Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane 0\\ u\left(0,\mathbf{x}\right) = u_{0}\left(\mathbf{x}\right),\mathbf{x\in D},\\ u\left\vert _{x_{i} = 0}\right. = h_{j}\left(t,x_{j}\right),\ x_{j}>0,\ j = 1,2,t>0, \end{array} \right. $?> {ut(t,x)−∇βu(t,x)=|u(t,x)|σu(t,x),x∈D,t>0u(0,x)=u0(x),x∈D,u|xi=0=hj(t,xj), xj>0, j=1,2,t>0, where D={x1>0,x2>0} , β∈(32,2), σ>0 and ∇β is a fractional Laplacian defined as ∇βu=∑j1Γ(2−β)∫0xjuyjyj(xj−yj)β−1dy. We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the L2− based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.

We study the main questions of the theory of IBV-problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution.We generalize the concept of the well-posedness of IBV-problem in the L 2 − based Sobolev spaces to the case of a multidimensional domain.We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong.
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Introduction
We study the nonhomogeneous Dirichlet initial-boundary value problem (IBV problem) for the Ginzburg-Landau equation with fractional Laplacian on a upper-right quarter plane    u t (t, x) − ∇ β u (t, x) = |u (t, x)| σ u (t, x) , x ∈ D, t > 0 u (0, x) = u 0 (x) , x ∈ D, u | xi=0 = h j (t, x j ) , x j > 0, j = 1, 2, t > 0, (1.1) where β ∈ 3  2 , 2 , σ > 0, and Many phenomena in nature can be described by nonlinear fractional differential equations, so their study has received wide attention.Nonlinear fractional differential equations are used in many fields, including control theory, biology, engineering, signal processing, acoustic waves, hydromagnetic waves, fractal dynamics, and many others [47].It was found that the fractional derivatives are more advantageous than integer-order derivatives, as they provide key tool for describing some important effects and properties of various materials, such as nonlocality, memory and heredity properties, which are mostly ignored using integer-order derivatives [44,46].Hence, there have been extensive study of fractional differential equations including the fractional Ginzburg-Landau equation.
This paper is the first attempt to give a method for studying the multidimensional IBV-problems for fractional equations using the Ginzburg-Landau equation with a fractional Laplacian as an example.When studying the multidimensional case of IBV-problems, new difficulties arise in comparison with the one-dimensional case due to the boundary.The method of separation of variables, which assumes finding solutions in partial derivatives in the two-dimensional case as a product of solutions in the one-dimensional case, cannot be used in the case of a fractional Laplacian.Therefore the theory of the IBV-problem in the multidimensional case differs from the same theory in the one-dimensional case.For example, in the case of IBV-problems, it is not clear how many boundary conditions are required for the correctness of the boundary value problem.The answer to this question is based on the construction of the Green's function for a linear fractional equation, which is interesting in itself.
It is also necessary to take into account the boundary effects that affect the behavior of solutions.So, for example, it is necessary to require as low regularity as possible from the solution, since this is connected with compatibility conditions : some relations both between the initial and boundary data, and between the boundary data themselves.This question is one of the main ones in the case of IBV-problems, and the answer to it essentially depends on the concept of well-posedness of the problem.Note that for the case of IBV-problems on a halfline there is no such complication.The large time asymptotic for solutions of (1.1) also was not studied previously.In the present paper we fill this gap.We prove the global in time existence of solutions.Also we are interested in the study of the influence of the Dirichlet boundary data on the asymptotic behavior of solutions.Comparing with the corresponding IBV-problem on a half-line, the solutions of the multidimensional IBV-problem can obtain rapid oscillations, grow or decay faster, and so on.In the book [26], it was proved that in the case of Dirichlet problem for dissipative one dimensional equations the solutions obtain more rapid time decay, comparing with the Cauchy problem.As we shown above for the fractional Ginzburg-Landau equation this is the same case: the asymptotic behavior of solutions essentially depends on the boundary data.
