Boundary Value Problems Governed by Superdiffusion in the Right Angle : Existence and Regularity

Fractional partial differential equations (FPDE) play a key role in the description of the so-called anomalous phenomena in nature and in the theory of complex systems (see, e.g., [1]). In particular, these equations provide a more faithful representation of the long-memory andnonlocal dependence ofmany anomalous processes.The signature of an anomalous diffusion species ⟨(Δx)2⟩ scales as a nonlinear power law in time; i.e., ⟨(Δx)2⟩ ∼ tα, α > 0. When α > 1, this is referred to as superdiffusion. Superdiffusion is used in modelling turbulent flow [2, 3], chaotic dynamics of classical conservative systems [4], model solute transport in underground aquifers [5–7], and rivers [8–10], biophysics [11], and physical and chemical models described by the Lévy processes [12, 13]. In the present paper, we focus on the boundary value problems to the stationary superdiffusion equation. Let Ω = {(x1, x2) : x1 ∈ (0, +∞), x2 ∈ (0, +∞)} be the first quarter with a boundary ∂Ω = Γ1 ∪ Γ2, Γ1 = {(x1, x2) : x2 = 0, x1 ≥ 0}, Γ2 = {(x1, x2) : x1 = 0, x2 ≥ 0}. For a fixed α ∈ (1, 2), we consider the linear equation in the unknown function u = u(x1, x2) : Ω 󳨀→ R, Dαx1u +Dαx2u = f (x1, x2) , (1) subject either to the Dirichlet boundary condition (DBC),


Introduction
Fractional partial differential equations (FPDE) play a key role in the description of the so-called anomalous phenomena in nature and in the theory of complex systems (see, e.g., [1]).In particular, these equations provide a more faithful representation of the long-memory and nonlocal dependence of many anomalous processes.The signature of an anomalous diffusion species ⟨(Δx) 2 ⟩ scales as a nonlinear power law in time; i.e., ⟨(Δx) 2 ⟩ ∼   ,  > 0. When  > 1, this is referred to as superdiffusion.
Here, the symbols D   1 and D   2 stand for the Caputo fractional derivative of order  with respect to  1 and  2 .
The goal of the present paper is the proof of the wellposedness and the regularity of solutions to boundary value problems (1)- (3) in weighted Hölder classes.It is worth noting that these classes allow one to control the behavior of the solution near the boundary including the corner point and at the infinity.
Outline of the Paper.In the next section, we introduce necessary functional spaces and state the main results (Theorems 3 and 4) along with the general assumptions.In Section 3, we construct the integral representation to ( 1 ,  2 ) in the case of homogenous boundary conditions.To this end, we apply Mellin transform and reduce problem (1)-(3) to the linear nonhomogenous difference equation of the first-order with variables coefficients in the two-dimensional case.Section 4 is devoted to some auxiliary results which will play a key role in the investigation.In Section 5, we estimate the seminorms of the minor derivatives D  1  1 D  2  2 , 0 ≤  1 +  2 ≤ 1, and in Section 6 we evaluate the Hölder coefficients of the major derivatives D  1  1 D  2  2 , 1 <  1 +  2 ≤ .In Section 7, using these estimates, we provide the proofs of Theorems 3 and 4.Moreover, in Remark 27 we show how results of Theorems 3 and 4 can be extend to the more general equation compared to (1).

