1Introduction VuTrongLuong and NguyenThiHue OntheasymptoticofsolutiontotheDirichletproblemforhyperbolicequationsincylinderswithedges 10 ,1–15; ElectronicJournalofQualitativeTheoryofDifferentialEquations2014,No.

On the asymptotic of solution to the Dirichlet problem for hyperbolic equations in cylinders with edges

Vu Trong Luong

and Nguyen Thi Hue

1Taybac University, Quyet Tam, Son La, Vietnam

2Saodo University, Chi Linh, Hai Duong, Vietnam

Received 31 August 2013, appeared 21 March 2014 Communicated by László Simon

Abstract. In this paper, we consider the Dirichlet problem for second-order hyperbolic equations whose coefficients depend on both time and spatial variables in a cylinder with edges. The asymptotic behaviour of the solution near the edge is studied.

Keywords: asymptotic behaviour of solutions, regularity of solutions, domains with edges, hyperbolic equations.

2010 Mathematics Subject Classification:35B65, 35B40, 35D30, 35L10.

1 Introduction

Let Ωbe a bounded domain inRⁿ, n ≥ 2, with the boundary∂Ωconsisting of two surfaces Γ1, Γ2 which intersect along a manifoldl₀. Assume that in a neighbourhood of each point of l0 the set Ω is diffeomorphic to a dihedral angle. Assume that in a neighbourhood of each point of l₀ the set Ω is diffeomorphic to a dihedral angle. For any P ∈ l₀, two half-spaces T₁(P), and T₂(P) tangent toΩ, and a two-dimensional plane π(P) normal to l₀ are defined.

We denote by ν(P)the angle in the plane π(P)(on the side ofΩ) bounded by the rays R₁ = T₁(P)∩π(P), R₂=T₂(P)∩π(P)and byβ(P)the aperture of this angle.

In the cylinder Q = _Ω×_R, we study a class of second-order hyperbolic equations. The Dirichlet boundary condition is given on the boundary∂Q= _∂Ω×_Rof the cylinder. Our goal is to describe the behaviour of the solutions near the edges. There are some approaches to this issue. For systems or equations dealt with in [13,6, 12] whose coefficients are independent of the time variable, B. A. Plamenevssky used Fourier transform to reduce the problem to an elliptic one with a parameter. In contrast to [13] and [12], in this paper, we consider equations with coefficients depending on both of time and spatial variables. We develop the approach suggested in [2] to demonstrate the asymptotic representation of the solution of the problem mentioned above near the edges. Furthermore, we investigate the unique solvability of the problem and the regularity of solutions in weighted Sobolev spaces.

BCorresponding author. Email: luongvt@utb.edu.vn

Let

L(x,t,∂)u= −

∑

∂

∂x_i

a_ij(x,t)^∂u

∂x_j

+

∑

b_i(x,t)^∂u

∂x_i +c(x,t)u,

be a second-order partial differential operator, wherea_ij(x,t),b_i(x,t)andc(x,t)are real-valued functions on Q belonging to C^∞(Q). Moreover, suppose that a_ij = a_ji, i,j = 1, . . . ,n, are continuous inx∈_Ωuniformly with respect tot∈_Rand

∑

a_ij(x,t)ξ_iξ_j ≥µ₀|ξ|² (1.1) for all ξ ∈ _Rⁿ\{0} and (x,t) ∈ Q, µ₀ = const > 0. In the present work, we consider the Dirichlet problem

u_tt+L(x,t,∂)u= f inQ, (1.2)

u|_∂Q =0. (1.3)

Let us introduce some functional spaces used in this paper. We denote by H^l(_Ω)and ˚H^l(_Ω) the usual Sobolev spaces as in [1]. Letα ∈ R, we introduce the space H_α^l(_Ω)as the weighted Sobolev space of all functionsudefined onΩwith the norm

kuk²_Hl

α(_Ω) =

∑

0≤|p|≤l

Z

Ω

r^{2(α+|p|−l)}|D^pu|²+|u|²dx,

in whichr²= x²₁+x²₂, andD^pu=∂^|p|u/∂x₁^p1. . .∂x^p_nⁿ,p= (p₁, . . . ,p_n).

