Chobanov, Georgi

Dimovski, Ivan

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c aNewSeries Vol. 26, 2012, Fasc. 1-2Exact Solutions of Nonlocal Pluriparabolic ProblemsGeorgi Chobanov 1, Ivan Dimovski 2Presented at 6th International Conference TMSF' 2011A generalization of the classical Duhamel principle for the pluriparabolic equation∂u∂t1+ · · ·+ ∂u∂tn=∂2u∂x2+ F (x, t1, . . . , tn) in 0 ≤ x ≤ a, 0 ≤ tk ≤ Tkwith time-nonlocal initial value conditions of the formχk,τ{u(x, t1, . . . , tk−1, τ, tk+1, . . . , tn)} = fk(x, t1, . . . , tk−1, tk+1, . . . , tn)with linear functionals χk on C[0, Tk] (k = 1, . . . , n), a space-local boundary value conditionof the formu(0, t1, . . . , tn) = ψ(t1, . . . , tn)and a space-nonlocal boundary value condition of the formΦξ{u(ξ, t1, . . . , tn)} = φ(t1, . . . , tn)with a linear functional Φ on C1[0, a] is proposed. To this end two non-classical convolutionsφt1...tn∗ ψ and F xt1...tn∗ G are used: the ﬁrst one for functions of t1, . . . , tn only and the second for functions of x, t1, . . . , tn. The corresponding Duhamel representation takes the followingform: If Ω(x, t1, . . . , tn) is a solution for the boundary value problem for the special choiceF ≡ 0, fk ≡ 0, ψ ≡ 0 and φ ≡ 1, then for ψ ≡ 0, fk ≡ 0, k = 1, . . . , n (under some additionalassumptions for smoothness of the boundary function φ and the function F )u(x, t1, . . . , tn) =∂n∂t1 . . . ∂tn(Ωt1...tn∗ φ) + ∂n∂t1 . . . ∂tn(Ωxt1...tn∗ F ).MSC 2010: 44A35, 44A40Key Words: operational calculus, nonlocal boundary conditions, pluriparabolic equa-tions54 G. Chobanov, I. Dimovski1. IntroductionIn the present paper it is proposed a generalization of the classical Duhamelprinciple for the pluriparabolic equation∂u∂t1+ · · ·+ ∂u∂tn=∂2u∂x2+ F (x, t1, . . . , tn) in 0 ≤ x ≤ a, 0 ≤ tk ≤ Tk (1)with local and nonlocal boundary value conditions (BVCs) of the form: nonlocalinitial conditionsχk,τ{u(x, t1, . . . , tk−1, τ, tk+1, . . . , tn)} = fk(x, t1, . . . , tk−1, tk+1, . . . , tn) (2)(k = 1, . . . , n), and local and nonlocal boundary conditionsu(0, t1, . . . , tn) = ψ(t1, . . . , tn), Φξ{u(ξ, t1, . . . , tn)} = φ(t1, . . . , tn). (3)Here χk, k = 1, . . . , n are linear functionals on C[0, Tk] and Φ is a linear func-tional on C1[0, a].Such problems for a pluriparabolic equation with energy functional of theformΦξ{u(ξ, t1, . . . , tn)} =∫ a0u(ξ, t1, . . . , tn)dξ (4)are considered by J.R. Cannon [3], A. Bouziani [2], S. Mesloub [9].2. ConvolutionsOur basic tool for obtaining explicit solutions of the problem consideredare some multidimensional non-classical convolutions. Their construction beginswith the simplest one-dimensional case (Dimovski [5]).Consider the elementary one-dimensional BVP in C[0, Tk]y′ − µy = f(t), χk{y} = 0. (5)For the sake of some technical simpliﬁcations we assume that the constant func-tion {1} does not belong to the kernel of the functional χk, k = 1, . . . , n, i.e.