Functional Setting and Main Results
Throughout this work, the symbol  will denote a generic positive constant, depending only on the structural quantities of the problem.We will carry out our analysis in the framework of the weighted Hölder spaces.Let  ∈ (0,1) be arbitrary fixed.We denote by () and () the distance from a point ( 1 ,  2 ) ∈ Ω to the origin (0, 0) and to the boundary Ω, correspondingly.Then for every  and  from Ω we define (, ) = min{(), ()} and (, ) = min{(), ()}.Note that if ,  ∈ Ω, then () = () and (, ) = (, ).
For fixed  ∈ R, we introduce the Banach spaces    (Ω) and C   (Ω) of the functions V with the norms for  ̸ = .(12) in the case of  ∈ (0, 1).
In a similar way we introduce the spaces    (Ω) and M +  (Ω).As for C   (Ω) and N +  (Ω), these spaces concave with    (Ω) and M +  (Ω).M +  and N +  defined above are Banach spaces.Indeed, the fact that they are normed spaces is easily seen, whereas the completeness follows from Theorem 2.7 [31] together with standard arguments (see, e.g., Remark 3.1.3in [32]).
We are now in the position to state our main results.
The remainder of the paper is devoted to the proof of Theorems 3 and 4 in the DBC case.The proof of these theorems for NBC is almost identical and is left to the interested reader.

Integral Representation for 𝑢(𝑥 1 ,𝑥 2 ) in the Special Case
We first dwell on the special case where   ≡ 0 and  is a finite function.Namely (2), (17), and ( 19) are replaced by the simpler conditions for some given positive  0 and We denote by  ⋆ ( 1 ,  2 ) the Mellin transform of the function ( 1 ,  2 ).Due to conditions (30)- (32) and assumptions of Theorem 3, we can apply, at least formally, the Mellin transformation to problem (1) and (2) (see for details § 1.4 and § 2.5 in [1]).Then simple calculations lead to the equation Introducing new variables and new functions we rewrite the equation in the more compact form Thus, we transform problem (1) and (2) to the linear nonhomogeneous difference equation of the first-order with variable coefficients.In order to solve this equation, we adapt the technique from Section 3 in [33] to our case.Proposition 6.Let  denote Euler-Mascheroni constant (see, e.g., Definition in [34]), and let E 1 ( 1 ) and E 2 ( 2 ) be arbitrary analytic functions such that Then the function solves homogenous equation (36)  Proof.In order to verify that the function  0 ( 1 ,  2 ) solves homogenous equation (36), it is enough to substitute  0 in (36) and take into account the properties of the Gammafunction: Γ( + 1) = Γ().Besides, the properties of the function  0 are simple consequences of the well-known properties of Gamma function: (i) Γ() has simple poles in the points Re  = −,  = 0, 1, 2, . . .
(ii) The Stirling asymptotic formula holds: as |Im | → +∞ while |Re | remains bounded We are now in the position to construct the solution of nonhomogeneous equation (36).
Till the end of the paper, we assume that  ∈ [0, 1] is an arbitrary fixed quantity and define the contour   in the complex plane  as (i) with the small positive number  0 ,  0 < /2.(iii) The contour  0 ( = 0) is obtained from  1 after its shifting to the right on Re  = 1.
Introducing the periodic function () with period 1, we assert the following results.
Besides, the following inequality holds: for  ∈ [0, 1] (iii) There is the asymptotic representation for the bounded Im  1 and Im  2 and if Re  1 , Re  2 meet requirement (45).
Note that statements (i) and (ii) are simple consequences of Proposition 6 and (32).As for assessment (iii), it can be easily drawn from (i) and (ii).
After that, we return to the representation of ( 1 ,  2 ) and obtain solution (36) as This proves Proposition 7 in the case   fl  −1 .
Recasting the arguments above in the case  ∈ [0, 1) and applying Cauchy theorem allow us to prove Proposition 7 in the case of arbitrary contour   ,  ∈ [0, 1).It completes the proof of Proposition 7.

Some Technical Results
First we introduce some equivalent norms.
The proof of this statement follows with direct calculations.Let  ≥ 1,  ∈ [0, 2] , ,  ∈ R. (62) Next we represent certain estimates for the functions which will be frequently used to evaluate the functions ( 1 ,  2 ) and D    ,  = 1, 2.
Proof.For simplicity consideration, we put  = 1.Here we prove this proposition for the function  1 + (, , ).The case of  1 − (, , ) is considered in the similar way.We start our consideration with the case of positive  (i.e.,  ̸ = 0).

