Asymptotics of solutions of second order parabolic equations near conical points and edges

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(1)Asymptotics of solutions of second order parabolic equations near conical points and edges. Vladimir Kozlov and Juergen Rossmann. Linköping University Post Print. N.B.: When citing this work, cite the original article.. The original publication is available at www.springerlink.com: Vladimir Kozlov and Juergen Rossmann, Asymptotics of solutions of second order parabolic equations near conical points and edges, 2014, Boundary Value Problems, 252. http://dx.doi.org/10.1186/s13661-014-0252-x Copyright: Hindawi Publishing Corporation / Springer Verlag (Germany) / SpringerOpen http://www.springeropen.com/ Postprint available at: Linköping University Electronic Press http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-114020.

(2) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. RESEARCH. Open Access. Asymptotics of solutions of second order parabolic equations near conical points and edges Vladimir A Kozlov1 and Jürgen Rossmann2* *. Correspondence: juergen.rossmann@uni-rostock.de 2 Institute of Mathematics, University of Rostock, Rostock, 18051, Germany Full list of author information is available at the end of the article. Abstract The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K × Rn–m , where K is an infinite cone in Rm , 2 ≤ m ≤ n. They obtain the asymptotics of the Green function near the vertex (n = m) and edge (n > m), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted Lp space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied.. 1 Introduction The present paper is concerned with an initial-boundary value problem for a second order parabolic equation in a n-dimensional domain with a (n – m)-dimensional edge M, n > m ≥ . In particular, we are interested in the asymptotics of the solution near the edge. The largest part of the paper deals with the problem ∂t u(x, t) – L(x, t, ∂x )u(x, t) = f (x, t) for x ∈ D, t > , u|∂ D×R+ = ,. (). u|t= = . in the domain D = K × Rn–m . Here K = x ∈ Rm : ω = x /x ∈ , is an infinite cone (angle if m = ), is a subdomain of the unit sphere Sm– with C , boundary ∂, and L(x, t, ∂x ) =. n i,j=. aij (x, t)∂xi ∂xj +. n . aj (x, t)∂xj + a (x, t),. (). j=. is a linear second order differential operator with variable coefficients. Initial-boundary value problems for parabolic equations in domains with angular or conical points and edges were studied in a number of papers. Most of these papers deal © 2014 Kozlov and Rossmann; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited..

(3) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 2 of 37. with the heat equation. Concerning the heat equation in domains with angular or conical points, we mention the papers by Grisvard [], Kozlov and Maz’ya [], de Coster and Nicaise [], where the asymptotics of the solutions near the singular boundary points was studied. For domains with edges, Solonnikov [, ] and Nazarov [] estimated the Green function and proved the existence of solutions of the Dirichlet and Neumann problems for the heat equation in weighted Sobolev spaces. Kozlov and Rossmann [, ] and Kweon [] investigated the asymptotics of solutions of the Dirichlet problem for the heat equation near an edge. A theory for general parabolic problems with time-independent coefficients in domains with conical points was developed in papers by Kozlov [–]. This theory includes the asymptotics of solutions in weighed L Sobolev spaces and a description of the singularities of the Green function near the conical points. The goal of the present paper is to extend these results to the case of time-dependent coefficients and to domains with edges. Moreover, we consider solutions in weighted Lp Sobolev spaces with arbitrary p ∈ (, ∞). However, we restrict ourselves to second order parabolic equations, and we consider only the first terms in the asymptotics. In our previous paper [], we obtained point estimates for the Green function. These estimates together with results of the theory of elliptic boundary value problems are used in the present paper in order to describe the behavior of solutions near the edge. We outline the main results of the present paper. For an arbitrary point x = (x , . . . , xn ) ∈ Rn , we put x = (x , . . . , xm ) and x = (xm+ , . . . , xn ). The same notation is used for multiindices α = (α , . . . , αn ). We assume that aij = aji are real-valued functions and that aij (x, t) – aij (, ) ≤ ,. ai (x, t) ≤ x – ,. a (x, t) ≤ x – ,. (). where is a small positive number. Besides this assumption, we impose some smoothness conditions on the coefficients aij and aj (see () and ()). The condition () ensures in particular that the difference of the operators L(x, t, ∂x ) and L (, , ∂x ) =. n . aij (, )∂xi ∂xj. i,j= , l,l is small in the operator norm Wp;β (DT ) → Lp;β (DT ). Here Wp;β (DT ) is the Sobolev space on DT = D × (, T) with the norm. uW l,l (D ) = p;β. T. T . . /p p(β–l+k+|α|) x ∂ k ∂ α u(x, t)p dx dt .. D k+|α|≤l. t. x. (). For l = , this space is denoted by Lp;β (DT ). In Sections  and , we deal with the asymptotics of the Green function near the edge M of D . In the case of constant coefficients aij , the asymptotics can easily be obtained by means of the asymptotics of the Green function for the heat equation which is given in [, ]. If the coefficients are variable, then the terms in the asymptotics contain the eigenvalues and eigenfunctions of the following operator pencil A(x , t; λ): –λ . λ A x , t; λ (ω) = x L , x , t, ∂x ,  x (ω),. ◦. ∈ W  ().. ().

(4) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 3 of 37. Let λ+ (x , t) be the smallest positive eigenvalue and let + (x , t; ω) be the corresponding eigenfunction. As was proved in [], the Green function G(x, y, t, τ ) of the problem () satisfies the estimate G(x, y, t, τ ) ≤ c(t – τ )–n/. . |x | √ |x | + t – τ. λ . |y | √ |y | + t – τ. λ. κ|x – y| exp – t–τ. √ for  < t – τ < T, |α | ≤ , |γ | ≤ , where λ < λ+ (, ) – C . Analogous estimates are valid for the derivatives of G (cf. Theorem .). Using this result, we show in Section  (see Theorem .) that G(x, y, t, τ ) admits the decomposition. λ+ (x ,t) + .  x , t; ω + R(x, y, t, τ ), G(x, y, t, τ ) = ψ x , y, t, τ x . (). where μ λ κ(|y | + |x – y | ) |y | R(x, y, t, τ ) ≤ c(t – τ )–n/ √|x | exp – √ t–τ t–τ t–τ √ for  < t – τ < T and |x | < t – τ . Here, μ is a certain number greater than sup λ+ (x , t). The coefficient ψ in () satisfies the estimate λ . κ(|y | + |x – y | ) ψ x , y, t, τ ≤ c(t – τ )–(n+λ+ (x ,t))/ √|y | exp – t–τ t–τ for  < t – τ < T. Moreover, ψ admits the decomposition. ψ x , y, t, τ = , x , y, t, τ + r x , y, t, τ , where , is the function () and λ . κ(|y | + |x – y | ) r x , y, t, τ ≤ c(t – τ )–(n–+λ+ (x ,t))/ √|y | exp – t–τ t–τ for  < t – τ < T. In Section , we apply the results of the foregoing section in order to describe the behavior of the solutions of the problem () near the edge M. By Theorem ., the following result holds. Suppose that f ∈ Lp;β (DT ), where. √ sup λ+ x , t <  – β – m/p < λ+ (, ) +  – C ,. √.  – β – m/p < inf λ+ x , t – .. Then the solution of the problem () admits the decomposition λ+ (x ,t) + . u(x, t) = (E h )(x, t)x   x , t; ω + v(x, t), where. h x , t =. t . D. ψ x , y, t, τ f (y, τ ) dy dτ ,.

