Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 56, pp. 1–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu A PARABOLIC BIPOLYNOMIAL FRACTIONAL DIRICHLET-LAPLACE PROBLEM DARIUSZ IDCZAK Abstract. We derive existence results for a parabolic bipolynomial abstract and classical problems containing fractional powers of the Dirichlet-Laplace operator on a bounded domain, in the sense of the Stone-von Neumann oper- ator calculus. The main tools are theorems on the existence and uniqueness of a weak solutions to an abstract problem, due to Friedman, and a general theorem on the equivalence of weak and strong solutions to some operator equation. 1. Introduction The basic model of diffusion processes is ut(t, x) + (−∆)u(t, x) = 0 where (−∆) is the Laplace operator in x. To represent some anomalous diffusions, one considers the so-called fractional model ut(t, x) + (−∆)βu(t, x) = 0 with (−∆)β being a fractional Laplacian of order β. Different kinds of such a Laplacian are used, e.g. spectral, restricted, censored (see [4, 5, 15]). In this article, we consider the non-homogenous parabolic equation ut(t, x) + k∑ i,j=0 αiαj(−∆)βi+βju(t, x) = f(t, x), a.e.x ∈ Ω, (1.1) containing the “bipolynomial” term ∑k i,j=0 αiαj(−∆)βi+βju(t, x), with initial con- dition u(0, x) = 0, x ∈ Ω, (1.2) and boundary condition u(t, ·) ∈ H1 0 (Ω), t ∈ (0, T ), (1.3) where αi > 0 for i = 0, . . . , k, 0 ≤ β0 < · · · < βk, (−∆)βi+βj is the spectral fractional lapalacian for i, j = 0, . . . , k , f : (0, T ) × Ω → R, 0 < T < ∞, Ω is an open and bounded set in RN with N ∈ N, H1 0 (Ω) is the classical Sobolev space of real-valued functions. Here, the spectral fractional Laplacian is a power of 2020 Mathematics Subject Classification. 35K10, 35R11. Key words and phrases. Parabolic equation; fractional Dirichlet-Laplace operator; existence of solutions. ©2022. This work is licensed under a CC BY 4.0 license. Submitted December 8, 2021. Published July 31, 2022. 1 2 D. IDCZAK EJDE-2022/56 the Dirichlet-Laplace operator in the Stone-von Neumann operator calculus sense. Particular cases of the above problem are: the classical parabolic Dirichlet-Laplace problem ut(t, x) + [(−∆)ω]u(t, x) = f(t, x), a.e x ∈ Ω, (1.4) the biharmonic parabolic problem ut(t, x) + [(−∆)ω]2u(t, x) = f(t, x), a.e. x ∈ Ω, (1.5) and the standard fractional (in state) parabolic problem ut(t, x) + [(−∆)ω]βu(t, x) = f(t, x), a.e. x ∈ Ω. (1.6) In [13], we derived the existence of a strong solution to the nonlinear non- autonomous partial elliptic system k∑ i,j=0 αiαj [(−∆)ω]βi+βju(x)− au(x) = DuF (x, u(x)), a.e. x ∈ Ω, with a ∈ R and Dirichlet boundary conditions. We applied there a direct variational method and some results based on the dual least action principle. The aim of this article is to prove existence of a solution to the parabolic problem (1.1)-(1.2)-(1.3) containing the bipolynomial term. An abstract form of the problem (1.1) -(1.2)-(1.3) is as follows u′(t)(x) + k∑ i,j=0 αiαj(−∆)βi+βju(t)(x) = f(t)(x), t ∈ [0, T ), a.e. x ∈ Ω, (1.7) u(0) = 0, (1.8) u(t) ∈ H1 0 (Ω), t ∈ (0, T ), (1.9) where u acts from [0, T ) to the domain of the operator ∑k i,j=0 αiαj(−∆)βi+βj . First, we prove existence of a solution to problem (1.7)-(1.8)-(1.9) and next apply this result to obtain existence of a solution to problem (1.1)-(1.2)-(1.3). In the proof of the existence of a solution to the abstract problem we shall use an existence result for an abstract parabolic variational problem, due to Friedman (see [9, Theorem 17, Part X.6]), and a general theorem on the equivalence of weak and strong solutions to some operator equation, obtained in [13] as a result of the Stone-von Neumann operator calculus. This equivalence theorem was also used in [13]. Our paper is the second part of a program of research of fundamental partial differential equations (elliptic, parabolic, hyperbolic) containing such a term. In the future, we plan to investigate a bipolynomial hyperbolic problem. Let us add that in the paper [14], using a global implicit function theorem, we studied an elliptic nonlinear problem containing the spectral fractional Laplacian and showed that for each functional parameter there exists a unique solution to the problem and that its dependence on this parameter is continuously differentiable. This article consists of three parts. In the first part, we recall and complete some facts concerning the different kinds of derivatives of abstract functions. Next, we formulate the Friedman theorem. In the second, main part of the paper, we derive an existence and uniqueness result for the abstract problem (1.7)-(1.8)-(1.9). As a corollary, under some additional assumptions, we obtain existence of a solu- tion to problem (1.1)-(1.2)-(1.3), for βk ≥ 1. To the best of our knowledge, such bipolynomial problems were not investigated up to now. The last part (Appendix) EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 3 contains the basics concerning the powers of a weak Dirichlet-Laplace operator and the mentioned theorem on the equivalence of weak and strong solutions to an operator equation. 