ByH^l,k(Q,γ),H_α^l,k(Q,γ) (γ∈ _R)we denote the weighted Sobolev spaces of functionsudefined onQwith the norms

kuk²_Hl,k(Q,γ) =

Z

Q

0≤|

∑

|D^pu|²+

∑

|u_tj|²

!

e^−γtdx dt <+_∞

and

kuk²

H_α^l,k(Q,γ) =

Z

Q

0≤|

∑

r^{2(α+|p|−l)}|D^pu|²+

∑

|u_tj|²

!

e^−γtdx dt <+_∞, whereu_tk = ^∂ku

∂t^k. The space ˚H^l,k(Q,γ)is the closure ofC₀^∞(Q)inH^l,k(Q,γ).

Finally, denote byH_α^l(Q,γ)the space of functionsu(x,t)defined onQwith the norm kuk²

H^l_α(Q,γ)=

∑

0≤|p|+k≤l

Z

Q

r^{2(α+|p|+k−l)}|D^pu_tk|²+|u|²e^−γtdx dt.

Let us denote B(u,v;t) =

∑

Z

Ω

a_ij(·_,t)^∂u

∂x_j

∂v

∂x_i dx+

∑

Z

Ω

b_i(·_,t)^∂u

∂x_iv dx+

Z

Ω

c(·_,t)uv dx,

the time-dependent bilinear form. Applying condition (1.1) and similar arguments as the proof of the Gårding inequality, it follows that

B(u,u;t)≥ µ₀kuk²_H1(Ω)−λ₀kuk²_L

2(_Ω), for a.e.t ∈_R (1.4)

for allu(x,t)∈ H^˚1,1(Q,γ), whereµ₀=const>0, λ₀ =const≥0.

We denote by(·,·)the inner product in L2(_Ω). Let f ∈ H_α⁰(Q,γ),γ > 0, a function u(x,t) is called the generalized solution in H^1,1(Q,γ) of problem (1.2)–(1.3) if and only if u(x,t) ∈ H˚^1,1(Q,γ), and for anyT>0 the equality

−

Z _T

−_∞(ut,vt)dt+

Z _T

−_∞B(u,v;t)dt=

Z _T

−_∞(f,v)dt, (1.5) holds for allv ∈ H^˚1,1(Q,−γ),v(x,t) =0,t≥ T.

The problems for nonstationary systems or equations in nonsmooth domains also have been investigated in [2,3,4,5,10,11], in which the authors obtain results on the regularity of solu- tions in weighted Sobolev spaces and asymptotic behaviour of solutions in the neighbourhood of the conical points. However, the problems are considered in domains with conical points and with initial conditions. Different from the above-mentioned papers, we consider the prob- lem without initial conditions in domains with edges. The paper is organized as follows. In Section 2, we present the results on the unique solvability of the problem. The regularity of the generalized solution is stated in Section 3. The main result, Theorem4.2, is given in Section 4.

2 The unique solvability

In this section, we will establish the unique solvability and the regularity in time variable of the solution for problem (1.2)–(1.3). Furthermore, some energy estimates of the solution are proven. The solvability condition of problem (1.2)–(1.3) is: the right hand side f(external force) of (1.2) belongs toH_α⁰(Q,γ)_whereγis a sufficiently large positive number. This condition is more applicable than f ∈ L₂(Q,γ), becauseL₂(Q,γ),→H_α⁰(Q,γ),α∈[0, 1].

Theorem 2.1. Suppose that f,ft∈ H_α⁰(Q,γ), γ>0, α∈ [0, 1], and the coefficients of the operator L satisfy

sup{|a_ij|,|a_ijt|,|b_i|,|c|:i,j=1, . . . ,n;(x,t)∈Q} ≤µ,µ=const.

Then for anyγ > γ₀ = ^2µ+e

min{1;2µ0−e},e∈ (0, 2µ₀), problem(1.2)–(1.3)has a generalized solution u in the spaceH˚^1,1(Q,γ)and

kuk²_H1,1(Q,γ) ≤C

kfk²_H0

α(Q,γ)+kftk²_H0 α(Q,γ)

, (2.1)

where C is a constant independent of u and f .

To prove the theorem, we construct an approximate sequenceu^hof solutionuof the prob- lem (1.2)–(1.3). It is known that there is a smooth functionχ(t)which is equal to 1 on[1,+_∞), is equal to 0 on (−_∞, 0]and assumes values from [0, 1]on [0, 1](see [14, Thm. 5.5] for more details). Moreover, we can suppose that all derivatives ofχ(t)are bounded. Leth ∈ (−_{∞, 0}] be an integer. Set f^h(x,t) =χ(t−h)f(x,t)then

f^h =

(f ift ≥h+_1, 0 ift ≤h.