(5) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 4 of 37. , E is the extension operator introduced in Section ., and v ∈ Wp;β (DT ). Note that the function h belongs to the anisotropic Sobolev-Slobodetski˘ı space Wps,s/ (Rn–m × (, T)), where s is the function s(x , t) =  – β – λ+ (x , t) – m/p. Section  in closing deals with the initial-boundary value problem in a bounded domain with an edge. Under some smoothness conditions on the coefficients of the differential operator, we obtain the same decomposition of the weak solution near an edge point as in the case of the previously considered domain D (see Theorem . at the end of the paper).. 2 Green function of parabolic equations with constant coefficients In this section, we assume that L (∂x ) =. n . ai,j ∂xi ∂xj = ∇xT A∇x ,. i,j=. where aij are real numbers, aij = aji for all i, j. Here A denotes the square matrix with the elements ai,j , ∇x is the nabla operator and ∇xT its transposed, i.e., ∇xT is the row vector with the components ∂xj . There exists a coordinate transformation ξ = Sx with a constant square matrix S such that the problem ∂t u(x, t) – L (∂x )u(x, t) = f (x, t) for x ∈ D, t ∈ R,. u|∂ D×R+ = ,. (). takes the form ∂t u – ξ u = f. for ξ ∈ D = K ∈ Rn–m , t ∈ R,. u|∂ D ×R = . (). in the new coordinates ξ , where K is a certain cone in Rm with vertex at the origin. This coordinate transformation can be constructed as follows. Let A be the matrix with the elements aij , i, j = , . . . , m, A the matrix with the elements aij , i, j = m + , . . . , n, and B the matrix with the elements aij , i = , . . . , m, j = m + , . . . , n. Furthermore, let ∇x and ∇x be the nabla operators in the coordinates x and x , respectively. Then the operator L can be written as. L (∂x ) = ∇xT A ∇x + B∇x + ∇xT BT ∇x + A ∇x . There exists an invertible matrix U such that UA U T = Im (the m × m identity matrix). This is true for the matrix U = –/ V , where is the diagonal matrix of the (positive) eigenvalues of the matrix A and the rows of V are the orthonormalized eigenvectors of A . For the coordinates y = Ux , y = x – BT A – x , we have ∇x = U T ∇y – A – B∇y , ∇x = ∇y and, consequently,. –. L (∂x ) = y + ∇yT A – BT A B ∇y . Obviously, the transformation (x , x ) → (Ux , x – BT A – x ) = (y , y ) maps K × Rn–m onto the set D = K × Rn–m , where K = UK is a cone in Rm . Since A – BT A – B is a symmetric matrix with only positive eigenvalues, there exists an invertible matrix W such that.

(6) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 5 of 37. W (A – BT A – B)W T = In–m . For ξ = y and ξ = Wy , we get L (∂x ) = ξ + ξ . Hence, the equation () has the form () after the coordinate transformation. –. ξ = W x – BT A x .. ξ = Ux ,. (). ˜  (ξ , η, t). This means that We denote the Green function of the problem () by G ˜  (ξ , η, t) = δ(ξ – η)δ(t) for ξ , η ∈ D , t ∈ R, (∂t – ξ )G ˜  (ξ , η, t) =  for ξ ∈ ∂ D , η ∈ D , G. ˜  (ξ , η, t)|t< = . G. Then the function ˜  (Sx, Sy, t) G (x, y, t) = | det S|G. (). is the Green function of the problem (). In order to describe the asymptotics of G near the edge, i.e., for small |x |, we introduce the following notation. We denote by {j } the nondecreasing sequence of eigenvalues of the Beltrami operator –δ on the subdomain = K ∩ Sm– of the unit sphere Sm– with Dirichlet boundary condition, and by {φj } an orthonormalized (in L ( )) system of eigenfunction to the eigenvalues j . Furthermore, let λ± j =. –m  ± ( – m) + j  . be the solutions of the quadratic equation λ(m –  + λ) = j . Then the functions λ+ u˜ j ξ = ξ j φj (ωξ ) and. . v˜ j ξ = –. λ+j. λ–  ξ j φj (ωξ ) +m–. are special solutions of the equation ξ u =  in K . We also introduce the functions. ˜ j η , t = w. . |η |  –λ+ j –m/ u ˜ η exp – (t) , j (λ+j + m/) t. ˜ , t) =  for η ∈ K and t > . which are special solutions of the heat equation (∂t – η )w(η Suppose that μ is a real number satisfying the inequalities λ+ < μ < λ+ +  and μ = λ+j for ˜  admits the decomposition all j. By [, Theorem .], the Green function G. . ˜  (ξ , η, t) = ξ – η , t g˜ ξ , η , t G . . ˜ j η , t u˜ j ξ + r˜ ξ , η , t , w = ξ – η ,t. (). λ+ j <μ . where (ξ , t) = (πt)(m–n)/ exp(– |ξt| ) is the fundamental solution of the heat equation in Rn–m and g˜ (ξ , η , t) is the Green function of the Dirichlet problem for the heat equation.

(7) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 6 of 37. in K . The remainder r˜ (ξ , η , t) in () satisfies the estimate λ+ –|γ |–ε μ–|α | k α γ |η | | ∂ ∂ ∂ r˜ ξ , η , t ≤ ct –k–(m+|α |+|γ |)/ |ξ √ √ t ξ η t |η | + t –ε –ε α γ κ|η | d(η ) d(ξ ) × exp – |ξ | |η | t. (). √ for |ξ | < c t, |α | ≤ , |γ | ≤ . Here, εα =  for |α | ≤ , while εα is an arbitrarily small positive real number if |α | = . Using the decomposition (), we obtain an analogous decomposition for the Green function G (x, y, t). For this, we introduce the functions . . uj x = u˜ j Ux ,. . . vj x = | det U|˜vj Ux. and . |W (x – y + BT A – y )| ˜ j Uy , t exp – ψj, x , y, t = | det S|(πt)(m–n)/ w t –λ+ –n/ (m–n)/ . q(y , x – y ) (t)  π y exp – u , = j | det A|/ (λ+ + m/) t. (). where q(y , y ) denotes the quadratic form.  –  q y , y = Uy + W y + BT A y . Note that the form q(y , y ) is independent of the coordinate transformation since U T U = A – and W T W = (A – BT A – B)– . Since U and W are invertible matrices, there exists a positive constant κ such that.  . q y , y ≥ κ y + y . for all y , y .. (). We furthermore note that both uj (x ) and vj (x ) are solutions of the equation L (∂x , )u =. m . aij ∂xi ∂xj u =  in K. i,j=. which have the form λ+ uj x = x j +j (ωx ),. λ– vj x = x j –j (ωx ).. ± This means in particular that λ± j are eigenvalues and j are eigenfunctions of the pencil A(λ) which is defined as. λ –λ A(λ) (ω) = x L (∂x , )x (ω),. ◦. ∈ W  ().. Theorem . Suppose that λ+ < μ < λ+ +  and μ = λ+j for all j. Then the Green function G (x, y, t) admits the decomposition G (x, y, t) =. λ+ j <μ. . ψj, x , y, t uj x + R (x, y, t),.

(8) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 7 of 37. where λ+ –|γ |–ε μ–|α | k α γ |y | | ∂ ∂ ∂ R (x, y, t) ≤ ct –k–(n+|α|+|γ |)/ |x √ √ t x y t |y | + t κ(|y | + |x – y | ) d(x) –εα d(y) –εγ × exp – |x | |y | t. (). √ for |x | < t, α = (α , α ), γ = (γ , γ ), |α | ≤ , and |γ | ≤ . Here, εα =  for |α | ≤ , while εα is an arbitrarily small positive real number if |α | = . Proof By () and (), we have. . ˜  (Sx, Sy, t) = | det S| ξ – η , t g˜ ξ , η , t G (x, y, t) = | det S|G. . – ˜ j η , t u˜ j ξ + R (x, y, t), w = | det S| ξ – η + WBT A x , t. (). λ+ j <μ. where ξ = Ux , η = Uy , ξ = W (x – BT A – x ), and η = W (y – BT A – y ), and . . – R (x, y, t) = | det S| ξ – η , t – ξ – η + WBT A x , t g˜ Ux , Uy , t. . – + | det S| ξ – η + WBT A x , t r˜ ξ , η , t . The right-hand side of () is equal to . . ψj, x , y, t uj x + R (x, y, t).. λ+ j <μ. Using (), one can easily check that R satisfies ().. . We derive another formula for the coefficient ψj, (x , y, t) in Theorem .. If t > , then. ξ g˜ ξ , η , t = ∂t g˜ ξ , η , t l for ξ , η ∈ K . Let Vp;β (K) denote the weighted Sobolev space (closure of C∞ (K\{})) with the norm. uV l. p;β. p(β–l+|α|) α p /p x ∂x u x dx . (K) = K |α|≤l.  It follows from () and () that ∂t g˜ (·, η , t) ∈ Vp;β (K ) for arbitrary p and β such that p(β + + ˜ j (η , t) in () is given by the formula λ ) > –m. Hence, the coefficient w. ˜ j η , t = w. K. . v˜ j ξ ξ g˜ ξ , η , t dξ . (cf. [, Theorem .]). Let. . U(ξ , η, t) = g˜ ξ , η , t ξ – η , t – ξ – η – WBT U T ξ , t .. ().

