Well-posedness of delay parabolic equations with unbounded operators acting on delay terms
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RESEARCH
Open Access
Well-posedness of delay parabolic equations with unbounded operators acting on delay terms Allaberen Ashyralyev1,2 and Deniz Agirseven3* * Correspondence: [email protected] 3 Department of Mathematics, Trakya University, Edirne, 22030, Turkey Full list of author information is available at the end of the article
Abstract In the present paper, the well-posedness of the initial value problem for the delay differential equation dv(t) + Av(t) = B(t)v(t – ω) + f (t), t ≥ 0; v(t) = g(t) (–ω ≤ t ≤ 0) in an dt 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 Keywords: delay parabolic equations; well-posedness; fractional spaces; coercive stability estimates
1 Introduction The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [–] and the references therein) and insight has developed over the last three decades. The theory of stability and coercive stability of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations (see [–]). It is well known that various initial-boundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form dv(t) + Av(t) = B(t)v(t – ω) + f (t), t ≥ , dt () v(t) = g(t) (–ω ≤ t ≤ ) 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)). Let A be a strongly positive operator, i.e. –A is the generator of the analytic semigroup exp{–tA} (t ≥ ) of the linear bounded operators with exponentially decreasing norm when t → ∞. That means the following estimates hold: exp{–tA}
E→E
≤ Me–δt ,
tA exp{–tA} ≤ M, E→E
t>
()
for some M > , δ > . Let B(t) be closed operators. ©2014 Ashyralyev and Agirseven; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ashyralyev and Agirseven Boundary Value Problems 2014, 2014:126 http://www.boundaryvalueproblems.com/content/2014/1/126
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A function v(t) is called a solution of the problem () if the following conditions are satisfied: (i) v(t) is continuously differentiable on the interval [–ω, ∞). The derivative at the endpoint t = –ω is understood as the appropriate unilateral derivative. (ii) The element v(t) belongs t
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