χk{1} 6= 0. Then without any loss of generality we can assume χk{1} = 1. Itssolution isy = rk(f, µ)(tk) =∫ tk0eµ(tk−σ)f(σ)dσ−χk{∫ σ0eµ(τ−σ)f(ξ)dξdσ}eµtkGk(µ), (6)Exact Solutions of a Non-local Pluriparabolic Problems 55where Gk(µ) = χk,τ{eµτ} is the exponential indicatrix of the functional χk. Ourassumption χk{1} 6= 0 is equivalent to Gk(0) 6= 0, i.e. µ = 0 is not an eigenvalueof BVP (5). Then instead of (6) we may consider the special caserk(f, 0) = lkf =∫ tk0f(σ)dσ − χk(∫ σ0f(ξ)dξdσ)(7)which deﬁnes a right inverse operator lk ofddtkon the space C[0, Tk] satisfyingthe following identitylkf′(tk) = f(tk)− χk,τf(τ). (8)Theorem 1. (Dimovski [5]) The operation(ftk∗ g)(tk) = χk,τ(∫ tkτf(tk + τ − σ)g(σ)dσ), (9)where the subscript τ means that χk acts on the variable τ only, is a commutativeand associative in C[0, Tk] such thatlkf(tk) = {1}tk∗ f(tk) (10)andrk(f, µ)(tk) ={eµtkGk(µ)}tk∗ f(tk). (11)Next we need an one-dimensional convolution, connected with d2dx2inC1[0, a]. Consider the elementary BVPy′′ + λ2y = f(x), y(0) = 0, Φ{y} = 0 (12)with a non-zero linear functional Φ on C1[0, a]. In order it to have a solution,it is necessary to assume Φξ{ξ} 6= 0. Again, without essential loss of generality,one can assume Φξ{ξ} = 1. The solution isy = R−λ2f(x) =1λ∫ x0sinλ(x−ξ)f(ξ)dξ−Φξ{1λ∫ x0sinλ(a− ξ)f(ξ)dξ}sinλxλE(λ),(13)where E(λ) = Φξ{sinλξλ}is the sine-indicatrix of the functional Φ. For asimpliﬁcation of the next consideration it is useful to assume that λ = 0 is not56 G. Chobanov, I. Dimovskian eigenvalue of (12). Since E(0) = Φξ{ξ} by the above assumptions we haveE(0) = 1. NowR0f(x) = Lf(x) =∫ x0(x− ξ)f(ξ)dξ − xΦξ{∫ ξ0(ξ − η)f(η)dη}(14)deﬁnes a right inverse operator L of d2dx2on the space C[0, a] satisfying theidentityLf ′′(x) = f(x) + [xΦ(1)− 1]f(0)− xΦξ[f(ξ)]. (15)Theorem 2. (Dimovski [5]) The operation(fx∗ g)(x) =− 12Φ˜ξ[∫ ξxf(ξ + x− η)g(η)dη −∫ ξ−xf(|ξ − x− η|)g(|η|)sgn (ξ − x− η)ηdη],(16)where Φ˜ξ = Φξ ◦∫ ξ0 , is commutative and associative in C[0, a] such thatLf(x) = {x} x∗ f(x) (17)andR−λ2f(x) ={sinλxλE(λ)}x∗ f(x). (18)Next the following multidimensional generalizations of the Duhamel con-volution are given.Theorem 3. The operation(φt1...tn∗ ψ)(t1, . . . , tn) =χn,τn . . . χ1,τ1[∫ tnτn. . .∫ t1τ1φ(t1 + τ1 − σ1, . . . , tn + τn − σn)ψ(σ1, . . . , σn)dσ1 . . . dσn](19)for φ, ψ ∈ C([0, T1]× · · · × [0, Tn]) is bilinear, commutative and associative andl1 . . . lnφ = {1} t1...tn∗ φ. (20)Using deﬁnition (19) of the operation φ t1...tn∗ ψ on C([0, T1] × · · · × [0, Tn]), weExact Solutions of a Non-local Pluriparabolic Problems 57deﬁne a (n+ 1)-dimensional convolution in C([0, a]× [0, T1]× · · · × [0, Tn]).Deﬁnition 1. For F,G ∈ C([0, a]× [0, T1]× · · · × [0, Tn]), let(Fxt1...tn∗ G)(x, t1, . . . , tn)= −12Φ˜ξ[∫ ξxF (ξ + x− η, t1, . . . , tn) t1...tn∗ G(η, t1, . . . , tn)dη−∫ ξ−xF (|ξ − x− η|, t1, . . . , tn) t1...tn∗ G(|η|, t1, . . . , tn)sgn (ξ − x− η)ηdη].