(9) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 8 of 37. In the integral. ˜ j η , t = w. K. . v˜ j ξ ξ U(ξ , η, t) dξ . (). one can integrate by parts for t > . Since ξ vj (ξ ) =  and g˜ (ξ , η , t) = vj (ξ ) =  for ξ ∈ ∂K , we conclude that the integral () vanishes. Hence, it follows from () that. . ˜ j η , t ξ – η , t = w. K. . . v˜ j ξ ξ g˜ ξ , η , t ξ – η – WBT U T ξ , t dξ .. We set ξ = Wx , η = Uy , η = W (y – BT A – y ), and we substitute ξ = Ux in the integral on the right-hand side. Then we obtain. –. ˜ j Uy , t W x – y + BT A y , t = w. . . ˜  (Sx, Sy, t) dx . vj x L (∂x , )G. K. Multiplying the last equality by | det S|, we arrive at the formula. ψj, x , y, t =. . . vj x L (∂x , )G(x, y, t) dx .. (). K. 3 Green function of parabolic equations with variable coefficients Now let L(x, t, ∂x ) be the operator () with x- and t-depending coefficients satisfying the condition (). We consider the Green function G(x, y, t, τ ) for the operator L = ∂t – L(x, t, ∂x ) in D = K × Rn–m , i.e., the solution of the problem. ∂t – L(x, t, ∂x ) G(x, y, t, τ ) = δ(x – y)δ(t – τ ) for x, y ∈ D, t, τ ∈ R,. G(x, y, t, τ ) =  for x ∈ ∂ D, y ∈ D, t ∈ R,. (). G(x, y, t, τ )|t<τ = .. In this section, we will employ an estimate for the Green function which was proved in []. For this end, we assume in the following that the coefficients of L satisfy some additional smoothness conditions. To be more precise, we suppose that ∂xγ aij ∈ C σ ,σ / (D × R) for |γ | ≤  and. a , aj , ∂xi aj ∈ C σ ,σ / (D × R). (). with some σ ∈ (, ) for i, j = , . . . , n and that n n γ α k γ α k α k ∂ ∂ ∂ aij + ∂ ∂ ∂ aj + ∂ ∂ a ≤ C x x t x x t t x i,j= |γ |≤. for α ≤ , k ≤ . (). j= |γ |≤. 3.1 Estimates of Green function Let n. . L , x , t, ∂x = aij , x , t ∂xi ∂xj i,j=. and. m. . L , x , t, ∂x ,  = aij , x , t ∂xi ∂xj . i,j=.

(10) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 9 of 37. Furthermore, let the pencil A(x , t; λ) be defined by (), and let λ± j (x , t) be its eigenvalues, where. · · · ≤ λ– < λ– <  – m ≤  < λ+ < λ+ ≤ · · · . The following estimate for the Green function G(x, y, t, τ ) is proved in [, Theorem .]. Theorem . Suppose that the coefficients of L satisfy the conditions (), (), and (). If √ T is a positive number and λ < λ+ (, ) – C , then G(x, y, t, τ ) satisfies the estimate k l α γ ∂ ∂ ∂ ∂ G(x, y, t, τ ) t τ x y λ–|α | λ–|γ | |x | |y | √ √ |x | + t – τ |y | + t – τ –εα –εγ d(x) κ|x – y| d(y) × exp – |x | |y | t–τ. ≤ c(t – τ )–(n+k+l+|α|+|γ |)/. . (). for  < t – τ < T, |α | ≤ , |γ | ≤ , |α | ≤ , |γ | ≤ , k, l ≤ . Here, εα denotes the same nonnegative number as in Theorem ... 3.2 Asymptotics of Green function Analogously to the matrix U in Section , let U(x , t) be a matrix such that. . . U x , t A x , t U T x , t = Im , where A (x , t) is the matrix with the elements aij (, x , t), i, j = , . . . , m. Under our assumptions on the coefficients, we may assume that the elements of U are C  -functions. Then the numbers j (x , t) = λ+j (x , t)λ–j (x , t) are eigenvalues of the Beltrami operator –δ (with Dirichlet boundary conditions) on the subdomain (x , t) = K (x , t) ∩ Sm– of the unit sphere, where K (x , t) = U(x , t)K . As in Section , we denote by {φj (x , t; ω)} an orthonormalized system of eigenfunctions corresponding to the eigenvalues j (x , t). Moreover, we set. λ+ (x ,t) . u˜ j x , t; ξ = ξ j φj x , t; ωξ. v˜ j x , t; ξ = –. λ+j. and. λ– (x ,t) .  ξ j φj x , t; ωξ . +m–. Then the functions. uj x , t; x = u˜ j x , t; Ux ,. . vj x = | det U|˜vj x , t; Ux. (). are special solutions of the equation L (, x , t, ∂x , )u(x ) =  for x ∈ K . By (), we have. . L , x , t, ∂x ,  G(x, y, t, τ ) = ∂t G + L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ). (). for x ∈ D , t > τ . Suppose that p and β are such that √ λ+ (, ) <  – β – m/p < λ+ (, ) +  – C ,.  – β – m/p = λ+j x , t. for all j, x , t,.

(11) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 10 of 37. where C is the same constant as in Theorem .. Then by Theorem ., the right-hand side  (K) for arbitrary fixed x ∈ Rn–m , y ∈ D , t > τ . Applying of () belongs to the space Vp;β [, Theorem .], we obtain . G x , x , y, t, τ =. ψj x , y, t, τ uj x , t; x + R(x, y, t, τ ),. λ+ j <–β–m/p  (K). The coefficients ψj (x , y, t, τ ) satisfy the equality (cf. ()) where R(·, x , y, t, τ ) ∈ Vp;β. . . ψj x , y, t, τ =. . vj x , t; x L , x , t, ∂x ,  G(x, y, t, τ ) dx .. (). K. In the following lemma, we give an estimate of these functions. √ Lemma . Suppose that sup λ+j (x , t) < λ+ () +  – C , where C is the same constant as in Theorem .. Then the function () satisfies the estimate λ–|γ | k l α γ . + d(y) –εγ ∂ ∂ ∂ ∂ ψj x , y, t, τ ≤ c(t – τ )–k–l–(n+|α |+|γ |+λj (x ,t))/ √|y | t τ x y |y | t–τ κ(|y | + |x – y | ) × exp – () t–τ for  < t – τ < T, k ≤ , l ≤ , and |α |, |γ |, |γ | ≤ . Here, κ is a certain positive number, √ λ is an arbitrary number less than λ+ () – C , and εγ is the same nonnegative number as in Theorem .. Proof We define Kt = {x ∈ K : |x | <. √ t} for t > . Then. ∂tk ∂τl ∂xα ∂yγ ψj x , y, t, τ . . ∂tk ∂τl ∂xα ∂yγ vj x , t; x L , x , t, ∂x ,  G(x, y, t, τ ) dx = Kt–τ. . + K\Kt–τ. . ∂tk ∂τl ∂xα ∂yγ vj x , t; x L , x , t, ∂x ,  G(x, y, t, τ ) dx .. We estimate the first integral on the right-hand side of the last equality using the decomposition (). Theorem . yields ν l σ γ . ∂ ∂ ∂ ∂ L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ) t τ x y λ–|γ | –ε |y | –ν–l–(n+|γ |+|σ |+λ+)/ λ– d(x) x ≤ c(t – τ ) √ |x | |y | + t – τ d(y) –εγ κ(|y | + |x – y | ) × exp – |y | (t – τ ) for |x | < t – τ < T, σ ≤ α , and ν ≤ k, where ε is an arbitrarily small positive number. Furthermore, k–ν α –σ – . ∂ ∂ vj x, t; x ≤ cx λj (x ,t)  + logx k+|α |–ν–|σ | . t x.