Our approach is based on developing a linear theory for a fractional Laplacian with a boundary and then using fixed point arguments.We remark that there are several ways for construct Green operator.For example, one can extend the boundary data into all space and reset the boundary conditions via suitable change of the dependent variable.However this approach can not be used in our case since fractional Laplacian is not even or odd.Another well-known approach is to use the theory of ordinary differential equations after performing the Laplace transform with respect to the time variable.It also cannot be used in our case due to the nonlocality of the fractional Laplacian.
To obtain a good linear theory of the IBV problem for the fractional Laplacian we develop a general method based on the Riemann-Hilbert approach and the theory of the Laplace transform in the two-dimensional case.This approach is new and non-standard, and its advantage is that it provides a powerful method for the analysis of nonlinear nonlocal systems.We believe that our approach can be used to deal with similar nonlocal equations involving other fractional derivatives on more general domains, in particular half-spaces, like x i ∈ R, x j > 0. This also works well for higher dimensions.Here we consider the case of two dimensions only for the convenience of the reader.Note that the fractional Ginzburg-Landau equation has a singularity, non-locality, and sometimes even includes a space-time connection, which creates many difficulties and problems for analysis.Thus, the analytical results obtained in this work are of considerable interest not only for the theory of the initial boundary value problem, but also for applications of numerical methods [13,42].Especially if, when implementing a numerical scheme, it becomes necessary to truncate the spatial domain.Also, the exact asymptotic formulas and error estimates obtained in this work can be used in numerical calculations by the method of successive approximations.We believe that the methods of this paper can also be used to consider the corresponding Schrödinger equations instead of the heat equations.We will probably need additional compatibility conditions.

Main results
To state precisely the results of the present paper we give some notations.
We denote a Laplace transform by where Re p j > 0. Then an integral formula for the inverse Laplace transform L * j ϕ , called the Mellin's inverse formula, is given by the line integral where integration is done along the vertical line Re p j = γ > 0 in the complex plane such that γ is greater than the real part of all singularities of ϕ(p j ).The generalization of the Laplace transform and inverse Laplace transform in two-dimensional case are where x =(x 1 , x 2 ), p =(p 1 , p 2 ), px = j p j x j , dx = Π 2 j =1 dx j and dp = j dp j .In additional to the notations introduced above, we will use the following notation : L t is Laplace transform with respect to time variables for Re ξ > 0 Laplace transform with respect to time and space variables ϕ(t, x) : Laplace transform with respect to time and space variables ϕ Laplace transform with respect to time and space variables ∇ β ϕ : The usual Lebesgue space in upper right-quarter plane D = (R + × R + ) : where the norm is given by In one-dimensional case we denote Sobolev space H s j (R + ) with the norm Here s + 1 2 is the integer part of the number s + 1 2 .Remark.Since this definition does not require that all boundary values of h vanish, the Sobolev space H s (R + ) differs from the usual H s 0 (R + ), which is defined as the subspace that is the closure of a function in H s (R) whose support lies in R + .

Also we denote the norm
with , where s ∈ 0, 3 2 .Remark.Note that this definition of Ẇs uses the mixed derivative in the usual Sobolev space.This is necessary to avoid a compatibility condition between the initial and the boundary data.Also note that where Here Γ j = {q j ∈ C; Re q j < 0}, such that q β j − p β j ̸ = 0 for any fixed Re p j = 0.
The usual approach in the theory of nonlinear IBV problem is to say that u solves the equation in the sense of Schwartz distributions.This, however, leads to the following question: in what sense is the nonlinear term a distribution and also in what sense does the solution u satisfy the given initial and boundary values.In this paper, we will use the following definition.Definition 2.1.Let K(p) = j p β j and h = {h 1 (t, x 1 ), h 2 (t, x 2 )} .We say that the problem (1.1) is locally wellposed in the space if for some T > 0 there is a unique mild solution If T = ∞ we say that the problem (1.1) is globally wellposed.