(21)Theorem 4. The operation deﬁned by (21) is bilinear, commutativeand associative in C([0, a]× [0, T1]× · · · × [0, Tn]) such thatLl1 . . . lnf = {x} xt1...tn∗ f. (22)S k e t c h o f t h e p r o o f. The proof of both Theorems 1 and 2 goesalong the same line. First, we verify the assertions for productsφ(t1, . . . , tn) = φ1(t1) . . . φn(tn) ψ(t1, . . . , tn) = ψ1(t1) . . . ψn(tn),orF (x, t1, . . . , tn) = f(x)φ1(t1) . . . φn(tn) G(x, t1, . . . , tn) = g(x)ψ1(t1) . . . ψn(tn)and reduce them to the one dimensional assertions. Next we approximate thearbitrary functions φ, ψ and F,G by products, e.g. by polynomials.The following analogues of the identities (8) and (15) hold for functionsu ∈ C([0, a]× [0, T1]× · · · × [0, Tn]):lkutk(x, t1, . . . , tn) = u(x, t1, . . . , tn)− χk,τk [u(x, t1, . . . , τk, . . . , tn)] (23)(k = 1, . . . , n) andLuxx(x, t1, . . . , tn)= u(x, t1, . . . , tn) + [xΦ(1)− 1]u(0, t1, . . . , tn)− xΦξ[u(ξ, t1, . . . , tn)]. (24)58 G. Chobanov, I. Dimovski3. Rings of multipliers of convolution algebrasIn what follows, let C = C([0, a]×[0, T1]×· · ·×[0, Tn]) and let(C, ∗) be therespective convolution algebra. We follow a standard procedure for constructingof an operational calculus for BVP (1)  (3) based on convolution (21) and itsmultipliers as outlined in Dimovski [5].Let us remind the notion of multiplier of the algebra (C, ∗) (Larsen, [7]).An operator M : C → C is said to be a multiplier of the convolution algebra(C, ∗), iﬀ the relationM(f ∗ g) = (Mf) ∗ gholds for arbitrary f, g ∈ C.The multipliers of (C, ∗) form a commutative ring M without annihilatorswith respect to the usual multiplication of operators. Let N be the multiplicativeset of the non-divisors of 0 of the ring M. N evidently is nonempty since at leastthe identity operator and the multiplier convolution operator L = {x}∗ are non-divisors of 0. Another examples are the operators lk.Consider the formal fractions A/B where A ∈M, B ∈ N.Deﬁnition 2. The ring M = N−1M of the multiplier fractions is thequotient of the ring M×N with respect to the equivalence relation(A,B) ∼ (C,D)⇔ AD = BC,i.e. M = M×N/ ∼.Theorem 5. The ring M of the multiplier fractions contains subringsisomorphic to: a) R, b)(C[0, a],x∗), c)(C[0, Tk],tk∗), d) (C, ∗).P r o o f. a) The correspondence α 7−→ αLL, α ∈ R is an embeddingR ↪→M;b) The correspondence f 7−→ (Lf)x∗Lis an embedding(C[0, a],x∗)↪→M;c) The correspondence ϕ 7−→ (lkϕ)tk∗lkis an embedding(C[0, Tk],tk∗)↪→M;d) The correspondence u 7−→ {u}∗Iwhere I is the identity operator of Cis an embedding (C, ∗) ↪→M.The veriﬁcation is immediate. Let us prove for example b). Let f, g ∈C[0, a]. We are to prove thatfx∗ g 7−→ (Lf)x∗L· (Lg)x∗LExact Solutions of a Non-local Pluriparabolic Problems 59Indeed,fx∗ g 7−→ {L(fx∗ g)}∗L=L[{L(f x∗ g)}∗]L2={(Lf)x∗ (Lg)}x∗L2=[(Lf)x∗L][(Lg)x∗L].Here we make use of the convolution propertyL(fx∗ g) = (Lf) x∗ g = f x∗ (Lg).For every φ ∈ C([0, T1]×· · ·× [0, Tn]) the partial convolution (19) deﬁnesa multiplier acting on F ∈ C as followsφt1...tn∗ F. (25)The corresponding equivalence class in M is called constant with respect to xand is denoted by[φ]x (26)Similarly, let f ∈ C([0, a]× [0, T1]×· · ·× [0, Tk−1]× [0, Tk+1]×· · ·× [0, Tn]). Thepartial convolution operatorfxt1...tk−1tk+1...tn∗ (27)deﬁnes a multiplier in an obvious manner. Its class is called constant with respectto tk and is denoted by[f ]tk . (28)4. Algebraization of the BVP (1)(3)Crucial for the algebraization of the problem are the reciprocal elementsto L and l1, . . . , ln in M. Let they be denoted by S, s1, . . . , sn, respectively.NowS{x} = SL = 1, sklk = 1 (k = 1, . . . , n) (29)where 1 denotes the unit of the algebra M. For a function u = u(x, t1, . . . , tn)this together with (23) and (24) givesutk = sku− [χk,τ{u(x, t1, . . . , τ . . . , tn)}]tk (k = 1, . . . , n). (30)60 G. Chobanov, I. Dimovskiand the last term is a constant with respect to the variable tk. Similarly,uxx = Su+ (xΦ(1)− 1)u(0, t1, . . . , tn) + [φ(t1, . . . , tn)]x (31)and the last term is a constant with respect to the variable x. Suppose u =u(x, t1, . . . , tn) is a solution to the boundary value problem (1)(3). Then (30)and (31) reduce the BVP (1)(3) to a simple linear algebraic equation for thefunction u:(s1 + · · ·+ sn − S)u = (xΦ(1)− 1)ψ(t1, . . . , tn) + [φ(t1, . . . , tn)]x+n∑k=1[fk(x, t1, . . . , tk−1, tk+1, . . . , tn)]tk + {F (x, t1, . . . , tn)}. (32)Deﬁnition 3. A function u ∈ C([0, a]× [0, T1]× · · · × [0, Tn]) is a weaksolution to BVP (1)(3) if it satisﬁes (32).In order to reveal the basic ideas, we restrict BVP (1)(3) to the caseχk,τ{u(x, t1, . . . , tk−1, τ, tk+1, . . . , tn)} = 0, k = 1, . . . , n (33)andu(0, t1, . . . , tn) = 0, Φξ{u(ξ, t1, . . . , tn)} = φ(t1, . . . , tn). (34)Suppose s1 + · · · + sn − S = Σ is a non-divisor of zero. Then 1Σ is welldeﬁned. If u is a weak solution of BVP (1),(33) and (34) then formally we obtainu =1Σ[φ(t1, . . . , tn)]x +1Σ{F (x, t1, . . . , tn)}. (35)In order to interpret (35) as a function, we need some algebraic manipulations:u=(s1 . . . sn)1(s1 . . . sn)Σ[φ(t1, . . . , tn)]x+(s1 . . . sn)1(s1 . . . sn)Σ{F (x, t1, . . . , tn)}.(36)Assuming that 1(s1 . . . sn)Σcan be interpreted as a continuous functionΩ(x, t1, . . . , tn) then it can be considered as a weak solution of the homogeneousproblem with φ ≡ 1. Indeed, the product l1 . . . ln can be interpreted as thenumerical multiplier [1]x, i.e.l1 . . . ln = {1} t1...tn∗ .HenceExact Solutions of a Non-local Pluriparabolic Problems 61Ω(x, t1, . . . , tn) = (s1 . . . sn)1(s1 . . . sn)Σl1 . . . ln. (37)Now we can formulate the following conditional theorem of existence (ageneralization of Duhamel principle).Theorem 6. If BVP (1),(33) and (34) has a weak solution Ω for F ≡ 0,φ ≡ 1, thenΩ(x, t1, . . . , tn) =1(s1 . . . sn)Σ(38)and the BVP (1),(33) and (34) with arbitrary F and φ also has a weak solutionof the formu(x, t1, . . . , tn) =∂n∂t1 . . . ∂tn(Ωt1...tn∗ φ) + ∂n∂t1 . . . ∂tn(Ωxt1...tn∗ F ), (39)provided F and φ have continuous partial derivatives∂nF∂t1 . . . ∂tnand∂nφ∂t1 . . . ∂tn.The proof requires some diﬀerentiation properties of the convolutions in-volved, but here we will not enter into details.5. Uniqueness of the solution for BVP (1)(3).Theorem 6 is a conditional theorem of existence of solution of BVP (1)(3). As for the uniqueness problem we can state a more deﬁnite assertion. To thisend, we study the uniqueness for BVP (1)(3) by means of the spectral propertiesof the one-dimensional problems that compose it, taking advantage from thefact that these problems are better studied. The eigenvalues µ(k)m (k = 1, . . . , n;m = 1, . . . ,∞) for (5) are the zeros of the indicatrices Gk(µ) = χk,τ {eµτ}. Theprojections on the respective eigenspaces