(12) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 11 of 37. The number λ can be chosen such that sup λ+j (x , t) < λ + . Consequently, . ∂tk ∂τl ∂xα vj. Kt–τ. . . γ x , t; x ∂y L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ) dx . ≤ c(t – τ )–k–l–(n+|α . |+|γ |+λ+)/. . |y | √ |y | + t – τ. λ–|γ |. κ(|y | + |x – y | ) exp – (t – τ ) λ–λ+ (x ,t)+–m l+|α | d(x) –ε j x dx ×  + log x |x | Kt–τ λ–|γ | |y | –k–l–(n+|α |+|γ |+λ+ j (x ,t))/ √ ≤ c(t – τ ) t–τ –εγ  κ(|y | + |x – y | ) d(y) . × exp – |y | (t – τ ) ×. d(y) |y |. –εγ . The same estimate holds for . γ k l α x ∂ ∂ ∂ ∂ v , t; x ∂ G(x, y, t, τ ) dx t y t τ x j . Kt–τ. Thus, we obtain . . ∂tk ∂xα vj x , t; x L , x , t, ∂x ,  ∂τl ∂yγ G(x, y, t, τ ) dx . Kt–τ. ≤ c(t – τ ) . –k–l–(n+|α |+|γ |+λ+ j (x ,t))/. d(y) × |y |. –εγ . . |y | √ t–τ. λ–|γ |. κ(|y | + |x – y | ) exp – . (t – τ ) . Since L (, x , t, ∂x , )vj (x , t; x ) =  for x ∈ K and ∂tk ∂τl ∂xα ∂y G(x, y, t, τ ) is exponentially decaying for large |x|, we get K\Kt–τ. γ. . ∂tk ∂xα vj x , t; x L , x , t, ∂x ,  ∂τl ∂yγ G(x, y, t, τ ) dx. . . = St–τ. ∂tk ∂xα. n . ai,j , x , t. i,j=. . × vj x , t; x ∂xj ∂τl ∂yγ G(x, y, t, τ ) – ∂τl ∂yγ G(x, y, t, τ )∂xj vj x , t; x cos(n, xj ) dσ , √ where St–τ is the intersection of K with the sphere |x | = t – τ and n is the normal vector to this sphere. By Theorem ., the integrand on the right-hand side of the last equality has the upper bound c(t – τ ) ×. –k–l–(n+|α |+|γ |+–λ– j (t))/. d(y) |y |. –εγ . . |y | √ |y | + t – τ. λ–|γ |. κ(|y | + |x – y | ) . exp – t–τ.

(13) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 12 of 37. Therefore, . . . ∂tk ∂xα vj x , t; x L , x , t, ∂x ∂τl ∂yγ G(x, y, t, τ ) dx . K\Kt–τ. ≤ c(t – τ ) . –k–l–(n+|α |+|γ |+λ+ j (x ,t))/. d(y) × |y |. –εγ . . |y | √ t–τ. λ–|γ |. κ(|y | + |x – y | ) . exp – (t – τ ) . This proves the lemma. For the estimation of the remainder R(x, y, t, τ ), we need the following lemma..  Lemma . Suppose that u ∈ Vp;β (K) is a solution of the problem L (, x , t, ∂x , )u = f in + K , u =  on ∂K , where λj (x , t) + ε <  – β – m/p < λ+j+ (x , t) – ε for a certain integer j ≥ . Then. uV . ≤. p;β (K). c(x , t) f V  (K) p;β ε. (). with a constant c(x , t) independent of ε. Proof First note that the eigenvalues λ+j (x , t) and λ+j+ (x , t) of the pencil A(x , t; λ) have no generalized eigenfunctions (see, e.g., [, Section .]). Let g(x , y ) be the Green function  of the Dirichlet problem for the operator L (, x , t, ∂x , ) in the cone K , ζ g(·, y ) ∈ Vp;β (K) for smooth ζ vanishing in a neighborhood of y . Then . u x =. . . g x , y f y dy . K. By [, Theorem .], the function g satisfies the estimates + + g x , y ≤ cx λj+ (x ,t) y –n–λj+ (t) for x < y , + + g x , y ≤ cx λj (x ,t) y –n–λj (t) for x > y . Moreover, in the case |y | < |x | < |y |, the estimates |g(x , y )| ≤ c|x – y |–m for m >  and |g(x , y )| ≤ c| log |x – y || for m =  are valid. For arbitrary integer ν, let χν (x ) =  for ν– ≤ |x | ≤ ν , χν (x ) =  else. Furthermore, let . uν x =. . . g x , y χν y f y dy .. K. Then it follows from the above estimates for g(x , y ) and from [, Lemmas .. and ..] that χμ uν V . ≤ c. χμ uν V . ≤ c. p;β– (K). p;β– (K). (μ–ν)(λ+ j (x ,t)–+β+m/p). χν f V . (μ–ν)(λ+ j+ (x ,t)–+β+m/p). p;β (K). χν f V . p;β (K). if μ ≥ ν, if μ < ν,.

(14) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 13 of 37. where c is independent of μ, ν, and f (cf. [, Lemma ..]). Consequently, p uV  (K) p;β–. =. μ. ≤c. p χμ uV  (K) p;β–.

(15) μ. . –ε|μ–ν|. ≤. μ. ν. p χμ uν V . p;β– (K). p. χν f V . .. p;β (K). ν. By Hölder’s inequality,

(16). . p –ε|μ–ν| χν f V . p;β (K). ν.

(17) ≤. . =. ν. p –ε|μ–ν| χν f V  (K) p;β. ε +  ε – . p– .

(18) . p– –ε|μ–ν|. ν p. –ε|μ–ν| χν f V . ν. p;β (K). .. Thus, we obtain p. ε  +  p– –ε|μ–ν| p ≤ c  χν f V  (K) ε –  p;β– (K) p;β μ ν ε ε  + p  + p p p =c ε χν f V  (K) = c ε f V  (K) .  –  –  p;β p;β ν. uV . The last inequality together with the estimate uV . p;β (K). ≤ c f V . p;β (K). + uV . p;β– (K). (see, e.g., [, Theorem ..]) implies ().. . Now we are able to prove the main result of this section. √ Theorem . Suppose that λ < λ+ () – C and √ λ+ () < μ < λ+ () +  – C ,. √ √ λ+j x , t ∈/ [μ – , μ + ] for all j, x , t,. (). where C is the same constant as in Theorem .. Then the Green function G(x, y, t, τ ) admits the decomposition G(x, y, t, τ ) =. . . ψj x , y, t, τ uj x , t; x + R(x, y, t, τ ),. (). λ+ j <μ. where uj is defined by (), and μ–|α | λ–|γ | k l α γ |y | ∂ ∂ ∂ ∂ R(x, y, t, τ ) ≤ c(t – τ )–k–l–(n+|α|+|γ |)/ √|x | √ t τ x y t–τ t–τ –εγ d(y) κ(|y | + |x – y | ) × exp – |y| t–τ. ().