We denote First, we formulate the local well-posedness in low-regularity settings result. , Then there exists T(r) > 0 depending only on size such that the problem (1.1) is locally wellposed in the space Remark.Since, when estimating in the W s -norm, we want to consider the case without any compatibility condition between the initial and boundary data, we need a requirement establishing a connection between s, ε and σ: Remark.Let us pay attention to the connection between the order of the Sobolev spaces ε and s in the definition of correctness.The spaces from which the boundary data will be drawn are not determined by the choice of initial data.If ε > 0 can be sufficiently small and independent of s, then, on the other hand, the order of the Sobolev space to which the boundary data belongs is equal to 2s − 1 2 .This result is unknown and is valid due to the strong dissipative smoothing mechanism of the fractional Laplacian.Without smoothness, these relations are effectively invalid.
The preceding results are local, which is to say the time interval [0, T) over which the solution is guaranteed to exist depends on the size of initial and boundary data.If T > 0 can be chosen independently of the size of the initial and boundary data, then the result is termed global well-posedness.Now we formulate the global existence results.We define the functional spaces and for some small γ > 0 with norm such that Theorem 2.2.Suppose β ∈ 3 2 , 2 and , where the norm where ϵ > 0 is a small enough.Also suppose, that the compatibility condition is fulfilled for t > 0. Then the problem (1.1) is globally wellposed in the space are valid, where Ah ∈ L ∞ given by (2.5).
Remark.In this result, we assume that lower order compatibility conditions between boundary data for s > 1 2 .We impose this compatibility conditions because of our requirement that for all t > 0the solution (2.4) should be continuous at the corner of the spatial domain, i.e. at x = 0. Note that we do not need any compatibility conditions between the initial and boundary data due to the strong dissipative smoothing mechanism of the fractional Laplacian.Indeed, the kernel e K(p)t of the Green operator M e K(p)t L(u 0 ) (x) is strongly dissipative, i.e.Re K(p) < 0 for any purely imaginary p and thus the integrand e K(p)t L(u 0 ) decreases enough to guarantee the convergence of M e K(p)t L(u 0 ) (x) .
Remark.The theorem 2.2 shows that if the boundary data h(t) ∼ t −θ decrease slowly (θ < 2 β ) then this significantly affects the asymptotic.The solutions scatter to a linear asymptotic state (short range case) and have the same decay rate as the boundary data.Also note that the condition 1 σ+1 < θ < 1 then includes the case of the non-linearity order σ ⩽ 1.This is a critical degree of nonlinearity in the case of the Cauchy problem, in the sense that small solutions do not behave as linear ones.The situation will be quite different in the case of rapid time decay rate of boundary data (θ ⩾ 2 β ).It can be shown that in this case the solution decay more slowly than the boundary data.
We organize our paper as follows.In section 3, we discuss previous results.In section 4, we present the sketch of proof.In section 5, we give the linear theory for (1.1).Section 6 is devoted to some preliminary estimates for the free Ginzburg-Landau evolution semigroup formulate on D. In section 7 we proved theorem 2.1.Section 8 is devoted to proof of theorem 2.2.In appendix we collect some important definitions used in this paper.

Previous results
The Ginzburg-Landau equation is one of the well-known nonlinear models in physics.Equations of this kind are of great interest and are widely used in physics, chemistry and mechanics [44,46,47].At this stage, it should be noted that there is obviously a huge amount of literature on the Ginzburg-Landau equations, so a review of this literature is beyond the scope of our paper.However, we want refer the interested reader to the some list of works [3-9, 14-25, 31, 37, 38, 40, 41, 45], which mostly treating Cauchy or half-line IBV problems for the Ginzburg-Landau equation and deals mainly with issues related to the correctness and blow-up of solutions.Note that this list contains relatively few papers dealing with nonhomogeneous IBV problems in the one-dimensional case.