2. Preliminaries 2.1. Derivatives of abstract functions. Details about the results presented in this section can be found in [9] (strong and weak derivatives) and [6] (Sobolev derivative). We recall them for the convenience of the reader. We prove the results that are not proved in the mentioned monographs. Let (a, b) ⊂ R be an interval, X - a real Hilbert space with scalar product (·, ·)X and the corresponding norm ‖ · ‖X . By Dk(a, b;X) we denote the space of functions f : (a, b) → X that are k- times continuously classic differentiable and f , f ′, . . . , fk ∈ L2(a, b;X). The norm in Dk(a, b;X) is ‖f‖Dk = ( k∑ i=0 ∫ b a ‖f i(t)‖2X )1/2 . (2.1) The completion of Dk(a, b;X) in L2(a, b;X) with respect to the norm ‖ · ‖Dk is denoted by Hk(a, b;X). We say that a function f : (a, b)→ X has k strong derivatives provided that f ∈ Hk(α, β;X) for any a < α < β < b. If f ∈ Hk(a, b;X), then by the strong derivative f js,(a,b)of f on (a, b) of order j = 1, . . . , k we mean the limit in L2(a, b;X) of the sequence (Djum) of classical derivatives of order j of functions um where (um) ⊂ Dk(a, b;X) is the Cauchy sequence with respect to the norm (2.1) converging in L2(a, b;X) to f (one can show that the strong derivatives do not depend on the choice of the sequence (um)). If f /∈ Hk(a, b;X), then by the strong derivative f js,(a,b) of f on (a, b) of order j = 1, . . . , k we mean the function that is the extension on (a, b) of each function f js,(α,β). Remark 2.1. Of course, if f ∈ Hk(a, b;X), then f ∈ Hk(α, β;X) for any a < α < β < b. It is easy to see that the function being the extension on (a, b) of each function f js,(α,β) coincides with the strong derivative f js,(a,b) for any j = 1, . . . , k. We say that a function f ∈ L2(a, b;X) has the weak derivative of order k if there exists a function v ∈ L2(a, b;X) such that (−1)k ∫ b a (f(t), ψ(k)(t))Xdt = ∫ b a (v(t), ψ(t))Xdt (2.2) for any function ψ : [a, b] → X continuously differentiable k times and such that ψ(j)(a) = ψ(j)(b) = 0 for j = 0, . . . , k − 1 (if a = −∞ we require ψ to be such that ψ(t) = 0 for sufficiently small t; similarly in the case of b =∞). By the weak derivative we mean the function v and denote it as f ′ω. We say that a function f ∈ L2(a, b;X) has the Sobolev derivative f ′Sob if there exists a function g ∈ L2(a, b;X) such that∫ b a f(t)ϕ′(t)dt = − ∫ b a g(t)ϕ(t)dt for any ϕ ∈ C∞c (a, b;R). We put in such a case f ′Sob = g. The set of all functions possessing the Sobolev derivative is denoted by W 1,2(a, b;X). From [16, Part I, 4 D. IDCZAK EJDE-2022/56 Thm. 2.1] it follows that, for any function f ∈W 1,2(a, b;X), there exists a sequence (fn) ⊂ C∞([a, b];X) such that fn → f and f ′n → f ′Sob in L2(a, b;X). (2.3) One proves that if f ∈ W 1,2(a, b;X) then there exists a continuous function f̃ : (a, b)→ X such that f = f̃ a.e. on I and f̃(t)− f̃(s) = ∫ t s f ′Sob(τ)dτ (2.4) for a.e. t, s ∈ (a, b). In the case of (a, b) bounded, it follows from the results presented in [6]. When (a, b) is unbounded one can repeat the proofs of Corollary 4.24, Lemma 8.1, Lemma 8.2 and Theorem 8.2 from the book [7] in the case of any Hilbert space X (in [7], the proofs are presented for X = R). In such a case, f has the classical derivative f ′ a.e. on I (in the sense of the representative a.e.) and f ′(t) = f ′Sob(t), a.e. t ∈ (a, b). (2.5) We have the following two lemmas containing a comparison of the strong deriva- tives with the weak and Sobolev ones. Lemma 2.2. If f ∈ Hk(a, b;X), then the strong derivatives of f are the weak ones. Proof. It is sufficient to consider the case of bounded interval (a, b) ⊂ R. Moreover, since k-th strong derivative is the first strong derivative of (k − 1)-th strong deriv- ative therefore it is sufficient to prove the lemma for k = 1. The proof consists of three steps. Step 1. First, let us assume that f ∈ C∞([a, b];X). We know that d dt (f(t), ψ(t))X = (f ′(t), ψ(t))X + (f(t), ψ′(t))X (2.6) for t ∈ [a, b] and any function ψ : [a, b] → X continuously differentiable with ψ(a) = ψ(b) = 0. So,∫ b a (f ′(t), ψ(t))Xdt+ ∫ b a (f(t), ψ′(t))Xdt = ∫ b a d dt (f(t), ψ(t))Xdt = (f(b), ψ(b))X − (f(a), ψ(a))X = 0. So, f has the weak derivative f ′ω and it is equal to the classical one f ′ which coincides with the strong derivative f ′s,(a,b). Step 2. Now, let us assume that f ∈ D1(a, b;X). In the same way as in the first case we assert that d dt (f(t)ϕ(t)) = f ′(t)ϕ(t) + f(t)ϕ′(t) for t ∈ (a, b), and any function ϕ ∈ C∞c (a, b;R). Clearly, the function f(t)ϕ(t) has compact support contained in (a, b). Consequently, d dt (f(t)ϕ(t)) also has the compact support and there exists ε > 0 such that f(t)ϕ(t) = 0 and d dt (f(t)ϕ(t)) = 0 for t ∈ (a, a+ ε) ∪ (b− ε, b). Continuity of d dt (f(t)ϕ(t)) implies the differentiability of the function F : (a, b) 3 t 7→ ∫ t a d ds (f(s)ϕ(s))ds ∈ X EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 5 everywhere on (a, b) and F ′(t) = d dt (f(t)ϕ(t)) for all t ∈ (a, b). Thus, there exists a constant c ∈ X such that∫ t a d ds (f(s)ϕ(s))ds = c+ f(t)ϕ(t) for t ∈ (a, b). So,∫ b a f ′(t)ϕ(t)dt+ ∫ b a f(t)ϕ′(t)dt = ∫ b a d dt (f(t)ϕ(t))dt = ∫ b−ε a+ε d dt (f(t)ϕ(t))dt = f(b− ε)ϕ(b− ε)− f(a+ ε)ϕ(a+ ε) = 0. From property (2.3) it follows that there exists a sequence (fn) ⊂ C∞([a, b];X) such that fn → f and f ′n → f ′ in L2(a, b;X). In consequence, using the first step we obtain∫ b a (f(t), ψ′(t))Xdt = ( f(·), ψ′(·) ) L2(a,b;X) = ( lim fn(·), ψ′(·) ) L2(a,b;X) = lim ( fn(·), ψ′(·) ) L2(a,b;X) = − lim ( f ′n(·), ψ(·) ) L2(a,b;X) = − ( lim f ′n(·), ψ(·) ) L2(a,b;X) = − ( f ′(·), ψ(·) ) L2(a,b;X) = − ∫ b a (f ′(t), ψ(t))Xdt (2.7) for any function ψ : [a, b]→ X continuously differentiable with ψ(a) = ψ(b) = 0. Step 3. To finish the proof assume that f ∈ H1(a, b;X). So, there exists a sequence (un) ⊂ D1(a, b;X) such that un → f and u′n → f ′s,(a,b) in L2(a, b;X). In the same way as in (2.7), we show that∫ b a (f(t), ψ′(t))dt = − ∫ b a (f ′s,(a,b)(t), ψ(t))dt for any function ψ : [a, b] → X continuously differentiable with ψ(a) = ψ(b) = 0. This means that f has the weak derivative f ′ω which is equal to the strong one f ′s,(a,b) and the proof is completed. � Lemma 2.3. If f ∈ H1(a, b;X), then the strong derivative f ′s,(a,b) is the Sobolev one. Proof. First, let us assume that (a, b) ⊂ R is bounded. In the second step of the proof of Lemma 2.2, it is proved that if f ∈ D1(a, b;X), then the classical derivative f ′ is the Sobolev one. Now, let f ∈ H1(a, b;X) and (un) ⊂ D1(a, b;X) be such that un → f and u′n → f ′s,(a,b) in L2(a, b;X). 6 D. IDCZAK EJDE-2022/56 We have ∫ b a f(t)ϕ′(t)dt = lim ∫ b a un(t)ϕ′(t)dt = − lim ∫ b a u′n(t)ϕ(t)dt = − ∫ b a f ′s,(a,b)(t)ϕ(t)dt for any ϕ ∈ C∞c (a, b;R) (the first and third equalities follow from the linear- ity and continuity of the operator L2(a, b;X) 3 v 7→ ∫ b a v(t)ω(t)dt ∈ X where ω ∈ C∞c (a, b;R)). It means that the strong derivative f ′s,(a,b) of the function f ∈ H1(a, b;X) is the Sobolev one. Now, let assume that (a, b) is unbounded. Fix a function ϕ ∈ C∞c (a, b;R) and let (α, β) ⊂ (a, b) be such a bounded interval that suppϕ ⊂ (α, β). Of course, f ∈ H1(α, β;X). Consequently, from the first part of the proof and Remark 2.1 we obtain∫ b a f(t)ϕ′(t)dt = ∫ β α f(t)ϕ′(t)dt = − ∫ β α f ′s,(α,β)(t)ϕ(t)dt = − ∫ b a f ′s,(a,b)(t)ϕ(t)dt. The proof is complete. � Theorem 2.4. Let (a, b) ⊂ R be a bounded interval and f : (a, b) → X. The following conditions are equivalent: (a) f ∈ H1(a, b;X); (b) f has the weak derivative of order 1; (c) f ∈W 1,2(a, b;X); (d) f ∈ L2(a, b;X) and∣∣ ∫ b a 〈f(t), ϕ′(t)〉X×X∗dt ∣∣ ≤ c‖ϕ‖L2(a,b;X∗) (2.8) for any function ϕ : (a, b) → X∗ infinitely differentiable with compact suppϕ contained in (a, b). Proof. (a)⇒(b). This implication follows from Lemma 2.2. (b)⇒(d). Let us fix a function ϕ : (a, b) → X∗ infinitely differentiable with the compact suppϕ ⊂ (a, b). Consider the function ψ : [a, b]→ X given by ψ(t) = { 0, t ∈ {a, b}, #ϕ(t), t ∈ (a, b), where #ϕ(t) is the unique element from X determining ϕ(t) according to Riesz theorem. It is easy to see that, for any t ∈ (a, b), the condition lim h→0 ‖ϕ(t+ h)− ϕ(t) h − ϕ′(t)‖X∗ = 0 implies lim h→0 ‖ψ(t+ h)− ψ(t) h −# ϕ′(t)‖X = 0. EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 7 Indeed, ‖ψ(t+ h)− ψ(t) h −# ϕ′(t)‖X = ‖ #ϕ(t+ h)−# ϕ(t) h −# ϕ′(t)‖X = ‖#( ϕ(t+ h)− ϕ(t) h − ϕ′(t))‖X = ‖ϕ(t+ h)− ϕ(t) h − ϕ′(t)‖X∗ → 0 as h → 0. So, function ψ is differentiable at t and ψ′(t) =# ϕ′(t). Moreover, if tn → t, then ‖ψ′(tn)− ψ′(t)‖X = ‖#(ϕ′(tn))−# (ϕ′(t))‖X = ‖#(ϕ′(tn)− ϕ′(t))‖X = ‖ϕ′(tn)− ϕ′(t)‖X∗ → 0. So, ψ′ is continuous at t. Case of t ∈ {a, b} is obvious because of the compactness of suppϕ ⊂ (a, b). Thus, condition (2.2) implies∫ b a ( f(t), ψ′(t) ) X dt = − ∫ b a (v(t), ψ(t))Xdt for some v ∈L2(a, b;X) and, consequently,∣∣ ∫ b a 〈f(t), ϕ′(t)〉X×X∗dt ∣∣ = ∣∣ ∫ b a (f(t),# ϕ′(t))Xdt ∣∣ = ∣∣ ∫ b a (f(t), ψ′(t))Xdt ∣∣ = ∣∣ ∫ b a ( v(t), ψ(t) ) X dt ∣∣ ≤ ‖v‖L2(a,b;X)‖ψ‖L2(a,b;X) = ‖v‖L2(a,b;X) (∫ b a ‖ψ(t)‖2Xdt )1/2 = ‖v‖L2(a,b;X) (∫ b a ‖#ϕ(t)‖2Xdt )1/2 = ‖v‖L2(a,b;X) (∫ b a ‖ϕ(t)‖2X∗dt )1/2 = ‖v‖L2(a,b;X)‖ϕ‖L2(a,b;X∗). (d)⇒(a). From condition (2.8) it follows (see [6, Propositions A6, A7 and Corol- laire A2]) that f ∈W 1,2(a, b;X). So, from property (2.3) it follows that there exists a sequence (ϕn) ⊂ C∞([a, b], X) such that ϕn → f and ϕ′n → f ′Sob in L2(a, b;X). It means that f ∈ H1(a, b;X). (d)⇔(c). This part of the theorem follows from [6, Propositions A6, A7 and Corollary A2]. � From the Theorem 2.4, Lemmas 2.2, 2.3 and from the uniqueness of weak, strong and Sobolev derivatives as well as from (2.5) the following corollary follows. Corollary 2.5. If (a, b) ⊂ R is bounded, then the sets W 1,2(a, b;X), H1(a, b;X) and the set of functions possessing weak derivative coincide and the notions of 8 D. IDCZAK EJDE-2022/56 Sobolev, strong, weak