Simple conclusions draw to representations in Ω
where Therefore, the main term in the integral Note that the second term here is a regular function of  and .In order to evaluate the first term in this representation, we apply Proposition 7.1 from [33] and obtain where  3 and  4 are twice continuously differentiated and bounded functions for || < 2.
In summary, we obtain the estimate Finally, this inequality together with (78) completes the proof of Proposition 9.
Recasting the proof of the previous proposition, we state the results, which will be used to evaluate |D     |,  = 1, 2, 0 <   < , in Sections 5 and 6.
The following results are related to the properties of the function where Proposition 11.Let  ∈ R,  ∈ (0, +∞).Then there are the following estimates for every fixed : (ii) (88) Proof.We will carry out the detailed proof in  2 + ( 1 ,  2 , , , , ) case.The arguments for  2 − ( 1 ,  2 , , , ) are almost identical and left to the interested reader.
Straightforward calculations lead to the representation First, we consider case  ∈ (0, 2].Asymptotic representation (70) with the decomposition in Figure 1 provides estimate in with  ∈ (0, 1) and in Then, this inequality together with (89) leads to statement (ii).
Let us verify point (ii).One can easily check that After that, recasting the proof of Proposition 9 with aid the last inequality arrives at the estimate Then we are left to evaluate the terms ∫ Ω ±   2 ,  = 1, 2, 7.
We restrict ourselves the estimate of ∫ Ω + 2  2 , the remaining terms are evaluated the same way.Asymptotic (70) with  = 0 and the change of variables,  −  = , provide the representation After that, following the proof of Proposition 9, we integrate by parts and deduce The standard calculations lead to In conclusion, we reach the estimate Collecting this inequality with (93), we deduce the first estimate in statement (ii) of Proposition 11.Further, to improve the estimate for  2 + ( 1 ,  2 , , 0, , ), we consider two different cases: (i) (ii) It is apparent that, in the first case there is and, therefore, keeping in mind the previous estimate of  2 ± ( 1 ,  2 , , 0, , ), we deduce that Coming to the second case (i.e.,  −1 > 4( −1 1 +  −1 2 ) −1 ), the change of the variable, ( 1 +  2 ) = V, leads to , ) The straightforward calculations ensure the uniformly boundedness of the first term in (102); moreover, this estimate is independent of  1 ,  2 , and .Further, we treat the second term in (102).Integrating by parts leads to After that, the properties of functions  1 ,  2 , and Φ 2 allow us to extend this estimate and conclude Hence, this estimate completes the proof of statement (ii).Finally, we are left to verify statement (iii).As for the first equality in (iii), it is a simple consequence of Lemma 5.3 [36].Let us check the second inequality in (iii).To this end, we rewrite  2 + (, ) as where we set Then representation (70) with  = 0 and direct calculations provide which implies Thus, the claim is proven.
Finally, we complete this preliminary section with two estimates that will be frequently used in the following sections.Next, we obtain the second inequality from this proposition.For the sake of clarity, we consider case  2 ||/ 1 || ≥ 1.In the opposite case, we exchange the function  ln( 2 ||/ 1 ||) by  −ln( 1 ||/ 2 ||) and repeat the arguments below.
The direct calculations together with the change of variables, V = 2ln( 2 ||/ 1 ||), reduce the function  1 to the form At this point, we consider the two different cases for  1 : (i) (ii) In the first case, we have Note that we apply integrating by parts in order to reach the last estimate.Further, we obtain the same estimate in the second case.To this end, we use the easily verified inequality, and deduce the bound In summary, we complete the proof of Proposition 12.