The Ginzburg-Landau equation was generalized into the fractional form in [21] from the variational Euler-Lagrange equation for fractal media.Theoretical analysis of Cauchy problem for the fractional Ginzburg-Landau equation has also well developed.Among them, [10,22,30,39,40,43].Due to the nonlocality of fractional operators, the study of a boundary value problem even in the case of a half-line is difficult.The first attempt to give a general theory of the IBV problem in the one-dimensional case is the book [26], where a pseudodifferential operator with an analytic symbol was defined.The well-known fractional derivatives of Caputo and Riemann-Liouville on the half-line were considered in [32], where a unified approach to the study of fractional partial differential equations on the half-line was given.
To determine more general fractional derivatives (a pseudodifferential operator on a halfline with a non-analytic symbol), the paper [35] developed a new research method based on the theory of sectional-analytic Cauchy functions and the Riemann-Hilbert problem.Using these powerful methods, IBV problems for nonlinear fractional equations were studied.
Here is a fairly complete list of these works [1,11,12,33,34], which addresses issues related to global correctness and long-term asymptotics.
The study of initial-boundary value problems with nonhomogeneous boundary conditions in the multidimensional case has not even gone as far as the case of the same problems on a half-line.There are only some results for nonlinear equations with integer order derivatives.For example, in the papers [27][28][29]36] considered IBV problems on D for the nonlinear Schrödinger equation with different boundary data types.In the paper [2], the global well-posedness of the Klein-Gordon equation on D with the Dirichlet type of boundary data was obtained.As far as we know, there are no results on the IBV problem for the fractional Ginzburg-Landau equation in the two-dimensional case.
We remark that bibliography mentioned above also contains some landmark papers as well as several recent studies on IBV issues that offer exciting and very important results and are thus considered useful for researchers in this field.

Sketch of proof
We briefly explain our strategy.To construct the Green's operator for the fractional Laplacian with boundary, we apply the two-dimensional Laplace transformation to corresponding linear equation (1.1) with respect space variables and transform Laplace with respect to time variables.We also use convolution properties of Laplace transform to get Thus, we get nonhomogeneous Riemann-Hilbert problem on the line Re p =0 for any Reξ = 0, where and u(ξ, p i ) and u xj (ξ, p i ) are Laplace transforms of boundary data u(t, x) xj=0 and u xj (t, x) xj=0 , j = 1, 2 correspondingly, i.e We make a cut in definition p β along negative axis p < 0. Note that we have four unknown functions û(ξ, p j ) and ûxj (ξ, p j ), j = 1, 2. We will find some of them using the theory of complex variables.Indeed, the term of left-side of equation ( 4 which belong to the right-sided complex plane Re p ∈ D. Therefore for the existence of unique solution of (4.1) the following conditions must be met Thus, we have to specify only two boundary data in the problem, for example, u | xi=0 = h j (t, x j ), j = 1, 2, and other boundary data u xj | xi=0 , j = 1, 2 will be found from (4.2).
Applying the L * -transformation in the equation (4.1) through the condition (4.2), we obtain that the Green's operator for the fractional Laplacian with boundary has the following form where transform M was defined by (2.3).This result is proved in proposition 5.3 below in section 5. Then the IBV problem (1.1) can be rewritten as the integral equation (2.4).To prove the local wellposed result we use the standard contraction method in Sobolev spaces.The difficulty we need to overcome is that the M-transform is not isomorphic to L 2 .This result, as well as other properties of the M-transform, are discussing in lemma 6.2 below in section 6.
In order to evaluate the solution in L 2 , we need to impose additional conditions on the initial and boundary data, or apply the Hausdorff-Young inequality in the case of the estimation of nonlinear term (lemma 6.3 in section 6 for more detail ).An essential place in the proof of the results is occupied by estimates of the boundary operator H(t)h.Note that, by the construction, the operator H(t)h has a singular kernel near the boundary x =0.To study the influence of boundary data on the properties of the solution for t > 1, we consider the case of a slow decrease in time of the functions h ∼t −θ , θ < 1, since the term depending on the initial data u 0 decays in time faster than t −1 .We obtain that the boundary operator H(t)h has the same damping in time as h.The proof of this result, as well as other interesting details about the behavior of the boundary operator, can be found in lemmas 6.4 and 6.5 in section 6 below.