and classical (in the sense of the representative a.e.) deriva- tives of the first order coincide in this set. 3. Fractional parabolic problem 3.1. Friedmann theorem. Let us assume that V , H are real Hilbert spaces with the scalar products (·, ·)V , (·, ·)H and the corresponding norms ‖ ·‖V , ‖ ·‖H , respec- tively, V = H and the embedding V → H is continuous. Let k be a fixed positive integer and a = a(t, u, v) : (−∞, T )× V × V → R with T > 0, be a function such that (i) for any t ∈ (−∞, T ), the form a(t, ·, ·) is bilinear; (ii) for any u, v ∈ V , the function a(·, u, v) is k times continuously differentiable in t ∈ (−∞, T ) and | d j dtj a(t, u, v)| ≤M‖u‖V ‖v‖V for some constant M and any t ∈ (−∞, T ), 0 ≤ j ≤ k; (iii) there exist constants α > 0, λ such that a(t, v, v) + λ‖v‖2H ≥ α‖v‖2V ∀v ∈ V. The following theorem is proved in [9, Part X.6, Theorem 17 and formula (6.30)]. Theorem 3.1. If a = a(t, u, v) satisfies (i)–(iii), a function f : (−∞, T ) → V has k strong derivatives on (−∞, T ) and f , f ′s,(−∞,T ), . . . , f k s,(−∞,T ) ∈ L 2(−∞, T ;V ), f(t) = 0, t < 0, then there exists a unique function u : (−∞, T )→ V which has k strong derivatives on (−∞, T ), such that u,u′s,(−∞,T ), . . . ,u k s,(−∞,T ) ∈ L 2(−∞, T ;V ), u(t) = 0, t < 0, a(t,u(t), v) + ( u′s,(−∞,T )(t), v ) H = ( f(t), v ) H , a.e. t ∈ (0, T ) for each v ∈ V . 3.2. Abstract fractional parabolic problem. Now, let A : D(A) ⊂ L2(Ω;R)→ L2(Ω;R) be a self-adjoint operator with non-empty resolvent set ρ(A). Theorem 3.2. If f : (−∞, T ) → D(A) is continuous, has all strong derivatives and f , f ′s,(−∞,T ), f ′′ s,(−∞,T ), · · · ∈ L 2(−∞, T ;D(A)), f(t) = 0, t < 0, then there exists a unique function ũ : [0, T ] → D(A) infinitely differentiable such that ũ(k)(0) = 0, k = 0, 1, . . . , ũ(t) ∈ D(A2), t ∈ (0, T ), (3.1) and, for t ∈ [0, T ), ũ′(t)(x) +A2ũ(t)(x) = f(t)(x), a.e. x ∈ Ω (3.2) EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 9 Proof. Existence. We denote V = D(A), H = L2(Ω;R) and (u, v)V = ∫ Ω u(x)v(x)dx+ ∫ Ω Au(x)Av(x)dx for u, v ∈ V , and (u, v)H = ∫ Ω u(x)v(x)dx for u, v ∈ H. We define a : (−∞, T )× V × V → R, by a(t, u, v) = ∫ Ω Au(x)Av(x)dx. It is clear that V is continuously embedded in H and a satisfies conditions (i)–(iii) with M = 1, λ = 1, α = 1. Using Theorem 3.1 we assert that there exists a unique function u : (−∞, T )→ D(A) which has all strong derivatives on (−∞, T ), u,u′s,(−∞,T ),u ′′ s,(−∞,T ), · · · ∈ L 2(−∞, T ;D(A)), u(t) = 0, t < 0, and for v ∈ D(A), a(t,u(t), v) + ( u′s,(−∞,T )(t), v ) L2(Ω;R) = ( f(t), v ) L2(Ω;R) , a.e. t ∈ (0, T ) i.e., ∫ Ω Au(t)(x)Av(x)dx = ∫ Ω (f(t)− u′s,(−∞,T )(t))(x)v(x)dx (3.3) for a.e. t ∈ (0, T ). Let us fix a positive integer k ≥ 2. Since u′s,(−∞,T ), . . . ,u (k) s,(−∞,T ) are the strong derivatives of u on (−∞, T ), it follows that u ∈Hk(0, T ;D(A)) and the strong derivatives u′s,(0,T ),. . . ,u (k) s,(0,T ) on (0, T ) safisfy u′s,(0,T ) = u′s,(−∞,T ) ∣∣ (0,T ) , . . . , u (k) s,(0,T ) = u (k) s,(−∞,T ) ∣∣ (0,T ) . Let a sequence (um) ⊂ Dk(0, T ;D(A)) be such that um → u in L2(0, T ;D(A)),u′m → u′s,(0,T ) in L2(0, T ;D(A)), u′′m → u′′s,(0,T ) in L2(0, T ;D(A)), (3.4) . . . (3.5) u(k) m → u (k) s,(0,T ) in L2(0, T ;D(A)). From Theorem 2.4 ((a)=⇒(c)) and (2.4) it follows that there exists a function ũ : [0, T ]→ D(A) such that u(t) = ũ(t), a.e. t ∈ [0, T ], ũ(t) = ũ(0) + ∫ t 0 g̃(s)ds, t ∈ [0, T ], (3.6) with g̃ ∈ L2(0, T ;D(A)) and∫ T 0 ũ(t)ϕ′(t)dt = − ∫ T 0 g̃(t)ϕ(t)dt for any ϕ ∈ C∞c ((0, T ),R). From (3.6) it follows that the function ũ is classically differentiable a.e. on [0, T ] and ũ′(t) = g̃(t) a.e. on [0, T ]. Of course, g̃(t) = u′Sob(t) 10 D. IDCZAK EJDE-2022/56 a.e. on [0, T ]. From Corollary 2.5 it follows that u′Sob(t) = u′s,(0,T )(t) a.e. on [0, T ]. Thus, u′s,(0,T )(t) = ũ′(t), a.e. t ∈ (0, T ) It is easy to see that u′′s,(0,T ) is the strong derivative of the first order of the function u′s,(0,T ), i.e. (u′s,(0,T )) ′ s,(0,T ) = u′′s,(0,T ) a.e. on (0, T ). Therefore, (3.2), (3.4) can be written down as u′m → ũ′ in L2(0, T ;D(A)), (3.7) (u′m)′ = u′′m → u′′s,(0,T ) = (u′s,(0,T )) ′ s,(0,T ) = (ũ′)′s,(0,T ) in L2(0, T ;D(A)). (3.8) Using (3.7) and (3.8), in the same way as above, we prove that there exists a function ˜̃u : [0, T ]→ D(A) such that ũ′(t) = ˜̃u(t), a.e. t ∈ [0, T ], ˜̃u(t) = ˜̃u(0) + ∫ t 0 ˜̃g(s)ds, t ∈ [0, T ]. (3.9) with ˜̃g ∈ L2(−∞, T ;D(A)), as well as (ũ′)′s,(0,T ) = ˜̃u′ a.e. on [0, T ]. Now, let us observe (see (3.9)) that, for t ∈ [0, T ], ũ(t) = ũ(0) + ∫ t 0 ũ′(t1)dt1 = ũ(0) + ∫ t 0 ˜̃u(t1)dt1. (3.10) Continuity of ˜̃u implies that ũ is continuously differentiable everywhere on [0, T ] and ũ′(t) = ˜̃u(t) for all t ∈ [0, T ]. In the same way as above, we check that ũ′ is continuously differentiable every- where on [0, T ]. So, ũ is twice continuously differentiable on [0, T ]. Repeating the above reasoning we assert that ũ is (k − 1)-times continuously differentiable. Since k ≥ 2 was arbitrary, ũ is infinitely