Moreover, if f ∈H_α⁰(Q,γ), then f^h ∈ H_α⁰(Q_h,γ), f^h ∈ H_α⁰(Q,γ),Q_h= _Ω×(h,∞), and kf^hk²_H0

α(Qh,γ)=kf^hk²_H0

α(Q,γ)≤ kfk²_H0

α(Q,γ),

whereH_α⁰(Q_h,γ)asH⁰_α(Q,γ), replacingQbyQ_h. Fixed f ∈ H_α⁰(Q,γ), we consider the follow- ing problem in the cylinderQ_h:

u_tt+L(x,t,∂)u= f^h(x,t)_inQ_h, (2.2)

u=_{0 on}S_h =∂Ω×(h,∞)_{, (2.3)}

u|_t=h=0,u_t |_t=h=0 onΩ. (2.4) This is the initial boundary value problem for hyperbolic equations in cylindersQ_h. A function u = u(x,t)is called the generalized solution in the space H^1,1(Q_h,γ)of problem (2.2)–(2.4) if and only ifu∈ H^˚1,1(Q_h,γ),u(x,h) =0, and for anyT >0 the equality

−

Z _T

h

(ut,vt)dt+

Z _T

h B(u,v;t)dt=

Z _T

h

(f,v)dt, (2.5)

holds for allv∈ H^˚1,1(Q_h,−γ), v(x,t) =0, t≥T.

Lemma 2.2. Suppose that the assumption of Theorem2.1 is satisfied. For any h fixed, there exists a solution u^hin the spaceH˚^1,1(Q_h,γ)of the problem(2.2)–(2.4)and the following estimate holds

ku^hk²_H1,1(Q

h,γ) ≤C

kf^hk²_H0

α(Qh,γ)+kf_t^hk²_H0 α(Qh,γ)

, (2.6)

where C is a constant independent of h.

Proof. We will prove the existence by Galerkin’s approximating method. Let{ω_k(x)}^∞_k=1be an orthogonal basis of ˚H¹(_Ω)which is orthonormal inL₂(_Ω). Put

u^N(x,t) =

∑

C_k^N(t)ω_k(x)

whereC_k^N(t),k=1, . . . ,N, is the solution of the following ordinary differential system:

(u_tt^N,ω_k) +B(u^N,ω_k;t) = (f^h,ω_k), k=1, . . . ,N, (2.7) with the initial conditions

C_k^N(h) =0, C_kt^N(h) =0, k=1, . . . ,N. (2.8) Let us multiply (2.7) byC_kt^N(t), then take the sum with respect tokfrom 1 toNto arrive at

(u_tt^N,u^N_t ) +B(u^N,u^N_t ;t) = (f^h,u^N_t ). Since(u_tt^N,u^N_t ) = _dt^{d 1}₂ku^N_t k²

L2(_Ω)

, we get d

dt

ku_t^Nk²_L

2(_Ω)

+2B(u^N,u_t^N;t) =2(f^h,u^N_t ). Integrating the equality above fromhtotwe find that

ku_t^N(t)k²_L

2(_Ω)+2 Z _t

h B(u^N,u_t^N;τ)dτ=2 Z _t

h

(f^h,u_t^N)dτ. (2.9)

Let us evaluate the right-hand side of (2.9). Integrating by parts, we have 2

Z _t

h

(f^h,u_t^N)dτ=2(f^h(t),u^N(t))−2 Z _t

h

(f_t^h,u^N)dτ.

By the Cauchy–Schwarz inequality and the Hardy inequality, for an arbitrary positive number α∈[_{0, 1}], it follows from the equality above that

2

Z _t

h

(f^h,u^N_t )dτ

≤2kr^αf^hk_L2(Ω)kr^−αu^Nk_L2(Ω)+2 Z _t

h

kr^αf_t^hk_L2(Ω)kr^−αu^Nk_L2(Ω)dτ

≤Ckf^hk_H0

α(_Ω)ku^Nk_H1(_Ω)+C Z _t

h

kf_t^hk_H0

α(_Ω)ku^Nk_H1(_Ω)dτ

≤C(e)kf^hk²_H0

α(_Ω)+eku^Nk²_H1(_Ω)+

Z _t

h C(e)kf_t^hk²_H0

α(_Ω)+eku^Nk²_H1(_Ω)dτ,

(2.10)

wheree>0 andC(e)is a constant independent ofN,h.