To get a locally well-posedness, we use the standard contraction mapping method in the W s space.Note that in the case s > 1  2 we need to modify the well-known Sobolev embedding theorem in the case of a quarter plane.The prove of local existence Theorem can be found in section 7. To get a global well-posedness, we use the following ideas.The nonlinear operator has the same structure as H(t)h, only without a singular kernel near the boundary x =0.Therefore it decays faster in time than H(t)h for t > 1.For this reason, we have that the boundary is significant and retains its influence for all t > 1.Also, in the case of slow damping of the boundary, one can obtain a global wellposed result in the case of nonlinearity in (1.1) with the order σ < 1, which is a subcritical ones in the case of the Cauchy problem or in the case of fast boundary damping.This is possible since time decay rate of nonlinear term is t − 2 βr , where nonlinearity |u| σ u belong to L r and we can use Hausdorf-Young inequality with maximum possible r ∼ 2. This result is given in section 8 below.

Fractional Green operator on a quarter plane
To obtain the fractional Green's operator on a quarter plane, we first prove some useful results.
Lemma 5.1.Let g(p) and ϕ(p) be analytic in the upper right quadrant of the complex plane.Assume that ϕ(p) ∈ L 1 and g(p) has first order zeros k i that belong to upper right quadrant of the complex plane , such that in the upper right quadrant of the complex plane.Then where γ > 0 is small enough.
Proof.Since g(p) and ϕ(p) are analytic in the upper right quadrant of the complex plane and ϕ(p) ∈ L 1 and g(p) has first order zeros k i that belong to upper right quadrant of the complex plane we use Cauchy's integral formula and take residue at the pole q j = k i to get where integration is done along the vertical line Re q j = γ > 0, j = 1, 2 in the complex plane such that γ < Re k i .
Therefore, making the changes of variables p j = q j , q j = p j we get pj= pj where where γ > 0 is a small enough.Again we use Cauchy's integral formula and take the residue at the point where γ 1 > 0, γ 2 > 0 we choose such that Re k i (q j ) < γ j .Therefore using g(k 2 (p 2 ), k 1 (p 1 )) = −g(p) and making changes of variables p j = q j , q j = p j , j = 1, 2, we get As a consequence, Lemma is proved. Then where M defined by (2.3).
Therefore, via lemma 5.1 where (5.2) We have Also by the direct calculation we obtain since due to analyticity of ϕ(p) in half plane Re p j > 0 via Cauchy Theorem for any fixed Req j = −γ and Reξ = 0. Thus, substituting (5.3) and (5.4) into (5.2),we end up with where Note that here we can choose the contour Re q j = −γ, such that Re (K(q j ) + K(p i )) > 0. Changing the order of the integration, applying the Jordan's lemma in the left-half of the complex plane and taking the residue at the point ξ = K(p) we get where Using analytic properties of integrand we can rewrite E(p, x) in the form where Γ j = {q j ∈ C; Re q j < 0}, such that q β j − p β j ̸ = 0 for any fixed Re p j = 0. Lemma is proved.
We consider the linearized version of the problem (1.1).