continuously differentiable. Now, (3.3) can be written down as∫ Ω Aũ(t)(x)Av(x)dx = ∫ Ω (f(t)− ˜̃u(t))(x)v(x)dx (3.11) for a.e. t ∈ (0, T ). Since ũ, f , ˜̃u are continuous on [0, T ), it follows that the mappings [0, T ) 3 t 7→ ∫ Ω Aũ(t)(x)Av(x)dx ∈ R, (3.12) [0, T ) 3 t 7→ ∫ Ω (f(t)− ˜̃u(t))(x)v(x)dx ∈ R (3.13) are continuous, too. Indeed, the continuity of (3.12) follows from the estimate∣∣ ∫ Ω Aũ(t1)(x)Av(x)dx− ∫ Ω Aũ(t2)(x)Av(x)dx ∣∣ ≤ ∫ Ω |Aũ(t1)(x)−Aũ(t2)(x)| |Av(x)|dx ≤ ‖Aũ(t1)−Aũ(t2)‖L2(Ω,R)‖Av‖L2(Ω,R) ≤ ‖ũ(t1)− ũ(t2)‖D(A)‖Av‖L2(Ω,R) EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 11 for any t1, t2 ∈ [0, T ). The continuity of (3.13) follows from the estimation (see [13]) ∣∣ ∫ Ω (f(t1)− ˜̃u(t1))(x)v(x)dx− ∫ Ω (f(t2)− ˜̃u(t2))(x)v(x)dx ∣∣ ≤ ∫ Ω (|f(t1)(x)− f(t2)(x)|+ |˜̃u(t1)(x)− ˜̃u(t2)(x)|)|v(x)|dx ≤ ( ‖f(t1)− f(t2)‖L2(Ω,R) + ‖˜̃u(t1)− ˜̃u(t2)‖L2(Ω,R) ) ‖v‖L2(Ω,R) ≤ ( ‖f(t1)− f(t2)‖D(A) + ‖˜̃u(t1)− ˜̃u(t2)‖D(A) ) ‖v‖L2(Ω,R) for any t1, t2 ∈ [0, T ). So, equality (3.11) holds everywhere on [0, T ). From Theorem 4.5 it follows that, for t ∈ [0, T ), ũ(t) ∈ D(A2) and˜̃u(t)(x) +A2ũ(t)(x) = f(t)(x), a.e. x ∈ Ω . Consequently (see (3.10)), ũ′(t)(x) +A2ũ(t)(x) = f(t)(x), a.e. x ∈ Ω (3.14) for t ∈ [0, T ). Initial conditions. Now, we shall show that ũ satisfies the zero initial conditions. Indeed, consider restrictions u′s,(−∞,T ) ∣∣ (a,T ) , u′′s,(−∞,T ) ∣∣ (a,T ) for some −∞ < a < 0. Clearly, u′s,(−∞,T ) ∣∣ (a,T ) = u′s,(a,T ), u′′s,(−∞,T ) ∣∣ (a,T ) = u′′s,(a,T ). Moreover, in the same way as in the case of the function ũ we obtain existence of a function û : [a, T ]→ D(A) such that u(t) = û(t), a.e. t ∈ [a, T ], û(t) = û(a) + ∫ t a ĝ(s)ds, t ∈ [a, T ] with ĝ ∈ L2(a, T ;D(A)). Since u(t) = 0 for t ∈ [a, 0), it follows that ĝ = 0 on [a, 0) and, consequently, û(a) = 0. So, û(t) = ∫ t a ĝ(s)ds = ∫ 0 a ĝ(s)ds+ ∫ t 0 ĝ(s)ds = ∫ t 0 ĝ(s)ds, t ∈ [0, T ]. Therefore, since ũ(t) = û(t) for t ∈ [0, T ], it follows that ũ(0) + ∫ t 0 g̃(s)ds = ∫ t 0 ĝ(s)ds, t ∈ [0, T ]. From this condition it follows that g̃ = ĝ a.e. on [0, T ] and ũ(0) = 0. Repeating the above reasoning with the function ũ replaced by ũ′,. . . ,ũ(k−2) re- spectively, we check that ũ′(0) = 0, . . . , ũ(k−2)(0) = 0. Since k ≥ 2 was arbitrary, ũ(k)(0) = 0 for k = 0, 1, . . . . 12 D. IDCZAK EJDE-2022/56 “Uniqueness”. The assertion follows from the fact that if a function ũ : [0, T ] → D(A) satisfies (3.2) and the conditions given in Theorem 3.2, then the function u : (−∞,T ) 3 t 7→ { 0, t < 0, ũ(t), t ∈ [0, T ) is continuously differentiable on (−∞, T ) and u,u′ ∈ L2(−∞, T ;D(A)). So, u ∈D1(−∞, T ;D(A)) ⊂ H1(−∞, T ;D(A)). It means that u has the first strong derivative u′s,(−∞,T ) which is equal to u′. From the fact that, for any t ∈ [0, T ), u(t) ∈ D(A2) and u′(t)(x) +A2u(t)(x) = f(t)(x), x ∈ Ω a.e. it follows (see Theorem 4.5) that, for any t ∈ [0, T ),∫ Ω Au(t)(x)Av(x)dx = ∫ Ω (f(t)− u′(t))(x)v(x)dx (3.15) for any v ∈ D(A). Clearly, the above equality is also satisfied for any t < 0 and v ∈ D(A). So, using our notations we can write a(t,u(t), v) + ( u′s(t), v ) H = ( f(t), v ) H (3.16) for all t ∈ (−∞, T ) and any v ∈ D(A). Uniqueness of the solution stated in Theorem 3.1 (with k = 1, V = D(A), H = L2) completes the proof of the theorem. � Theorem 3.3. Let the assumptions of Theorem 3.2 be satisfied. If, additionally, f(t) = 0, t < ε for some ε > 0, then the solution described in Theorem 3.2 satisfies ũ(t) = 0, 0 ≤ t < ε. Proof. Let ũ : [0, T ]→ D(A) be the solution of (3.2), given in Theorem 3.2. Then the function u described in the “uniqueness” part of the proof of Theorem 3.2 is the solution to (3.16), corresponding to f . Since the zero function defined on (−∞, ε) is the solution to (3.15), it satisfies also (3.16). From the uniqueness of the solution to (3.16) on (−∞, ε) it follows that ũ ∣∣ [0,ε) ≡ 0. � Applying Theorems 3.2 and 3.3 in the case of A = w((−∆)ω) (see Proposition 4.4 in subsection 4.1) we obtain the following result. Corollary 3.4. Assume that f : (−∞, T ) → D(w((−∆)ω)) is continuous, has all strong derivatives and f , f ′s,(−∞,T ), f ′′ s,(−∞,T ), · · · ∈ L 2(−∞, T ;D(w((−∆)ω))), f(t) = 0, t < 0. Then there exists a unique function ũ : [0, T ] → D(w((−∆)ω)) infinitely differen- tiable such that ũ(k)(0) = 0, k = 0, 1, . . . , ũ(t) ∈ D(w2((−∆)ω)), t ∈ (0, T ), and, for t ∈ [0, T ), ũ′(t)(x) + w2((−∆)ω)ũ(t)(x) = f(t)(x), a.e. x ∈ Ω EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 13 If, additionally, f(t) = 0 for t < ε, with some ε > 0, then the solution ũ satisfies ũ(t) = 0, 0 ≤ t < ε. 3.3. Application to classical fractional parabolic problems. Let us recall that if (a) N ≥ 1, Ω ⊂ RN is an open bounded set with the boundary of class C1,1, or (b) Ω ⊂ R2 is an open bounded convex polygon, then (−∆)ω = −∆ (it follows from the results contained in [10] and [11]) and, consequently, D((−∆)ω) = D(−∆) = H1 0 (Ω) ∩H2(Ω). Lemma 3.5. If γ > 0, one of the conditions (a) or (b) is satisfied, and f belongs