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

Ashyralyev and Agirseven Boundary Value Problems 2014, 2014:126 http://www.boundaryvalueproblems.com/content/2014/1/126 RESEARCH Open Access Well-posedness of delay parabolic equations with unbounded operators acting on delay terms Allaberen Ashyralyev1,2 and Deniz Agirseven3* * Correspondence: 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 Page 2 of 15 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 to D(A) for all t ∈ [–ω, ∞), and the function Av(t) is continuous on the interval [–ω, ∞). (iii) v(t) satisfies the equation and the initial condition (). A solution v(t) of the initial value problem () is said to be coercive stable (well-posed) if       Av(t) ≤ max Ag(t) + sup f (t) E E –ω≤t≤ ≤s≤t E () for every t, –ω ≤ t < ∞. We are interested in studying the coercive stability of solutions of the initial value problem under the assumption that   B(t)A–  E→E ≤ () holds for every t ≥ . We have not been able to obtain the estimate () in the arbitrary Banach space E. Nevertheless, we can establish the analog of estimates () where the space E is replaced by the fractional spaces Eα ( < α < ) under an assumption stronger than (). The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained. The present paper is organized as follows. Section  is introduction. In Section , two main theorems on well-posedness of the initial value problem () are established. In Section , the coercive stability estimates in Hölder norms for the solutions of the initialboundary value problem for delay parabolic equations are obtained. Finally, Section  is our conclusion. 2 Theorems on well-posedness The strongly positive operator A defines the fractional spaces Eα = Eα (E, A) ( < α < ) consisting of all u ∈ E for which the following norms are finite:   u Eα = supλ–α A exp{–λA}uE . λ> We consider the initial value problem () for delay differential equations of parabolic type in the space C(Eα ) of all continuous functions v(t) defined on the segment [, ∞) with values in a Banach space Eα . First, we consider the problem () when A– and B(t) commute, i.e. A– B(t)u = B(t)A– u, u ∈ D(A). () Theorem . Assume that the condition   B(t)A–  E→E ≤ ( – α) M–α () Ashyralyev and Agirseven Boundary Value Problems 2014, 2014:126 http://www.boundaryvalueproblems.com/content/2014/1/126 Page 3 of 15 holds for every t ≥ , where M is the constant from (). Then for every t, (n – )ω ≤ t ≤ nω, n = , . . . , we have the following coercive stability estimate:   v (t)   + Av(t)Eα    ≤ M(α) max Ag(t) Eα –ω≤t≤ Eα + n–  k= max –(k–)ω≤s≤kω   f (s) Eα where M(α) does not depend on g(t) and f (t). Here, we put +      max f (s) Eα , (n–)ω≤s≤t () m k= ak =  when m < . Proof It is clear that v(t) = u(t) + w(t), () where u(t) is the solution of the problem  du(t) + Au(t) = B(t)u(t – ω), dt t ≥ , () u(t) = g(t) (–ω ≤ t ≤ ), and w(t) is the solution of the problem  dw(t) + Aw(t) = B(t)w(t – ω) + f (t), dt t ≥ , w(t) =  (–ω ≤ t ≤ ). () First, we consider the problem (). Using the formula  t u(t) = exp{–tA}g() + exp –(t – s)A B(s)g(s – ω) ds, ()  the semigroup property, condition (), and the estimates (), (), we obtain   λ–α A exp{–λA}Av(t)E   ≤ λ–α A exp –(λ + t)A Ag()E   t    λ+t–s  –α   B(s)A–  +λ A exp –  A  E→E  E→E     λ+t–s  × A exp –  A Ag(s – ω) ds E  t –α     λ Mλ–α –α Ag() +  – α ≤ ds max Ag(s – ω)Eα Eα (λ + t)–α M–α  (λ + t – s)–α ≤s≤ω   ≤ max Ag(t)Eα –ω≤t≤ for every t,  ≤ t ≤ ω and λ, λ > . This shows that     Au(t) ≤ max Ag(t) Eα Eα –ω≤t≤ () Ashyralyev and Agirseven Boundary Value Problems 2014, 2014:126 http://www.boundaryvalueproblems.com/content/2014/1/126 Page 4 of 15 for every t,  ≤ t ≤ ω. Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality   Au(t)   ≤ max Ag(s)Eα Eα –ω≤s≤ is true for t, (n – )ω ≤ t ≤ nω, n = , , , . . . for some n. Letting t = s + nω, we have  du(s + nω) + Au(s + nω) = B( (...truncated)