arepk,µ(k)m(φ) = − 12pii∫Γµ(k)mrk(φ, µ)dµ = − 12ipi∫Γµ(k)meµtkdµGk(µ) tk∗ φ, (40)where Γµ(k)mis a small contour around the eigenvalue µ(k)m (see [6]).We will prove a theorem for uniqueness of the solution of BVP (1)(3)under some additional restrictions on the time-functionals χk (k = 1, . . . , n).Deﬁnition 4. A linear functional χk on C[0, Tk] is called stronglynonlocal if its support includes the endpoints of the interval [0, Tk], i.e. 0, Tk ∈suppχk. The corresponding BVPs are called strongly nonlocal.62 G. Chobanov, I. DimovskiFurther we consider only strongly nonlocal BVPs with respect to the timevariables. In the case of simple eigenvalues µ(k)m the respective eigenspaces areone-dimensional and spanned on the functions eµ(k)m tk and, moreover,pk,µ(k)m(φ) = φtk∗{− eµ(k)m tkG′k(µ(k)m )}. (41)Similarly the eigenvalues λl (l = 1, . . . ,∞) for (12) are the zeros of E(λ) =Φ{sinλξλ}.In order to state the uniqueness result, we need a lemma:Lemma 1. The following equalities holdS{sinλlx} = −λ2l sinλlx and sk{eµ(k)m tk} = µ(k)m eµ(k)m tk , (42)k = 1, . . . , n; l,m = 1, . . . ,∞.P r o o f. It is enough to apply (8) and (15).It is easily seen thatΣ{sinλlxeµ(1)m1t1 . . . eµ(n)mn tn}= (µ(1)m1+· · ·+µ(n)mn+(λl)2){sinλlxeµ(1)m1t1 . . . eµ(n)mn tn}(43)and if µ(1)m1 + · · ·+ µ(n)mn + (λl)2 = 0, then Σ is a divisor of zero.If there is no dispersion relation of this form, i.e ifµ(1)m1 + · · ·+ µ(n)mn + (λl)2 6= 0 (44)for each combination of eigenvalues µ(1)m1 , . . .µ(n)mn , λl, then Σ is a non-divisor of0.Lemma 2. (Multidimensional Schwartz-Leontiev theorem) If φ ∈C([0, T1]× . . . [0, Tk]) andn∏k=1pk,µ(k)m(φ) = 0 (45)for all combinations of eigenvalues, then φ ≡ 0.P r o o f. The proof of Lemma 2 follows from the one-dimensional Schwartz-Leontiev theorem (see [1], p. 198, [6], pp. 92-93, [10] and [8], pp. 260-261).P r o o f o f P r o p o s i t i o n 1. Suppose that Σ is a divisor of 0, i.e. thatfor some u[s1 + · · ·+ sn − S]u(x, t1, . . . , tn) = 0.Letum1...mn(x, t1, . . . , tn) = u(x, t1, . . . , tn)t1...tn∗ {eµ(1)m1 t1 . . . eµ(n)mn tn}.Exact Solutions of a Non-local Pluriparabolic Problems 63Now, from (41) it followsum1...mn(x, t1, . . . , tn) = U(x)eµ(1)m1t1 . . . eµ(n)mn tnand the function U(x) must satisfy(µ(1)m1 + · · ·+ µ(n)mn − S)U = 0Now (44) and [5], Theorem 1.4.1 guarantee the existence of (µ(1)m1 + · · ·+ µ(n)mn −S)−1 and hence U = 0. Equivalently the function U must satisfyd2Udx2− (µ(1)m1 + · · ·+ µ(n)mn)U = 0and U(0) = 0, Φ(U) = 0 and since (44) implies that µ(1)m1 + · · · + µ(n)mn is not aneigenvalue of (12), then U ≡ 0.Remark. The assumption that the eigenvalues are simple is a technicalone and is used only to simplify the above argument. The proof for multipleeigenvalues µ(k)mk is somewhat more involved, but the conclusion is the same.Now we can stateTheorem 7. The conditions that (44) holds for all possible combina-tions of eigenvalues of the one-dimensional eigenvalue problems are suﬃcient foruniqueness of the solution of BVP (1)(3), provided all the time nonlocal BVCsare strong.Example. Consider the BVP∂u∂t1+∂u∂t2=∂2u∂x2+ F (x, t1, t2) in 0 ≤ x ≤ 1, 0 ≤ t1, t2 ≤ T (46)with nonlocal