SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

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

Allaberen Ashyralyev12 and Deniz Agirseven3*

Author Affiliations

1 Department of Mathematics, Fatih University, Istanbul, 34500, Turkey

2 Department of Mathematics, ITTU, Gerogly Street, Ashgabat, 74400, Turkmenistan

3 Department of Mathematics, Trakya University, Edirne, 22030, Turkey

For all author emails, please log on.

Boundary Value Problems 2014, 2014:126  doi:10.1186/1687-2770-2014-126

Published: 20 May 2014

Abstract

In the present paper, the well-posedness of the initial value problem for the delay differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M4">View MathML</a>) in an arbitrary Banach space E with the unbounded linear operators A and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5">View MathML</a> in E with dense domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6">View MathML</a> is studied. Two main theorems on well-posedness of this problem in fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> 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.

Keywords:
delay parabolic equations; well-posedness; fractional spaces; coercive stability estimates