We consider the second term in the left-hand side of (2.9), we can write B(u^N,u_t^N;t) =

∑

Z

Ω

a_ij∂u^N

∂x_j

∂u^N_t

∂x_i dx+

∑

Z

Ω

b_i∂u^N

∂x_i u_t^Ndx+

Z

Ω

cu^Nu^N_t dx

=: B₁+B₂.

(2.11)

It is easy to see that

B₁= ^d dt

1

2A[u^N,u^N,t]−¹ 2

Z

Ω

∑

a_ijt∂u^N

∂x_j

∂u^N

∂x_i dx, (2.12)

for the symmetric bilinear form

A[_u^N_,u^N_,t] =

Z

Ω

∑

a_ij∂u^N

∂x_j

∂u^N

∂x_i dx.

The equality (2.12) implies Z _t

h B₁dτ≥ µ₀ku^N(t)k²_H1(_Ω)−µ/2 Z _t

h

ku^Nk²_H1(_Ω)dτ, (2.13) and we also note

Z _t

h B₂dτ

≤µ/2^Z t h

ku^Nk²_H1(_Ω)+ku_t^Nk²_L

2(_Ω)dτ

. (2.14)

Combining estimates (2.10), (2.13) and (2.14), we obtain ku_t^N(t)k²_L

2(_Ω)+ku^N(t)k²_H1(Ω) ≤ ^2µ+e min{_{1; 2µ0}−e}

Z _t

h

ku^N_t k²_L

2(_Ω)+ku^Nk²_H1(Ω)dτ +C

kf^hk²_H0

α(_Ω)+

Z _t

h

kf_t^hk²_H0 α(_Ω)dτ

,

(2.15)

where we used (2.12) and 0<e<_2µ0.

Thus the Gronwall–Bellman inequality yields the estimate ku_t^N(t)k²_L

2(_Ω)+ku^N(t)k²_H1(Ω)

≤C

kf^h(t)k²_H0

α(_Ω)+

Z _t

h

kf_t^h(τ)k²_H0

α(_Ω)dτ

+Cγ₀

t

Z

h

e^γ0(t−τ)

kf^h(τ)k²_H0 α(_Ω)+

Z _τ

h

kf_t^h(s)k²_H0 α(_Ω)ds

dτ,

(2.16)

whereγ₀ = ^2µ+e

min{1;2µ0−e}, e ∈ (0, 2µ₀). Now multiplying both sides of this inequality by e^−γt, γ>γ₀, then integrating them with respect totfromhto∞, we get

ku^Nk²_H1,1(Q_h,γ) ≤C Z _∞

h e^−γtkf^h(t)k²_H0

α(_Ω)dt+

Z _∞

h e^−γt Z _t

h

kf_t^h(τ)k²_H0

α(_Ω)dτdt

+Cγ₀ Z _∞

h e^−γt Zt

h

e^γ0(t−τ)kf^h(τ)k²_H0

α(_Ω)dτdt +Cγ₀

Z _∞

h e^−γt

t

Z

h

e^γ0(t−τ) Z _τ

h

kf_t^h(s)k²_H0

α(_Ω)ds dτdt.

By the Fubini theorem andγ>γ₀, we conclude that ku^Nk²_H1,1(Qh,γ) ≤C

kf^hk²_H0

α(Q_h,γ)+kf_t^hk²_H0 α(Q_h,γ)

, (2.17)

whereCis a constant.

From the inequality (2.17), by standard weak convergence arguments, we can conclude that the sequence{u^N}^∞_N=1possesses a subsequence convergent to a functionu^h∈ H^˚1,1(Q_h,γ), which is a generalized solution of problem (2.2)–(2.4).

Proof of Theorem2.1: Letk be a integer less thanh, denoteu^k a generalized solution of the problem (2.2)–(2.4) with the replacement ofhbyk. We defineu^h in the cylinderQ_k by setting u^h(x,t) =_{0 for}k ≤t ≤ h. Putu^hk = u^h−u^k, f^hk = f^h− f^k, sou^hkis the generalized solution of the following problem

u^hk_tt +L(x,t,∂)u^hk= f^hk(x,t)inQ_k, u^hk =0 onS_k, u^hk|_t=k=0,u^hk_t |_t=k=0 onΩ.