(19) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 14 of 37. √ for  < t – τ < T, |x | < t – τ , |α | ≤ , |α |, |γ |, |γ | ≤ , k, l ≤ . Here, εγ is the same constant as in Theorem .. The coefficients ψj (x , y, t, τ ) satisfy the estimate (). Proof Let ζ be a smooth function on the interval (, ∞), ζ (r) =  for r <  and ζ (r) =  for r > . Furthermore, let χ(x , t) = ζ (t –/ |x |) for x = (x , x ) ∈ D and t > . It follows from the equality. L , x , t, ∂x ,  R(x, y, t, τ ) = L , x , t, ∂x ,  G(x, y, t, τ ) . = L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ) + ∂t G(x, y, t, τ ) that. . L , x , t, ∂x ,  χ x , t – τ ∂τl ∂yγ R(x, y, t, τ ) = f (x, y, t, τ ) for t > τ , where. . f = χ x , t – τ ∂τl ∂yγ L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ) + ∂t G(x, y, t, τ ) . . . . ψj x , y, t, τ uj x , t; x . + L , x , t, ∂x ,  , χ x , t – τ ∂τl ∂yγ G(x, y, t, τ ) – γ. Here, [L , χ] = L χ – χL denotes the commutator of L and χ . Furthermore, ∂τl ∂y R(x, y, γ  t, τ ) =  for x ∈ ∂K . We estimate the Vp;β (K)-norm of the function χ(·, t – τ )∂τl ∂y R(·, x , y, t, τ ) for  – β – m/p = μ. By [, Theorem .], . χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ  τ y V. p;β (K). . ≤ cf ·, x , y, t, τ V . p;β (K). .. (). Here, the constant c is independent of x , y, t, τ . Indeed, by Lemma ., we have . χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ  τ y V. p;β (K). c ≤ √ L (, , ∂x , )χ(·, t – τ )∂τl ∂yγ R ·, x , y, t, τ V  (K) , p;β with a constant c independent of x , y, t, τ . Furthermore, under the condition (), the inequality . L , x , t, ∂x ,  – L (, , ∂x , ) χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ τ y. . ≤ c χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ .  (K) Vp;β.  (K) Vp;β. τ y. holds. Thus, . χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ  τ y V. p;β (K). √ ≤ c χ(·, t – τ )∂τl ∂yγ R ·, x , y, t, τ V . p;β (K). c + √ f ·, x , y, t, τ V  (K) , p;β .

(20) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 15 of 37.  which implies () if is sufficiently small. Next, we estimate the Vp;β -norm of f (·, x , y, t). By Theorem .,. . χ x , t – τ ∂ l ∂ γ L , x , t, ∂x ,  – L(x, t, ∂x ) G(x, y, t, τ ) τ y –l–(n+|γ |+λ+)/ λ–. . –ε . |y | x ≤ c(t – τ ) √ t–τ –εγ   d(y) κ(|y | + |x – y | ) × exp – |y | t–τ d(x) |x |. λ–|γ |. √ for  < t – τ < T, where λ < λ+ () – C . Here, λ can be chosen such that p(β + λ – ) > –m. Therefore, . . χ(·, t – τ )∂ l ∂ γ L , x , t, ∂x ,  – L ·, x , t, ∂x G ·, x , y, t, τ  τ y V. p;β (K). λ–|γ | |y | ≤ c(t – τ )–l+(β–n–|γ |–+m/p)/ √ t–τ –εγ  d(y) κ(|y | + |x – y | ) . × exp – |y | t–τ Analogously, we obtain . χ(·, t – τ )∂t ∂ l ∂ γ G ·, x , y, t  τ y V. p;β (K). ≤ c(t – τ ) ×. –l+(β–n–|γ |–+m/p)/. d(y) |y |. –εγ . . |y | √ t–τ. λ–|γ |. κ(|y | + |x – y | ) . exp – t–τ. Since [L (, x , t, ∂x ), χ(x , t – τ )]G(x, y, t, τ ) vanishes for |x | < obtain the estimate. √. √ t – τ and |x | >  t – τ , we. . . . L , x , t, ∂x ,  , χ x , t – τ ∂ γ G ·, x , y, t, τ y.  (K) Vp;β. λ–|γ | |y | ≤ c(t – τ )–l+(β–n–|γ |–+m/p)/ √ t–τ –εγ d(y) κ(|y | + |x – y | ) × exp – |y| t–τ  (K)by means of Theorem .. Using Lemma ., we get the same estimate for the Vp;β γ norm of the functions [L (, x , t, ∂x , ), χ(·, t – τ )]∂τl ∂y ψj (x , y, t, τ )uj (x , t; ·). Consequently, () implies. . χ(·, t – τ )∂ l ∂ γ R ·, x , y, t, τ  τ y V. p;β (K). λ–|γ | |y | ≤ c(t – τ )–l+(β–n–|γ |–+m/p)/ √ t–τ –εγ d(y) κ(|y | + |x – y | ) × exp – |y | t–τ. ().

(21) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 16 of 37. for  < t – τ < T. We prove an analogous estimate for the x - and t-derivatives of ∂τl ∂y R. Obviously, γ. . L , x , t, ∂x ,  χ x , t – τ ∂xj ∂τl ∂yγ R(x, y, t, τ ). . = ∂xj f (x, y, t, τ ) – ∂xj L , x , t, ∂x ,  χ x , t – τ ∂τl ∂yγ R(x, y, t, τ ) γ. for j ≥ m + , where f is the same function as above. Since, moreover, ∂xj ∂τl ∂y R(x, y, t, τ ) =  for x ∈ ∂K and j ≥ m + , we get . χ(·, t – τ )∂x ∂ l ∂ γ R ·, x , y, t, τ  j τ y V. ≤ c ∂xj f V . p;β. p;β (K). . l γ  (K) + χ(·, t – τ )∂τ ∂y R ·, x , y, t, τ V. p;β (K).  (K)-norms of ∂xj f can be estimated in the same way as f . This tofor j ≥ m + . The Vp;β gether with () leads to the estimate. . χ(·, t – τ )∂x ∂ l ∂ γ R ·, x , y, t, τ  j τ y V. p;β (K). ≤ c(t – τ ) ×. –l+(β–n–|γ |–+m/p)/. d(y) |y |. –εγ . . |y | √ t–τ. λ–|γ |. κ(|y | + |x – y | ) exp – t–τ. for j ≥ m + . Analogously, the inequality . χ(·, t – τ )∂ k ∂ α ∂ l ∂ γ R ·, x , y, t, τ  t x τ y V. p;β (K). ≤ c(t – τ )–k–l+(β–n–|α . d(y) × |y |. –εγ . |–|γ |–+m/p)/. . |y | √ t–τ. λ–|γ |. κ(|y | + |x – y | ) exp – t–τ. holds for |α | ≤  and k ≤ . Applying the estimate β–+|α |+m/p . x ∂ α v(x, y, t, τ ) ≤ cv ·, x , y, t, τ  x V. p;β (K). |α |≤. . for v(x, y, t, τ ) = χ(x , t – τ )∂tk ∂xα ∂τl ∂y R(x, y, t, τ ), p > m (cf. [, Lemma ..]), we get γ. μ–|α | λ–|γ | k l α γ |y | ∂ ∂ ∂ ∂ R(x, y, t, τ ) ≤ c(t – τ )–k–l–(n+|α|+|γ |)/ √|x | √ t τ x y t–τ t–τ –εγ d(y) κ(|y | + |x – y | ) × exp – |y| t–τ for |x | <. √. t – τ ,  < t – τ < T, |α | ≤ , |α |, |γ |, |γ | ≤ . This proves ().. .

