyev,et al.  Preface: Second International Conference on Analysis and Applied Mathematics (ICAAM 2014) , 2012 .

[19]  Allaberen Ashyralyev,et al.  New Difference Schemes for Partial Differential Equations , 2004 .

[20]  Haydar Akça,et al.  Stability estimates of difference schemes for neutral delay differential equations , 2001 .

[21]  Ali Fuat Yeniçerioglu,et al.  On the stability of the second-order delay differential equations with variable coefficients , 2004, Appl. Math. Comput..

[22]  A. Iacob,et al.  Well-Posedness of Parabolic Difference Equations , 1994 .