Estimate on Maximum of |𝑢(𝑥
(ii) If  = 0 or  = 1, the following representations hold: where () is a Dirac delta function and Here the constant  is independent on  1 ,  2 and .
Proof.First of all, we verify statement (i Then the results of Proposition 9 with aid of (125) provide statement (i).
As for statement (ii), we confirm ourselves the case of  = 0, due to the arguments in the case  = 1 are similar.
First, we represent L  ,  ∈ (0, 1), as ( Here we use the same reasons as in the proof of Proposition 9. Then, as mentioned in the proof of Proposition 7 (see (ii) there) the function (− − /) has the simple poles if  = 0, 1.Thus, following Cauchy's residue theorem, we rewrite   as where we set Concerning the estimate of L 1 0 ( 1 ,  2 ), we apply Lemma 5.3 from [36] and deduce Then the asymptotic representation of Gamma-function (see, e.g., (1.5.15) in [1]), where we put Treating the first term in (132) via Proposition 9 and inequalities (125), we arrive at Concerning  2 , we get and therefore After that, statement (ii) of Proposition 12 (with  =  1 ,  =  2 ) provides In summary, we can conclude that Finally, coming to  3 , we first rewrite it as At last, the proof of statement (iii) is simple consequences of statements (i) and (ii) from this proposition.Thus, the claim is proven.
We are ready now to state estimates of the function ( 1 ,  2 ).Lemma 14.Let assumptions (h1), (31), and (32) hold.Then the function ( 1 ,  2 ) represented with (119) satisfies inequalities sup Proof.At the beginning, we verify the first inequality in this lemma.To this end, we evaluate the term  −− ()||.Putting  ∈ (1/, 1) in (119), we conclude here we use the simple inequality Concerning  1 , we apply statement (i) from Lemma 13 with  >  to deduce with the constant is independent of  1 ,  2 , and .
Coming to the term  2 , applying Proposition 11 to the function and keeping in mind Proposition 9 with inequality (125), we get In conclusion, we have if  > .
Proof.Taking into account (154), we will carry out the detailed proof of the inequality sup The remaining terms in (i) and (ii) are estimated in the same way and with the recasting of the corresponding arguments of Lemma 14.
We begin to evaluate the first term in (163).To this end, we represent where we put This finishes the proof of the lemma.
The following result is a simple consequence of Lemmas 14 and 16, interpolation inequalities, and Proposition 8.

Estimates on
We begin our consideration with the estimates of the function D   1 .Here, we use representation (119) to the function ( 1 ,  2 ) with  = 1; i.e., L  = L 1 .
Recasting the arguments leading to (127) arrives at where we put After that, straightforward calculations arrive at Thus, we can rewrite ( 1 ,  2 ) in the form We are now in the position to calculate D  1  1 ( 1 ,  2 ).Indeed, formula (2.1.17)[1] and direct calculations provide the representation where we put with After that, introducing new variables and new functions we rewrite the functions R 1 , R 2 , and U  in the more comfortable forms In order to reach the last equality in (181), we recast the arguments of Lemma 13 which lead to representations (132), (134), and (138).Indeed, and where the positive constant  is independent of   and   .Hence, we obtain    ,0,Ω follow from the corresponding estimates of the functions U  ( 1 ,  2 ).
First, we describe the properties of the kernels R 2 and R 3 .
where the positive constant  is independent of ,  1 , and  2 .
Proof.First, applying Proposition 11 with  = 0 and  = +∞ and using definition (85), we have where Then, applying statement (iii) in Propositions 11 and 12 with aid (125) arrives at the first inequality of this proposition.Further, we proceed with a detailed proof of statement (ii).The proof of statement (iii) is almost identical and is left to the interested reader.
At the beginning, we evaluate the first term in the lefthand side of the inequality in (ii) Simple calculations lead to where we set At this point, we estimate each term R 3 separately.By Proposition 9 and estimates (125), where the positive constant  is independent of   and   .Concerning R 32 , keeping in mind Propositions 9 -10 and estimates (125), we deduce where the positive constant  is independent of   and   .In summary, we can conclude In order to obtain the last inequality, we use the condition  > .
Then, we are left to evaluate the second term in inequality (ii) To this end, coming to representation (191), we deduce with Finally, Propositions 9 and 10 provide inequality          ∫ where we put It is apparent that the estimates of each U 3 are simple consequence of Lemma 18.Thus, we deduce   ,0,Ω .Next we evaluate the term ⟨U 2 ⟩ ()  1 ,0,Ω .For simplicity consideration, we assume and put In virtue of representation (181) and Lemma 18, we have Then we are left to tackle the second term in the right-hand side of the last inequality.Simple calculations lead to At this point, recasting the proof of statement (i) in Lemma 18 arrives at In conclusion, we obtain the estimate In order to reach the last inequality we use property (32) to the function .Summarizing, we have inequality The same arguments in the case of the difference This completes the proof of Proposition 20.
Next we obtain the same results to the function U 3 ( 1 ).
Since the proof of this result is technically tedious and repeating certain steps in the proof of Lemma 18, we provide one in the Appendix.
Then, based on Lemma 21, we can conclude the following.
Lemma 22.Let assumptions of Lemma 14 hold.Then there is the estimate Proof.We provide here the estimate of ⟨U 3 ⟩ ()  1 ,0,Ω .The case of ⟨U 3 ⟩ ()  2 ,0,Ω is treated in similar arguments.It should be noted that it is enough to evaluate ⟨U 3 ⟩ ()  1 ,0,Ω in case of (223).Indeed, in the opposite case, its estimate follows from Lemma 19.Thus, we assume that (223) hold.Introducing the domain we rewrite the function U 3 in a more suitable form: Then we consider the difference where we set Then putting and making change of variables (179), we reduce the domain At this point, we estimate each term V  separately.Change of variables (179) leads to Applying decomposition like (164) to each term in the representation above, then statements (i) and (ii) in Lemma 21 provide estimate Indeed, let us verify this with the example of the first term in the representation to V 1 .Standard calculations lead to Then, properties of the function  and Lemma 21 with To reach the last inequality we apply (223) to (233).The remaining terms in the representation of V 1 are evaluated in the same way.Concerning V 2 , we repeat the arguments above with changing (