Proof.We suppose that there exists some function u(t, x), which satisfy problem (5.6) in classical sense.To construct the Green's operator for the fractional Laplacian with boundary, we apply the two-dimensional Laplace transformation to corresponding linear equation (5.6) with respect space variables and transform Laplace with respect to time variables to get nonhomogeneous Riemann-Hilbert problem on the line Re p = 0 We make a cut in definition p β along negative axis p < 0. We rewrite (5.8) in the form with We fixed p j , Re p j > 0 and ξ, Re ξ > 0. There exists only one root k j = ξ − p β Since, by construction, u(ξ, p) must be an analytic function in Re p ∈ D, Re ξ = 0, this means that for the solubility of problem (5.6) it is necessary that the following conditions are satisfied (5.10) Thus, if we consider Dirichlet boundary data u(t, x) xj=0 = h i (t, x i ), j = 1, 2, the another unknown boundary data u xj (t, x) xj=0 we find from (5.10) as ) (5.12) Therefore, the sum of (5.11) and (5.12) implies Also taking in (5.11) Therefore, as result of (5.13) and (5.14) Substituting (5.15) into (5.9)we obtain where with l = 1, 2 and The fundamental importance of the proven fact, that the solution u constitutes an analytic function in Re p ∈ D, Re ξ > 0 and, as a consequence, taking inverse Laplace transform of (5.16) we obtain where operators G(t, x) given by (5.17) Therefore via lemma 5.2 we have ) into (5.17) and using convolution property of Laplace transform we obtain Thus we get On other hand by direct calculation M e K(p)t u 0 (t, x) + H(t)h satisfy problem (5.6) in classical sense.Proposition is proved.

Lemmas
First, we prove the main properties E j (p j , x j ) and M j ϕ given by E j (p j , x j ) = βp j 2π i ˆΓj dq j e qjxj − e pjxj q β j − p β j q α j =   e pjxj + βp j 2π i ˆΓj dq j e qjxj q β j − p β j q α j   and Lemma 6.1.The following estimates are valid E j (p j , x j ) = 0, x j < 0, Re p j = 0, (6.3) lim xj→0 + E j (p j , x j ) = 0, Re p j = 0, (6.4) , where Indeed, we use Cauchy's integral formula and take residue at the pole q j = p j (Rep j = 0) to get Again we use Cauchy's integral formula and take residue at the pole q j = p j (Rep j = 0) and q j = s j (Res j ⩾ 0) to get Therefore, recalling L j {E j } (s j ) = I 1 + I 2 we prove (6.1).Now we prove that Indeed, since L j {ϕ} is analytic for Re p j >0 and q β j − p β j ̸ = 0 for any q j ∈ Γ j we use Cauchy Theorem to get As consequence Now we prove E j (p j , x j ) = 0, x j < 0. Indeed, since e q j x j (q β j −p β j )q α j has first order zeros p j (Rep j = 0)we use Cauchy's integral formula and take residue at the pole q j = p j to get and as a consequence for any Re p j = 0 and x j < 0 using |e pjxj − 1| ⩽ p δ j x δ j we get δ ∈ (0, 1) Here q j = |p j | q j , q β j − p β j |p j | −β ̸ = 0. Also from (6.6) follows lim xj→0 + E j (p j , x j ) = 0, Re p j = 0. Lemma is proved.

Boundary operator
Denote where and the compatibility condition Then following estimates Proof.Using the convolution properties of the Laplace transform, we rewrite H(t)h in the following form First we get estimates in the case of s ∈ 0, 1 2 .Via lemma 6.2 we obtain where Making the change of a variables .
Making the change of a variables p i = |ξ| which implies Thus, via Hölder inequality from (6.13) we get for κ To get estimate in the case of s ∈ 1 2 , 3 4 we use where where To estimate J 1 we use (6.14) with λ = s − 3 2 to get since making the changes of a variables p i = |ξ| 1 β p i , we have for small γ > 0 As consequence, ˆiR Now we estimate J 2 .We have and as a consequence from (6.16) and (6.17) it follows To get L 2 −norm estimate we rewrite Also for any pure imaginary ξ we have since via lemma 6.1 M j {L j ϕ} = ϕ(x j ).We return to estimate ∥H(t)h∥ L 2 .According to (6.18) and (6.19) we divide the estimation of ∥H(t)h∥ L 2 into two parts : where and Making the changes of variables p j = |ξ| 1 β p j we get By the same way we get Thus we get for λ = 1 2 (1 since via lemma 6.1 using Therefore, using from (6.20) we get As a consequence, via lemma 6.2 we obtain where Making the change of a variables As a consequence, via Hölder inequality, we have where γ > 0 small.Now we estimate ∥H(t)h−h i ∥ L ∞ i .Via lemma 6.2 we obtain where Lemma is proved.