According to Lemma2.2, we have ku^hkk²_H1,1(Q,γ) =ku^hkk²_H1,1(Q

k,γ)≤C kf^h− f^kk²_H0

α(Q_k,γ)+kf_t^h− f_t^kk²_H0

α(Q_k,γ)

. It is easily seen that

kf^h− f^kk²_H0

α(Q,γ)=kf^h− f^kk²_H0

α(Q_k,γ) =

Z _h+1

k e^−γtkf^h− f^kk²_H0 α(_Ω)dt

≤2 Z _h+1

k e^−γtkfk²_H0

α(_Ω)dt.

As f ∈ H_α⁰(Q,γ)_{, we have}

Z _h+1

k e^−γtkfk²_H0

α(_Ω)dt→0, h,k → −_∞.

Therefore,

limkf^h− f^kk²_H0

α(Q,γ) =0, h,k→ −_∞.

Repeating this argument, we get

kf_t^h− f_t^kk²_H0

α(Q,γ) →0, h,k→ −_∞.

This shows that {u^h}is a Cauchy sequence in ˚H^1,1(Q,γ). Hence, {u^h}is convergent to uin H˚^1,1(_Q,γ)_{. Since u}^h is a generalized solution of problem (2.2)–(2.4), for any T > _{0, we have} that the equality

−

Z _T

h

(u_ht,v_t)dt+

Z _T

h B(u_h,v;t)dt=

Z _T

h

(f^h,v)dt, (2.18) holds for allv ∈ H^˚1,1(Q,−γ)_, v(x,t) = _0, t ≥ T. Using (2.18) whenh → −∞, we obtain (1.5).

It means thatuis a generalized solution of the problem (1.2)–(1.3). Using (2.6), we get ku^hk²_H1,1(Q,γ) ≤C

kfk²_H0

α(Q,γ)+kf_tk²_H0

α(Q,γ)

.

From this inequality, sendingh → −∞, we get (2.1). The proof of the theorem is completed.

Theorem 2.3. If γ > 0and|a_ijt|,|b_i|,b_ixi|,|c| ≤ µ₁e^2γt, a.e.(x,t) ∈ Q,µ₁ = const,then problem (1.2)–(1.3)has no more than one solution inH˚^1,1(Q,γ).

Proof. It suffices to prove that the only solution of (1.2)–(1.3) with f ≡0 isu≡0. To verify this, for anyT>0, fix 0≤s≤ Tand set

v(x,t) =





 Z _t

s u(x,τ)dτ, −_∞<t <s

0, s ≤t< +_∞.

Thenv∈ H^˚1,1(Q,−γ)for anyγ>0. From the definition of generalized solution, we get

−

Z _s

−_∞(ut,vt)dt+

Z _s

−_∞B(u,v;t)dt=0.

Asv_t= −u(t≤s)_{, so}

Z _s

−_∞(ut,u)dt−

Z _s

−_∞B(vt,v;t)dt=0.

Therefore, Z _s

−_∞

d dt

1 2kuk²_L

2(_Ω)−¹

2A[v,v;t]

dt= −

Z _s

−_∞C(u,v;t) +D(v,v;t)dt, where

C(u,v;t) =

Z

Ω

∑

b_iuv_xi+b_ixiuv dx−

Z

Ωcuv dx and

D(v,v;t) = ¹ 2

Z

Ω

∑

a_ijtv_xiv_xj dx.

Hence, 1

2ku(s)k²_L

2(_Ω)+ ¹ 2 lim

t→−_∞A[v(t),v(t);t] =−

Z _s

−_∞C(u,v;t) +D(v,v;t)dt.

Using inequality (1.1) and the Cauchy inequality, we arrive at ku(s)k²_L

2(_Ω)+ lim

t→−_∞kv(t)k²_H1(Ω) ≤C Z _s

−_∞e^2γt

ku(t)k²_L

2(_Ω)+kv(t)k²_H1(Ω)dt. (2.19) Let us write

w(t) =

Z _t

−_∞u(x,τ)dτ,t ≤s, then

t→−lim_∞v(t) =−w(s),v(t) =w(t)−w(s). It follows readily from (2.19) that

ku(s)k²_L

2(_Ω)+ (1−Ce^2γs)kw(s)k²_H1(_Ω) ≤C Z _s

−_∞e^2γt

ku(t)k²_L

2(_Ω)+kw(t)k²_H1(_Ω)

dt.