to C∞c ((0, T )× Ω,R), then the function f : (−∞, T )→ D((−∆)γ), f(t) = { 0, t ≤ 0, f(t, ·), t ∈ (0, T ) has compact support contained in (0, T ), belongs to Dk(−∞, T ;D((−∆)γ) for any k ∈ N, and, consequently, f has all strong derivatives f js,(−∞,T ) on (−∞, T ) that are equal everywhere on (−∞, T ) to classical ones f j. Proof. First, let us notice that for any t ∈ (0, T ), f(t, ·) ∈ C∞c ⊂ D((−∆)γ). So, f is well defined. It is also clear that supp f ⊂ (0, T ). Now, we shall show that f ∈ D1(−∞, T ;D((−∆)γ). So, let us fix a point t ∈ (0, T ) and let l be a positive integer such that γ ≤ l. We know that ‖u‖˜γ ≤ const · ‖u‖˜l = const · ‖[(−∆)lu‖L2 . Thus, to show that ‖ f(t+ h)− f(t) h − f ′(t)‖2˜γ → h→0 0 for some f ′(t) ∈ D((−∆)γ) it is sufficient to prove that ‖f(t+ h, ·)− f(t, ·) h − ∂f ∂t (t, ·)‖2˜l → h→0 0. (of course, ∂f ∂t (t, ·) ∈ C∞c ⊂ D((−∆)γ)). Indeed (below, (hm) is a sequence con- verging to 0), ‖f(t+ hm, ·)− f(t, ·) hm − ∂f ∂t (t, ·)‖2˜l = ∫ Ω ∣∣∣ (−∆)lf(t+ hm, x)− (−∆)lf(t, x) hm − (−∆)l ∂f ∂t (t, x) ∣∣∣2dx = ∫ Ω ∣∣∣ (−∆)lf(t+ hm, x)− (−∆)lf(t, x) hm − ∂ ∂t (−∆)lf(t, x) ∣∣∣2dx Since f ∈ C∞c ((0, T )×Ω,R), the above sequence of integrands converges pointwise to zero function. Moreover, using the fact that the function (−∆)lf is Lipschitzian on (0, T )× Ω (with a constant L) we obtain∣∣∣ (−∆)lf(t+ hm, x)− (−∆)lf(t, x) hm − ∂ ∂t (−∆)lf(t, x) ∣∣∣ ≤ L+ C 14 D. IDCZAK EJDE-2022/56 where C is such that |∂f∂t (−∆)l(t, x)| ≤ C for (t, x) ∈ (0, T ) × Ω. So, from the Lebesgue dominated convergence theorem it follows that∫ Ω ∣∣∣ (−∆)lf(t0 + hm, x)− (−∆)lf(t0, x) hm − ∂ ∂t (−∆)lf(t0, x) ∣∣∣2dx → m→∞ 0. Thus, there exists the derivative f ′(t) ∈ D([(−∆)ω]γ) at any point t ∈ (0, T ) and f ′(t)(x) = ∂f ∂t (t, x) for all x ∈ Ω. Let us also observe that if tj → t > 0 then f ′(tj)→ f ′(t) in D((−∆)γ). Indeed, ‖f ′(tj)− f ′(t)‖˜γ ≤ const · ‖f ′(tj)− f ′(t)‖˜l = const · ∫ Ω |(−∆)l ∂f ∂t (tj , x)− (−∆)l ∂f ∂t (t, x)|2dx → j→∞ 0. The above convergence follows from the above mentioned Lebesgue theorem. Thus, we have just proved that f is continuously differentiable on (0, T ). The relations supp f ⊂ (0, T ), supp f ′ ⊂ (0, T ), and the continuity of f , f ′ imply that f ∈D1(−∞, T ;D((−∆)γ)). Consequently, f has one (first) strong derivative f ′s,(−∞,T ) on (−∞, T ) which is equal to the classical derivative f ′. Now, since ∂f ∂t ∈ C ∞ c ((0, T )×Ω,R), we can repeat our reasoning for the function f ′ : (−∞, T )→ D((−∆)γ), f ′(t) = { 0, t ≤ 0, ∂f ∂t (t, ·), t ∈ (0, T ), to state that f ′∈D1(−∞, T ;D((−∆)γ)). So, f ∈D2(−∞, T ;D((−∆)γ)) and the classical derivatives f ′, f ′′ : (−∞, T )→ D((−∆)γ), f ′′(t) = { 0, t ≤ 0, ∂2f ∂t2 (t, ·), t ∈ (0, T ) are the strong ones. Repeating our reasoning infinitely many times we complete the proof. � It is known (see [1]) that if N = 1, 2, 3 and Ω ⊂ RN is an open set which has the cone property, then H2(Ω) ⊂ CB(Ω) continuously, where CB(Ω) is the space of continuous and bounded functions on Ω, with norm ‖u‖CB(Ω) = sup x∈Ω |u(x)| of uniform convergence on Ω. So, we assume that one of the conditions (A1) N = 1, 2, 3, Ω ⊂ RN is an open bounded set with the boundary of class C1,1 and with the cone property (A2) Ω ⊂ R2 is an open bounded convex polygon is satisfied. Clearly, in the case (A2), Ω has the cone property. Theorem 3.6. If βk ≥ 1, one of the conditions (A1), (A2) is satisfied and f ∈ C∞c ((0, T )×Ω,R), then there exist ε > 0 and a function ũ : [0, T ]×Ω→ R satisfying ∂ũ ∂t (t, x) + k∑ i,j=0 αiαj(−∆)βi+βj ũ(t, x) = f(t, x), a.e. x ∈ Ω (3.17) EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 15 for t ∈ (0, T ) and such that ũ(t, x) = 0, t ∈ [0, ε), x ∈ Ω, (3.18) ũ(t, ·) ∈ H1 0 (Ω) ∩H2(Ω) ⊂ CB(Ω), t ∈ (0, T ], (3.19) (−∆)ũ(t, ·) ∈ H1 0 (Ω) ∩H2(Ω) ⊂ CB(Ω), t ∈ (0, T ). (3.20) Proof. Let ũ : [0, T ] → D(w(−∆)) = D((−∆)βk) be the solution to (3.2) given by Corollary 3.4, corresponding to function f given in Lemma 3.5. Define the function ũ : [0, T ]× Ω 3 (t, x) 7→ ũ(t)(x) ∈ R. For a t ∈ [0, T ], we have lim h→0 ‖ ũ(t+ h)− ũ(t) h − ũ′(t)‖D([(−∆)ω ]βk ) = 0. Since the convergence in D([(−∆)ω]βk) implies the uniform convergence on Ω (be- cause under our assumptions D([(−∆)ω]βk) ⊂ D([(−∆)ω]1) = D((−∆)1) = H1 0 (Ω) ∩H2(Ω) ⊂ CB(Ω) (3.21) and these embeddings are continuous. Recall that the norm ‖ · ‖∼1 and the norm induced from H2(Ω) are equivalent in D((−∆)1) (see [8, Theorem VII.1.3.2 - Ba- nach’s theorem on compatible norms]). Therefore, for t ∈ [0, T ], lim h→0 ũ(t+ h, x)− ũ(t, x) h = ũ′(t)(x) uniformly on Ω. In particular, the partial derivative ∂ũ ∂t (t, x) exists for all t ∈ [0, T ], x ∈ Ω and ∂ũ ∂t (t, x) = ũ′(t)(x) for all t ∈ [0, T ], x ∈ Ω. Thus, (3.17) holds for t ∈ (0, T ). Condition (3.18) follows from Theorem 3.3. Condition (3.19) follows from the fact that ũ(t) ∈ D((−∆)βk) ⊂ H1 0 (Ω) ∩H2(Ω) ⊂ CB(Ω) for t ∈ [0, T ]. Condition (3.20) follows from the relation ũ(t) ∈ D((−∆)2βk) ⊂ D((−∆)2) for t ∈ (0, T ) (see (3.1)). � 4. Appendix 4.1. Functions