(22) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 17 of 37. Comparing the representation () with the estimate (), we conclude that. √ inf λ+ x , t ≥ λ+ (, ) – C ,. (). x ,t. where C is the same constant as in Theorem ... 3.3 Asymptotics of the coefficients ψj (x , y, t, τ ) Let G (x , t; z, y, s) be the Green function of the first boundary value problem for the operator n. ∂ . ∂ – ai,j , x , t ∂s – L , x , t, ∂z = ∂s i,j= ∂zi ∂zj. with constant coefficients ai,j (, x , t) depending on the parameters x and t. This means that. . ∂s – L , x , t, ∂z G x , t; z, y, s = δ(z – y)δ(s) for z, y ∈ D, s ∈ R,. G x , t; z, y, s |s< = . G x , t; z, y, s =  for z ∈ ∂ D, y ∈ D, s ∈ R,. We write the operator L (, x , t, ∂z ) in the form. . L , x , t, ∂z = ∇zT A x , t ∇z + B x , t ∇z + ∇zT BT x , t ∇z + A x , t ∇z , where ∇z and ∇z denote the nabla operators in the z - and z -variables, respectively. As in Section , let U = U(x , t) and W = W (x , t) be square and continuously differentiable (with respect to x and t) matrices such that UA U T = Im and W (A – BT A – B)W T = In–m . By Theorem ., the function G admits the decomposition . G x , t; z, y, s =. . ψj, x , t; z , y, s uj x , t; z + R x , t; z, y, s. λ+ j (x ,t)<μ. if λ+ (x , t) < μ < λ+ (x , t) +  and μ = λ+j (x , t) for all j. Here,. ψj, x , t; z , y, s =. . . . vj x , t; x L , x , t, ∂x ,  G x , t; x , z , y, s dx. (). K. (cf. ()), the functions uj (x , t; ·) and vj (x , t; ·) are defined by (), and R satisfies the estimate in Theorem .. A more explicit formula for the function ψj, is. ψj,. +. . –λ (x ,t)–n/. π (m–n)/ (s) j x , t; z , y, s = uj x , t; y + / | det A| (λj + m/) q(x , t; y , z – y ) × exp – s . . (cf. ()), where A(x , t) is the coefficients matrix of the operator L (, x , t, ∂z ), and.  –  q x , t; y , y = Uy + W y + BT A y . ().

(23) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 18 of 37. is a quadratic form with respect to y and y satisfying the inequality (). We define j, (x , y, t, τ ) = ψj, (x , t; x , y, t – τ ), i.e.,. j,. –λ+ (x ,t)–n/. π (m–n)/ (t – τ ) j x , y, t, τ = uj x , t; y + / | det A| (λj + m/) q(x , t; y , x – y ) × exp – (t – τ ) . (). for x ∈ Rn–m , y ∈ D , τ < t. Theorem . The coefficients ψj (x , y, t, τ ) in Theorem . admit the decomposition. ψj x , y, t, τ = j, x , y, t, τ + rj x , y, t, τ , where rj satisfies the estimate λ . + κ(|y | + |x – y | ) rj x , y, t, τ ≤ c(t – τ )–(n–+λj (x ,t))/ √|y | exp – t–τ t–τ √ for  < t – τ < T, λ < λ+ () – C . Proof For shortness, we write λ+j instead of λ+j (x , t) in the proof of this theorem. Since (∂s – L (, x , t, ∂y ))G (x , t; x, y, s) =  for x, y ∈ D , s > , we have. . ∂s – L , x , t, ∂y ψj, x , t; z , y, s = . for y ∈ D , s > , x , z ∈ Rn–m , t ∈ R. This means that the function j, satisfies the equation. –∂τ – L , x , t, ∂y j, x , y, t, τ = . for y ∈ D, τ < t.. On the other hand, it follows from () that. . –∂τ – L∗ (y, τ , ∂y ) ψj x , y, t, τ =  for y ∈ D, τ < t.. Here L∗ denotes the formally adjoint differential operator to L. Consequently,. . . –∂τ – L∗ (y, τ , ∂y ) rj x , y, t, τ = L∗ (y, τ , ∂y ) – L , x , t, ∂y j, x , y, t, τ. for y ∈ D and τ < t. Furthermore, rj (x , y, t, t) =  for x ∈ Rn–m , y ∈ D . This follows from the representation. rj x , y, t, τ =. . . vj x , t; x L , x , t, ∂x ,  G(x, y, t, τ ) – G x , t; x, y, t – τ dx K. of the function rj = ψj – j, (cf. () and ()) and from the equality G(x, y, t, t) = G (x , t; x, y, ) = δ(x – y). Thus,. rj x , y, t, τ =. t τ. D. G(z, y, s, τ ) L∗ (z, s, ∂z ) – L , x , t, ∂z j, x , z, t, s dz ds..

(24) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 19 of 37. Since j, has the form (), we get ∗. L (z, s, ∂z ) – L , x , t, ∂z j, x , z, t, s . . α ≤c ∂zα j, x , z, t, s + z + z – x + t – s ∂z j, x , z, t, s |α|≤. |α|=. κ(|z | + |x – z | ) – –λ+ +(–n)/ λ+ z j exp – ≤ c(t – s) j t–s. for  < t – s < T, where c and κ are positive constants. The last estimate together with () implies . rj x , y, t, τ ≤. t τ. D. (s – τ ). –n/. (t – s). λ+ – –λ+ j +(–n)/ j. . |y | √ |y | + s – τ. z. λ κ|z – y| |z | exp – √ s–τ |z | + s – τ   κ(|z | + |x – z | ) dz ds × exp – t–s . λ. ×. √ for  < t – τ < T, where λ < λ+ () – C . Using the equalities |z – y | |z | |(t – τ )z – (t – s)y | |y | + = + s–τ t–s (t – τ )(t – s)(s – τ ) t–τ and |z – y | |z – x | |(t – τ )(z – x ) – (t – s)(y – x )| |y – x | + = + , s–τ t–s (t – τ )(t – s)(s – τ ) t–τ we obtain   t . – –λ+ +(–n)/ λ+ rj x , y, t, τ ≤ c exp – κ(|y | + |x – y | ) (s – τ )–n/ (t – s) j z j t–τ K τ λ λ |z | |y | |(t – τ )z – (t – s)y | × exp –κ √ √ (t – τ )(t – s)(s – τ ) |y | + s – τ |z | + s – τ |(t – τ )(z – x ) – (t – s)(y – x )| × dz dz ds. exp –κ (t – τ )(t – s)(s – τ ) Rn–m The inner integral over Rn–m is equal to (t – s)(n–m)/ (s – τ )(n–m)/ (t – τ )(m–n)/ . Substituting √ z = ξ t – s,. √ y = η t – τ ,. t – s = s (t – τ ),. s – τ =  – s (t – τ ),. we obtain   . + rj x , y, t, τ ≤ c(t – τ )–(λj +n–)/ exp – κ(|y | + |x – y | ) t–τ . × F ξ , η , s dξ ds , . K. ().

(25) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 20 of 37. where √ λ |ξ | s √ √ |ξ | s +  – s √ λ |η | κ|ξ – η s| × exp – . √ –s |η | +  – s. λ+ –. –(λ+ +)/. ( – s)–m/ ξ j F ξ , η , s = s j. . √ √ Let K = {ξ ∈ K : |ξ – η s| < |η | s} and K = K\K . We may assume that λ > . Then obviously . . |η | λ / (λ–λ+ –)/ λ+λ+ j – ξ F ξ , η , s dξ ds ≤ c s j dξ ds √ |η | +  K |ξ |<|η | s  λ λ+λ+ –+m |η | η j . ≤c |η | + . / . . √ √ √ If ξ ∈ K , then |ξ – η s| < |ξ | < |ξ – η s|. Therefore, the substitution ξ – η s = ζ yields . / . F ξ , η , s dξ ds. K. . |η | λ / (λ–λ+ –)/ λ+λ+ j – ≤c ζ s j exp –κ ζ dζ ds. |η | +   Rm. The number λ can be chosen such that λ+j (x , t) < λ +  for all x , t. Thus, . |η | λ F ξ , η , s dξ ds ≤ c . |η | +  K. / . . Next, we consider the integral of F(ξ , η , s) for the interval / ≤ s ≤ . Obviously, . . . /. F ξ , η , s dξ ds K. ≤c.  /. K. √ λ λ+ – κ|ξ – η s| |ξ | j |η | dξ ds. exp – √ ( – s)m/ |η | +  – s –s. We define √ K = ξ ∈ K : ξ – η s < η , √ K = ξ ∈ K : ξ – η s > η ,. K = K\K \K .. √ √ If ξ ∈ K and / ≤ s ≤ , then (  – )|η | < |ξ | < |η |. Thus, the substitution ξ – η s = √ ζ  – s yields . . /. K. F ξ , η , s dξ ds. λ+ – ≤ cη j.  /. |η | √ |η | +  – s. λ .  –κ ζ dζ ds. exp √ |ζ | –s<|η |.