We define Lemma 6.5.Let s ∈ 1 2 , 3  4 and h ∈ Y θ .Also let the compatibility condition h 1 (t, 0) = h 2 (t, 0) is valid.Then the following estimates and following asymptotic are valid Proof.After integration by parts in the domain of τ ∈ t 2 , t , we get where , where Making the change of variables, we have for − 1 2 < λ < β − 1 2 and for some fixed pure imaginary p j .By the direct calculation, we have since h i (t, 0) = h j (t, 0).Indeed, we have Also we have lim Using (6.23), we obtain Therefore, via lemma 6.2, we get where via Hölder inequality Therefore, applying (6.22) with By the same way we obtain Also we have for ∂ τ h = ∂h ∂τ .As a consequence and Thus, finally we have Now we estimate ∥H(t)h∥ L 2 .We use (6.21) via lemma 6.2 we get since making the change of variables p = pt − 1 β we have Asymptotic.Again we use (6.21) to get where Also, via lemma 6.2, we have and and Thus, Lemma 6.5 is proved.

Nonlinear problem (proof of theorem 2.1)
Let Also in the case of s ∈ 1 2 , β 2 we have compatibility condition We introduce We prove, that there exists a unique local in time solution of the initial-boundary value problem by a standard contraction mapping principle.We define mapping in X T as where u ∈ X T and lim From lemma 6.3, we have From lemma 6.4 we have To estimate ´t 0 G(t − τ )N u(τ )dτ we use lemma 6.3 to get Also we adopt embedding Sobolev Theorem In the case of s > 1 2 we make the modification of embedding Sobolev Theorem to get , where λ > 1 2 for s > 1 2 .Therefore, substituting 2 r = (1 − s) (σ + 1) into (7.2),we get Theorem is proved.

Nonlinear problem (proof of theorem 2.2)
We define the functional spaces Also we define , 0 < θ < 1, γ > 0 is a small enough .
We have T > 1, since ∥h∥ σ+1 Y θ + ∥u 0 ∥ H ε is a small enough.Therefore we consider the case of t > 1.
From lemma 6.5 if the compatibility conditions h 1 (t, 0) = h 2 (t, 0) are valid then Also from lemma 6.3 we have Using and θ ⩾ 1 σ+1 .Therefore by a standard contraction mapping principle we prove that there exists a unique global solution of the initial-boundary value problem (1.1) u ∈ X.
Also via lemmas 6.3 and 6.
for t > 1.Now we turn to the proof of the asymptotic formula for the solution u of the problem (1.1).In the case θ < 1 the rate of decay of solutions in time depends only on the boundary data.Via (8.5), we have , where δ = 2 β − θ > 0. From lemma 6.5 Theorem is proved.

. 1 )
u(ξ, p) is analytic in Re p ∈ D by the construction.Therefore for wellposed of the problem (4.1) the right-side of equation (4.1) must be analytic Re p ∈ D and has discontinuities only on the line Re p =0.On the other hand, note that 1 ξ−K(p) in (4.1) is not an analytic Re p ∈ D because the equation ξ − K (p) = 0 has two roots = 0, where u (ξ, p) = L t L {u (t, x)} , u 0 (p) = L {u 0 } , and u(ξ, p j ) and u xj (ξ, p j ) are Laplace transforms of boundary data u(t, x) xj=0 and u xj (t, x) xj=0 , j = 1, 2 correspondingly

j 1 β
, j = 1, 2 of equation K (p) = ξ such that Re k j > 0 for all fixed Re ξ > 0 and Re p j > 0. Therefore, in the expression for the function u(ξ, p) the factor 1 ξ−K(p) has a pole in the point p j = k j , j = 1, 2.