of self-adjoint operators. This subsection contains the results from the theory of self-adjoint operators in real Hilbert space. Results presented in this section can be found, in a more general setting, in [13]. We give them here for the convenience of the reader. In [2, 17], these results are derived in the case of complex Hilbert space but their proofs can be moved without any or with small changes to the case of real Hilbert space (one can also consult the book [12]). Let H be a real Hilbert space with a scalar product 〈·, ·〉 : H × H → R and E : B → Π(H) - a spectral measure where B is the σ-algebra of Borel subsets of R, Π(H) - the set of all projections of H on closed linear subspaces. Let b : R → R be a Borel measurable function, defined E - a.e. By ∫∞ −∞b(λ)E(dλ) we denote the operator ∫ ∞ −∞ b(λ)E(dλ) : D ⊂ H → H 16 D. IDCZAK EJDE-2022/56 given by (∫ ∞ −∞ b(λ)E(dλ) ) x = ∫ ∞ −∞ b(λ)E(dλ)x, where D is the set of points x ∈ H such that∫ ∞ −∞ |b(λ)|2‖E(dλ)x‖2 <∞ . (4.1) The above integral is taken with respect to the nonnegative measure B 3P 7→ ‖E(P )x‖2 ∈ R+ 0 ; the integral ∫∞ −∞ b(λ)E(dλ)x is taken with respect to the vector measure B 3P 7→ E(P )x ∈ H. One proves that the operator ∫∞ −∞ b(λ)E(dλ) is self-adjoint. Remark 4.1. To integrate a Borel measurable function b : B → R where B is a Borel set containing the support of the measure E, it is sufficient to extend b on R to a whichever Borel measurable function (putting, for example, b(λ) = 0 for λ /∈ B). If b : R→ R is Borel measurable and σ ∈ B, then by the integral∫ σ b(λ)E(dλ) we mean the integral ∫ ∞ −∞ χσ(λ)b(λ)E(dλ) where χσ is the characteristic function of the set σ. The integral ∫ σ b(λ)E(dλ) can be also defined with the aid of the restriction of E to the set σ. The next theorem plays a fundamental role in the spectral theory of self-adjoint operators (below, σ(A) denotes the spectrum of an operator A : D(A) ⊂ H → H). Theorem 4.2. If A : D(A) ⊂ H → H is self-adjoint and the resolvent set ρ(A) is non-empty, then there exists a unique spectral measure E with the closed support Λ = σ(A), such that A = ∫ ∞ −∞ λE(dλ) = ∫ σ(A) λE(dλ). The basic notion in the Stone-von Neumann operator calculus is a function of a self-adjoint operator. Namely, if A : D(A) ⊂ H → H is self-adjoint and E is the spectral measure determined according to the above theorem, then, for any Borel measurable function b : R→ R, one defines the operator b(A) by b(A) = ∫ ∞ −∞ b(λ)E(dλ) = ∫ σ(A) b(λ)E(dλ). Proposition 4.3. If E is the spectral measure for a self-adjoint operator A : D(A) ⊂ H → H with non-empty resolvent set, then αkA k + · · ·+ α1A+ α0I = ∫ ∞ −∞ (αkλ k + · · ·+ α1λ 1 + α0)E(dλ) and, for any Borel measurable function b : R→ R, (b(A))n = bn(A) (4.2) with any fixed positive integer n ≥ 2. EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 17 Now, let β > 0 and σ(A) ⊂ [0,∞). According to the Remark 4.1 by Aβ we mean the operator Aβ = ∫ ∞ −∞ b(λ)E(dλ) where b : R 3 λ→ { λβ , λ ≥ 0, 0, λ < 0. We have the following generalization of Proposition 4.3. Proposition 4.4. If E is the spectral measure for a self-adjoint operator A : D(A) ⊂ H → H with σ(A) ⊂ [0,∞), then αkA βk + · · ·+ α1A β1 + α0A β0 = ∫ ∞ −∞ w(λ)E(dλ). where w : R 3 λ→ { αkλ βk + · · ·+ α1λ β1 + α0λ β0 , λ ≥ 0, 0, λ < 0, (4.3) and 0 ≤ β0 < β1 < · · · < βk. Moreover, Aβ2 ◦Aβ1 = Aβ2+β1 (4.4) for β2, β1 > 0. If the numbers β0, β1,. . . ,βk are positive integers (including zero), then one can omit the assumption σ(A) ⊂ [0,∞) and consider the function w(λ) = αkλ βk + · · ·+ α1λ β1 + α0λ β0 , λ ∈ R (assuming that the resolvent set ρ(A) is nonempty). 4.2. Equivalence of weak and strong solutions. The results presented in this section with additional remarks and comments can be found in [13]. Let E be the spectral measure for a self-adjoint operator A : D(A) ⊂ H → H with non-empty resolvent set and b : R → R - a Borel measurable function, defined E - a.e. Fact that the operator b(A) is self-adjoint means that its domain satisfies the equality D(b(A)) = { u ∈ H : there exists z ∈ Hsuch that∫ Ω u(t)b(A)v(t)dt = ∫ Ω z(t)v(t)dt for any v ∈ D(b(A)) } (4.5) and b(A)u = z, u ∈ D(b(A)) . (4.6) From Proposition 4.3 it follows that b(A)(b(A)u) = b2(A)u. (4.7) In particular, u ∈ D(b2(A)) if and only if u ∈ D(b(A)) and b(A)u ∈ D(b(A)). Using this fact and (4.5), (4.6), we obtain the following result. Theorem 4.5. If g ∈ L2, then u ∈ D(b2(A)) and b2(A)u = g (4.8) if and only if u ∈ D(b(A)) and∫ Ω b(A)u(t)b(A)v(t)dt = ∫ Ω g(t)v(t)dt (4.9) 18 D. IDCZAK EJDE-2022/56 for any v ∈ D(b(A)). Consequently, if A : D(A) ⊂ H → H is self-adjoint with σ(A) ⊂ [0,∞), w is given by (4.3), then we have the following corollary, where clearly, w2(A) =∑k i,j=0 αiαjA βi+βj . Corollary 4.6. If g ∈ L2, then u ∈ D(w2(A)) and w2(A)u = g (4.10) if and only if u ∈ D(w(A)) and∫ Ω w(A)u(t)w(A)v(t)dt = ∫ Ω g(t)v(t)dt (4.11) for any v ∈ D(w(A)). Remark 4.7. The above theorem states that u is the strong solution to problem (4.10) if and only if it is the weak one in the sense of (4.11). 