(26) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 21 of 37. λ . |η | exp –κ ζ dζ ds √ √ |η | + s  |ζ | s<|η | |η | ∞ λ  |η | λ+ – j dζ ds + exp –κ ζ dζ ds ≤ c|η| √ √ s |η |  Rm |ζ | s<|η | λ+ (x ,t) . = cη j λ+ – ≤ cη j. . ∞. √ √ If ξ ∈ K and / ≤ s ≤ , then |η | < |ξ | and |ξ – η s| < |ξ | < |ξ – η s|. Substituting √ √ ξ – η s = ζ  – s, we get . . K. /. F ξ , η , s dξ ds. . . ≤c. . /. √ |ζ | –s>|η |. ( – s). (λ+ j –)/. . |η | √ |η | +  – s. λ. λ+ – –κ|ζ | ζ j e dζ ds.. We denote the integrand on the right-hand side of the last inequality by H(ζ , η , s). Obviously, –. . λ+ / λ+ H ζ , η , s ≤ cη ( – s) j ζ j exp –κ ζ for. √. √  – s < |η | and |ζ |  – s > |η |. On the other hand,. . . |η | λ λ+j – (λ+ –)/ ζ exp –κ ζ H ζ , η , s ≤ c( – s) j √ –s for. √.  – s > |η |. Consequently,. . . K. /. –. F ξ , η , s dξ ds ≤ cη . .  –|η |. ( – s). λ+ j /. λ+ (x ,t) ds = c η j. for |η | > /. For |η | < / we obtain . . K. /. F ξ , η , s dξ ds. – ≤ cη . .  –|η |. ( – s). λ+ j /. λ ds + cη . . –|η |. ( – s). (λ+ j –λ–)/. ds. /. λ+ λ . ≤ c η j + η logη . Finally, since |ξ | < |η | for ξ ∈ K , we get . . /. K. F ξ , η , s dξ ds. . λ+ – . / . ≤c . |ξ |<|η |. λ+λ+ +m– ≤ cη j. λ κ|η | |η | dξ ds exp – √ s |η | + s √ λ+ (x ,t) κ|η | (|η | + s)–λ ds = c η j exp – . m/ s s. |ξ | j sm/. . ∞.

(27) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 22 of 37. The above obtained estimates for the integrals of F(ξ , η , s) together with () imply   . + rj x , y, t, τ ≤ c(t – τ )–(n–+λj (x ,t))/ η λ exp – κ(|y | + |x – y | ) , t–τ √ where λ < λ+ () – C . This proves the desired estimate.. . 4 Asymptotics of solutions of the problem (1) Now, we consider the solution t u(x, t) =. D. . G(x, y, t, τ )f (y, τ ) dy dτ. of the problem (), where G(x, y, t, τ ) denotes the Green function introduced in the last section. We assume that the coefficients of the operator L(x, t, ∂x ) satisfy the same conditions  (D)), (), (), and () as in the foregoing section and that f ∈ Lp;β (DT ) = Lp (, T; Vp;β where p, β are such that μ =  – β – m/p satisfies the inequalities (). Then by Theorem ., the function G has the representation G(x, y, t, τ ) =. . ψj x , y, t, τ uj x , t; x + R(x, y, t, τ ). λ+ j <μ. with a remainder R(x, y, t, τ ) satisfying the estimate (). Let ζ be an infinitely differentiable function on R+ = (, ∞) which is equal to one on the interval (, ) and to zero on (, ∞). Furthermore, we define . |x | χ x , y = ζ , |y |. . |x | . χ x , t, τ = ζ √ t–τ. Obviously, u(x, t) =. . Hj (x, t)uj x , t; x + v(x, t),. (). λ+ j <μ. where t. . . χ x , y χ x , t, τ ψj x , y, t, τ f (y, τ ) dy dτ. (). t . v(x, t) = ψj x , y, t, τ uj x , t; x f (y, τ ) dy dτ . G(x, y, t, τ ) – χ χ. (). Hj (x, t) = . D. and. . D. λ+ j <μ. We estimate the remainder v and the coefficients Hj in the decomposition ().. 4.1 An estimate for a weighted Lp Sobolev norm of the remainder l,l Let l be a nonnegative integer, and let p, β be real numbers, p > . Then the space Wp;β (D T ) is defined as the set of all functions u(x, t) on DT = D × (, T) with finite norm (). An.

(28) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 23 of 37. equivalent norm is

(29) . T. u =. /p l pβ k α p p(β–l+k) k p x ∂ ∂ u + x ∂ u dx dt D k=. . t. x. t. |α|=l–k. (see, e.g., [, Lemma ..]). In order to estimate the first order x-derivatives of the remainder v, we employ the following lemma (cf. [, Lemma A.]). Lemma . Let K be the integral operator (Kf )(x, t) =. t D. . K(x, y, t, τ )f (y, τ ) dy dτ. with a kernel K(x, y, t, τ ) satisfying the estimate . |x | √ |x | + t – τ –κ|x – y| exp t–τ. a+r . |K| ≤ c(t – τ )–(n+–r)/ ×. |x |β–r |y |β. |y | √ |y | + t – τ. b. for  < τ < t < T and x, y ∈ D , where κ > ,  < r ≤ , a + b > –m, – mp – a < β < m – Then K is bounded on Lp (DT ).. m p. + b.. Analogously to [, Lemma .], we prove the following lemma. Lemma . Suppose that f ∈ Lp;β (DT ), where p and β are such that μ =  – β – m/p satisfies (). Furthermore, let v be the function (). Then ∂xα v ∈ Lp;β–+|α| (DT ) for |α| ≤  and ∂ α v. Lp;β–+|α| (DT ). x. |α|≤. ≤ cf Lp;β (DT ). with a constant c independent of f . Proof Obviously,. v=.  t j=. . D. Vj (x, y, t, τ )f (y, τ ) dy dτ ,. where. V (x, y, t, τ ) = χ x , t, τ R(x, y, t, τ ),. V (x, y, t, τ ) =  – χ x , t, τ G(x, y, t, τ ) and. . . V (x, y, t, τ ) =  – χ x , y χ x , t, τ ψj x , y, t, τ uj x , t; x . λ+ j <μ.

(30) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 24 of 37. Using Theorem ., we obtain the estimate . μ–|α|–ε |x | √ |x | + t – τ λ κ|x – y| |y | × exp – √ t–τ |y | + t – τ. α ∂ V (x, y, t, τ ) ≤ c(t – τ )–(n+|α|)/ x. √ for  < t – τ < T and |α| ≤ , where  < λ < λ+ () – C and ε is a sufficiently small positive number. The same estimate holds for ∂xα V and ∂xα V by means of Theorem . and Lemma ., respectively. Consequently by Lemma ., the integral operators with the kernels |x |β–+|α| |y |–β ∂xα Vj (x, y, t, τ ) are bounded in Lp (DT ) for |α| ≤ , j = , , . This proves the lemma. Next, we estimate the Lp;β norm of (∂t – L)v. Lemma . Suppose that f ∈ Lp;β (DT ), where p and β are such that μ =  – β – m/p satisfies the condition (). Then the function () satisfies the estimate . ∂t – L(x, t, ∂x ) v. Lp;β (DT ). ≤ cf Lp;β (DT ). with a constant c independent of f . Proof By the definition of v, we have. Hj (x, t)uj x , t; x . ∂t – L(x, t, ∂x ) v(x, t) = f (x, t) – ∂t – L(x, t, ∂x ) λ+ j <μ. Here,. ∂t – L(x, t, ∂x ) Hj (x, t)uj x , t; x t . . . . ∂t – L(x, t, ∂x ) χ x , y χ x , t, τ ψj x , y, t, τ uj x , t; x f (y, τ ) dy dτ . = –∞. D. By Lemma ., . . . . ∂t χ x , y χ x , t, τ ψj x , y, t, τ uj x , t; x λ+j (x ,t) λ κ(|y | + |x – y | ) |x | |y | ≤ c(t – τ )––n/ √ , exp – √ t–τ t–τ t–τ √ where λ is an arbitrary positive number less than λ+ () – C . Using the fact that |x | < √  t – τ on the support of χ , we obtain β –β . . . . x y ∂t χ x , y χ x , t, τ ψj x , y, t, τ uj x , t; x λ+λ+j (x ,t)+ β–λ– κ|x – y| |x | |x | ≤ c(t – τ )–n/ . exp – √ |y |β–λ t–τ |x | + t – τ.