Proofs of Theorems 3 and 4
Under restrictions (30) and (31), the proof of Theorems 3 and 4 follows from the arguments of Sections 3-6.Indeed, Proposition 17 and Lemmas 16, 19, and 23 with Remark 24 provide existence of a solution  which satisfies estimates (22), ( 23) and ( 26), (27).Next, direct calculations with aid of representation (176) allow one to verify that the constructed solution  with (57) satisfies (1) and the corresponding boundary conditions.Further, the uniqueness of the solution follows from the coercive estimates.This completes the proof of Theorems 3 and 4 if ( 30) and ( 31) hold.At this point, we remove restriction (30).To this end, we introduce the new function After that, we look for the solution to (1), (2) in the form where the unknown function W solves the boundary value problem with homogenous boundary condition with It is apparent that the function  satisfies conditions (h3), (19).Thus, arguments of Sections 3-6 guarantee the onevalued solvability to (248) and, besides, the function W satisfies properties (22), (23), and ( 27) and (26).Then, we return the function  and obtain the one-to-one solvability of the original problem, where the solution  possesses the same properties as W. Thus, Theorems 3 and 4 are proven in the absence of (30).Finally, we are left to remove restriction (31).To this end, it is worth to remark that a function  ∈   − (Ω) which satisfies condition (31) belongs to  ∈   −− (Ω) with any positive  and hence also with  =  1 ,  1 ∈ (0, 2−).Further, it is enough to repeat the arguments from Sections 3-7 in order to prove Theorem 3, if  satisfies (17).If  meets requirement (18), the same reasons hold.This finishes the proof of Theorems 3 and 4. Remark 27.Actually, with inessential modifications in the proofs, the very same results hold for the more general equation where 1 <  1 ,  2 < 2. The details are left to the interested reader.

Appendix Proof of Lemma 21
To prove statement (i), we estimate the first term in this inequality.The second one is evaluated with the same arguments.Let us rewrite the first term in the form (A.12) Collecting this inequality with (A.11), we obtain the first inequality in (iv).This completes the proof of Lemma 21.
independent of  1 and  2 .The last inequality together with the analogous estimate for |J 1 | completes the proof of statement (ii).This finishes the proof of Lemma 18.Now we estimate the function |D   1 |.