initial condition of the form1µ− 1 [µu(x, t1, 0)− u(x, t1, T )] = 01µ− 1 [µu(x, 0, t2)− u(x, T, t2)] = 0 (47)(Dezin nonlocal conditions, see [4]) and local and nonlocal energy boundaryconditionsu(0, t1, t2) = 0 and∫ 10u(x, t1, t2)dx = φ(t1, t2). (48)The spectral properties of the last problem (Samarskiy-Ionkin spectral problem)are studied from operational calculus point of view in [5], Theorem 3.4.4. Theeigenvalues are 2npi with multiplicity two and the corresponding eigenspaces arespanned on the functions sin 2npix and x cos 2npix.Since the eigenvalues of the corresponding one dimensional problems withDezin condition are of the form µ(k)m =1T(ln |µ|+2m(k)pii), k = 1, 2, m(k) ∈ Z wecan have dispersion relation only if the imaginary part (of the sum corresponding64 G. Chobanov, I. Dimovskito (44)) is zero i.e. if m(1) = m(2) = 0 or m(1) = −m(2). In both cases thecondition 2Tlnµ 6= (2pin)2 for all n ∈ N guarantees that the real part is nonzeroand hence that there is no dispersion relation of the form µ(1)m1+µ(2)m2+(2npi)2 = 0.Acknowledgements. The authors are supported by Grant No D ID02/25/2009 "ITMSFA" of the National Science Fund  Ministry of Education,Youth and Science of Bulgaria and by Grant No 174, of NSF of Bulgaria.References[1] N.S. Bozhinov, Convolutional Representations of Commutants and Multi-pliers, Pub. House of BAS, Soﬁa (1988).[2] A. Bouziani, On the solvability of nonlocal pluriparabolic problems, Elec-tronic Journal of Diﬀerential Equations, Vol. 2001, No 21 (2001), 116;http://ejde.math.swt.edu or http://ejde.math.unt.edu.[3] J.R. Cannon, The solution of heat equation subject to speciﬁcation of en-ergy, Quart. Appl. Math., 21 (1963), 155-160.[4] G. Chobanov, I. Dimovski, Extension of the Duhamel principle for the heatequation with Dezin's initial condition, Annuaire de l'Univ. de Soﬁa "St.Kliment Ohridski", Fac. de Mathematiques et Informatique, Livre 1 - Math-ematiques et Mecanique, 93 (1999), 73-92.[5] I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht - N.Y. - London(1990).[6] I.H. Dimovski, R.I. Petrova, Finite integral transforms for non-local bound-ary value problems. In: Generalized Functions and Convergence (Eds.: P.Antosik, A. Kaminski), World Scientiﬁc, Singapore (1990), 89-103.[7] R. Larsen, An Introduction to the Theory of Multipliers, Berlin-Heidelberg-New York (1972).[8] A.F. Leontiev, Series of Exponents. Moscow, Nauka (1976) (Russian).[9] S. Mesloub, On a nonlocal problem for a pluriparabolic equation Proc. ofthe Institute of Mathematics of NAS of Bielorussia, 6 (2000), 116.[10] L. Schwartz, Approximation d'une fonction quelconque par des sommesd'exponentielles imaginaires, Ann. Fac. Sci de l'Univ. de Toulouse, 6 (1943),110-175.Institute of Mathematics and Informatics, Bulgarian Academy of Sciences"Acad. G. Bontchev" Str., Block 8Soﬁa  1113, BULGARIA Received: October 21, 2011e-mails: 1 chobanov@math.bas.bg , 2 dimovski@math.bas.bg

Exact Solutions of Nonlocal Pluriparabolic Problems

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Exact Solutions of Nonlocal Pluriparabolic Problems

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