4.3. Weak Dirichlet-Laplace operator and its fractional power. Details about the results presented in this section can be found in [13]. Let Ω ⊂ RN be an open bounded set, H1 0 = H1 0 (Ω,R) - the closure of C∞c = C∞c (Ω,R) in H1 = H1(Ω,R), where H1(Ω,R) = W 1,2(Ω,R) is the classical Sobolev space, C∞c (Ω,R) is the space of test functions. We say ([3]) that u : Ω → R has a weak (minus) Dirichlet-Laplacian if u ∈ H1 0 and there exists a function g ∈ L2 = L2(Ω,R) such that ∫ Ω ∇u(x)∇v(x)dx = ∫ Ω g(x)v(x)dx for any v ∈ H1 0 . The function g is called the weak Dirichlet-Laplacian (or shortly, Laplacian) of u and denoted by (−∆)ωu. In [3], g is named Dirichlet-Laplacian and denoted by (−∆)u. The operator (−∆)ω : D((−∆)ω) ⊂ L2 → L2 where D((−∆)ω) = {u : Ω→ R; u has the weak Dirichlet-Laplacian} is bijective and self-adjoint. We call it the weak Dirichlet-Laplace operator. Remark 4.8. In fact, (−∆)ω is the Fridrich’s extension of the classical Dirichlet- Laplace operator (−∆)class : C∞c ⊂ L2 → L2. Moreover, (−∆)class ⊂ (−∆) ⊂ (−∆)ω where −∆ : H1 0 ∩H2 ⊂ L2 → L2 is the Dirichlet-Laplace operator defined by generalized partial derivatives of second order of Sobolev type. The spectrum σ((−∆)ω) of (−∆)ω contains only the eigenvalues. More precisely, σ((−∆)ω) is a sequence 0 < λ1 ≤ λ2 ≤ · · · ≤ λj →∞ (each λj is repeated kj times where kj is the multiplicity of λj). Moreover, to each eigenvalue λj corresponds an eigenfunction ej and the system {ej} is the Hilbertian basis in L2 (see [3, Theorem 8.3.2]; in [3, Proposition 8.5.3], the spectrum and eigenfunctions of (−∆)ω are given EJDE-2022/56 FRACTIONAL DIRICHLET-LAPLACE PROBLEMS 19 in the case of Ω = (0, π)N (cube in RN )). Thus, for any u ∈ L2, there exist real numbers aj , j ∈ N, such that u(x) = ∑ ajej(x) in L2 and ‖u‖2L2 = ∑ |aj |2. Now, let us fix a number β > 0 and consider the operator [(−∆)ω]β : D([(−∆)ω]β) ⊂ L2 → L2 given by ([(−∆)ω]βu)(x) = ∑ λβj ajej(x) where D([(−∆)ω]β) = {u = u(x) = ∑ ajej(x) ∈ L2 : ∑ ((λj) β)2a2 j <∞}, and the convergences of the functional series are meant in L2. In fact, [(−∆)ω]β is the power of the operator (−∆)ω in the sense of Stone-von Neumann operator calculus. So, [(−∆)ω]β is self-adjoint, the spectrum σ([(−∆)ω]β) consists of eigen- values λβj , j ∈ N, and eigenspaces corresponding to λβj -s are the same as eigenspaces for (−∆)ω, corresponding to λj-s. (These properties follow from the general results concerning the power of the self-adjoint operator in the sense of Stone-von Neumann operator calculus.) D([(−∆)ω]β) with the scalar product (u, v)β = (u, v)L2 + ( [(−∆)ω]βu, [(−∆)ω]βv ) L2 is a Hilbert space; the corresponding norm ‖u‖β = (‖u‖2L2 + ‖[(−∆)ω]βu‖2L2)1/2. Moreover, the scalar product (u, v)∼β = ( [(−∆)ω]βu, [(−∆)ω]βv ) L2 determines the equivalent norm ‖u‖∼β = ‖[(−∆)ω]βu‖L2 . It is easy to see that if 0 < β1 < β2, then D([(−∆)ω]β2) ⊂ D([(−∆)ω]β1). (4.12) continuously and C∞c ⊂ D([(−∆)ω]β) for any β > 0. The continuity of the embedding (4.12) follows from the estimations ∞∑ j=1 ((λj) β1)2a2 j = ((λ1)β1)2 ∞∑ j=1 ((λj) β1)2 ((λ1)β1)2 a2 j ≤ ((λ1)β1)2 ∞∑ j=1 ((λj) β2)2 ((λ1)β2)2 a2 j = ((λ1)β1)2 ((λ1)β2)2 ∞∑ j=1 ((λj) β2)2a2 j . Acknowledgement. I want to thank professors Bogdan Przeradzki and Robert Stegliński for their valuable guidance on the continuity of the embedding (4.12). 20 D. IDCZAK EJDE-2022/56 References [1] R. A. Adams; Sobolev Spaces, Academic Press, New York, 1975. [2] A. Alexiewicz; Functional Analysis, PWN, Warsaw, 1969 (in Polish). [3] H. Attouch, G. Buttazzo, G. Michaille; Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, SIAM-MPS, Philadelphia, 2006. [4] K. Bogdan, K. Burdzy, Z. 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[12] B. Helffer; Spectral Theory and its applications, Cambridge, United Kingdom, 2013. [13] D. Idczak; A bipolynomial fractional Dirichlet-Laplace problem, Electronic Journal of Dif- ferential Equations, 2019 (2019), No. 59, pp. 1-17. [14] D. Idczak; Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator, Electronic Journal of Differential Equations, 2021 (2021), No. 64, pp. 1-19. [15] M. Kwaśnicki; Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), No. 1, pp. 7–51. [16] J.-L. Lions, E. Magenes; Problemes aux Limites Non Homogenes et Applications, Dunod, Paris, 1968. [17] W. Mlak; An Introduction to the Hilbert Space Theory, PWN, Warsaw, 1970 (in Polish). Dariusz Idczak Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland Email address: dariusz.idczak@wmii.uni.lodz.pl 1. Introduction 2. Preliminaries 2.1. Derivatives of abstract functions 3. Fractional parabolic problem 3.1. Friedmann theorem 3.2. Abstract fractional parabolic problem 3.3. Application to classical fractional parabolic problems 4. Appendix 4.1. Functions of self-adjoint operators 4.2. Equivalence of weak and strong solutions 4.3. Weak Dirichlet-Laplace operator and its fractional power Acknowledgement References