(31) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 25 of 37. By () and (), we have inf λ+j > μ – . Therefore, we can apply Lemma . (with a = λ + inf λ+j and b = ) and conclude that the operator with the kernel |x |β |y |–β ∂t χ χ ψj uj is bounded in Lp (DT ). Furthermore, we obtain the estimate β –β . . . . . x y L(x, t, ∂x ) – L , x , t, ∂x ,  χ x , y χ x , t, τ ψj x , y, t, τ uj x , t; x ≤ c(t – τ ). –n/. |x | √ |x | + t – τ. λ+λ+j (x ,t)+. κ|x – y| |x |β–λ– exp – |y |β–λ t–τ. by means of Lemma .. Again Lemma . (with a = λ + inf λ+j –  and b = ) implies the boundedness of the integral operator with the kernel |x |β |y |–β (L(x, t, ∂x ) – L (, x , t, ∂x , ))χ χ ψj uj . Using the equality L (, x , t, ∂x , )uj (x , t; x ) = , one can show analogously that the integral operator with the kernel |x |β |y |–β L (, x , t, ∂x , )χ χ ψj uj is bounded in Lp (DT ). Hence the mapping. Lp;β (DT ) f → ∂t – L(x, t, ∂x ) Hj (x, t)uj x , t; x ∈ Lp;β (DT ) . is bounded. This proves the lemma.. For the estimation of the second order derivatives of v, we need the following lemma. Lemma . Let u be a solution of the problem (). If u ∈ Lp;β– (DT ), ∂xj u ∈ Lp;β– (DT ) for , (DT ) and j = , . . . , n and f ∈ Lp;β (DT ), then u ∈ Wp;β

(32) uW , (DT ) ≤ c f Lp;β (DT ) + uLp;β– (DT ) + p;β. n . ∂xj uLp;β– (DT ) ,. (). j=. where c is independent of u. Proof Let ζν be infinitely differentiable functions on D depending only on r = |x | such that supp ζν ⊂ x : ν– < r < ν+ ,. +∞ . ζν = ,. α ∂ ζν (x) ≤ cα –ν|α| x. ν=–∞. for all α, where cα is independent of ν and x. Then ζν u satisfies the equations. ∂t – L(x, t, ∂x ) ζν u = fν. ζν u = . in D × (, T),. on ∂ D × (, T),. ζν u|t= = .. where fν = ζν f – [L(x, t, ∂x ), ζν ]u. By [, Theorem .], the operator ∂t – x of the heat equation realizes an isomorphism from the space , u ∈ Wp;γ (DT ), u =  on ∂ DT , u(x, t) =  for t =  onto Lp;γ (DT ) for γ + m/p =  + m/. Using the coordinate transformation (), we obtain the same result for the operator ∂t – L (, ∂x ). Under the condition () on the coefficients.

(33) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 26 of 37. , of L(x, t, ∂x ), the operator L(x, t, ∂x ) – L (, ∂x ) is small in the operator norm Wp;γ (D T ) → Lp;γ (DT ). Consequently, the function ζν u satisfies the estimate. ζν uWp;γ , (DT ) ≤ cfν Lp;γ (DT ) with a constant c independent of f and ν. Multiplying this inequality by ν(β–γ ) , we obtain ζν uW , (DT ) ≤ cfν Lp;β (DT ) p;β. (). with a constant c independent of u and ν. Obviously,. fν Lp;β (DT ) ≤ ζν f Lp;β (DT ) + c ην uLp;β– (DT ) + ην ∇uLp;β– (DT ) , where ην = ζν– + ζν + ζν+ and c is a constant independent of f and ν. Hence, () implies p. ζν u. , (D T ) Wp;β. p p p ≤ c ζν f Lp;β (DT ) + ην uLp;β– (DT ) + ην ∇uLp;β– (DT ) .. Summing up over all ν, we get ().. . Using the last three lemmas, we can easily prove the following theorem. Theorem . Suppose that f ∈ Lp;β (DT ), where p and β are such that μ =  – β – m/p satisfies the condition (). Then the solution u of the problem () admits the decomposition , () with a remainder v ∈ Wp;β (DT ). The coefficients Hj (x, t) depend only on |x |, x , t, and satisfy the estimates Hj Lp;β+λ+ – (DT ) ≤ cf Lp;β (DT ). (). l α ∂ ∂ Hj t x L. (). j. and (D T ) p;β+λ+ j +l+|α|–. ≤ cf Lp;β (DT ). for  ≤ l + |α| ≤ . The constant c in () and () is independent of f . Proof By Lemma ., the solution u has the representation (), where ∂xα v ∈ Lp;β–+|α| (DT ) for |α| ≤ . Furthermore, by Lemma ., (∂t – L(x, t, ∂x ))v ∈ Lp;β (DT ). Applying Lemma ., , (DT ) and we conclude that v ∈ Wp;β vW , (DT ) ≤ cf Lp;β (DT ) . p;β. In order to prove (), we have to show that the integral operator with the kernel β+λ+ +l+|α|– –β l α . . y ∂ ∂ χ x , y χ x , t, τ ψj x , y, t, τ K(x, y, t, τ ) = x j t x is bounded in Lp (DT ). Using the estimates l α . ∂t ∂ χ x , y χ x , t, τ ≤ cx –|α | (t – τ )–l x.

(34) Kozlov and Rossmann Boundary Value Problems 2014, 2014:252 http://www.boundaryvalueproblems.com/content/2014/1/252. Page 27 of 37. and l α . + ∂t ∂ ψj x , y, t, τ ≤ c(t – τ )–l –(n+|α |+λ+λj (x ,t))/ y λ x κ(|y | + |x – y | ) × exp – t–τ (cf. Lemma .), we obtain l+|α |+λ+λ+j (x ,t) β–λ– |x | K(x, y, t, τ ) ≤ c(t – τ )–n/ √|x | |y |β–λ t–τ κ(|y | + |x – y | ) . × exp – t–τ √ Since |x | ≤  t – τ on the support of χ , we can replace the term √|xt–τ| by |x |+|x√|t–τ . Applying Lemma ., we get the boundedness of the integral operator with the kernel K(x, y, t, τ ). This proves (). The estimate () holds analogously. . 4.2 On the coefficient in the asymptotics We consider the coefficients Hj in () and their traces. hj x , t =. t . D. ψj x , y, t, τ f (y, τ ) dy dτ. (). on M × (, T). In the next lemma, we show that hj belongs to the anisotropic SobolevSlobodetski˘ı space Wps,s/ (Rn–m × (, T)) with the norm hW s,s/ (Rn–m ×(,T)) p. =. T. . h(·, t)p s n–m dt + W (R ) p. Rn–m. p h x , · s/ dx W ((,T)) p. /p ,. where s is a certain function on Rn–m × (, T) between  and . Lemma . Suppose that f ∈ Lp;β (DT ), where p and β are such that μ =  – β – m/p satisfies the condition (). Then the trace hj of the function () belongs to the space Wps,s/ (Rn–m × (, T)), where s(x , t) =  – β – λ+j (x , t) – m/p, and it satisfies the estimate hj W s,s/ (Rn–m ×(,T)) ≤ cf Lp;β (DT ) p. (). for λ+j < μ. Moreover, t –s/ hj ∈ Lp (Rn–m × (, T)) and . T. Rn–m. t –ps(x. ,t)/. p hj x , t dx dt ≤ cf p Lp;β (DT ). with a constant c independent of f .. ().

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