ChooseT1so small that 1−Ce^2γT1 ≥1/2, then we have ku(s)k²_L

2(_Ω)+kw(s)k²_H1(_Ω) ≤2C Z _s

−_∞e^2γt

ku(t)k²_L

2(_Ω)+kw(t)k²_H1(_Ω)

dt

for alls ≤ T₁. Consequently the Gronwall inequality implies u ≡ 0 on(−_∞,T₁]. In view of the uniqueness of the solution of the problem with initial condition (2.2)–(2.4),u ≡0 holds on R.

By the same arguments as in the proof of Theorem2.1together with inductive arguments (cf. [2]), we obtain the following theorem:

Theorem 2.4. Let h∈ _N^∗, assume that

(i) sup{|a_ijtk+1|,|b_itk|,|c_tk|:i,j=1, . . . ,n;(x,t)∈Q,k≤ h} ≤µ, (ii) f_tk ∈ H_α⁰(Q,γ₀), k≤ h+1.

Then for an arbitrary real numberγ satisfyingγ > γ₀, the generalized solution u ∈ H^˚1,1(Q,γ)of problem(1.2)–(1.3)has derivatives with respect to t up to order h belonging to H˚^1,1(Q,γ), and

ku_thk²_H1,1(Q,γ) ≤C

h+1

∑

kf_tjk²_H0

α(Q,γ) (2.20)

where C is a constant independent of u and f .

3 Regularity of the generalized solution

We reduce the operator with coefficients atP∈l₀, t ∈_R L⁽₀²⁾ := −

∑

a_ij(P,t) ^∂

2

∂x_i∂x_j,

to its canonical form. After a linear transformation of coordinates, it can be realized that via this reductionT₁andT2go over into hyperplanesT₁⁰ andT₂⁰, respectively. The angleβat(P,t) is transformed to

ω(P,t) =arctan[a₁₁(P,t)a₂₂(P,t)−a²₁₂(P,t)]^1/2 a₂₂(P,t)cotβ−a₁₂(P,t) ^.

Clearly, the valueω(P,t)does not depend on the method by whichL⁽₀²⁾is reduced to its canon- ical form. The functionω(P,t)is infinitely differentiable andω(P,t) > 0. Then, we have the following theorem.

Theorem 3.1. Let the assumptions of Theorem 2.4 be satisfied for a given positive interger h+1.

Furthermore, assume α ∈ [0, 1] and 1−α < ^π

ω. Then the generalized solution u ∈ H^˚1,1(Q,γ) of problem(1.2)–(1.3)has derivatives with respect to t up to order h, u_th ∈ H_α^2,0(Q,γ)and

ku_thk²

Hα^2,0(Q,γ) ≤C

h+2 k

∑

kf_tkk²_H0

α(Q,γ), where C is a constant independent of u,f .

Proof. We will prove the assertion of the theorem by induction onh. Firstly, we consider the case h = 0. It is easy to see thatu(·,t₀), t₀ ∈ R, is the generalized solution of the following problem:

L(x,t₀,∂)u= F(x,t₀)inΩ, u

∂Ω =0,

whereF(x,t₀) = f(x,t₀)−u_tt(x,t₀)∈H_α⁰(_Ω). From [8, Thm. 2], we getu(x,t₀)∈ H²_α(_Ω)and ku(·,t₀)k²_H2

α(_Ω) ≤Ch

kF(·,t₀)k²_H0

α(_Ω)+kuk²_L

2(_Ω)

i

≤Ch kfk_H0

α(_Ω)+ku_ttk²_L

2(_Ω)+kuk²_L

2(_Ω)

i .

(3.1)

Multiplying both sides of the above inequality bye^−t0γ, then integrating with respect tot₀on Rand using estimates from Theorem2.4, we obtain

kuk²

Hα^2,0(Q,γ) ≤C

∑

kf_tkk²_H0 α(Q,γ). Thus, the assertion of the theorem is valid forh=0.

Next, suppose that the assertion of the theorem is true forh−1, we will prove that it also holds fork=h. Differentiatinghtimes both sides of (1.2) with respect tot, we find

Lu_th = _fth −u_th+2−

h−1 k

∑

k h

L_th−ku_tk := _{F. (3.2)} By using the assumptions of the theorem and the inductive assumption, we obtain f_th ∈ H_α⁰(_Ω), u_th+2 ∈ L₂(_Ω) ⊂ H⁰_α(_Ω)_, α ∈ [_{0, 1}]_{, and}u_tk ∈ H_α⁰(_Ω)_, k ≤ h−1. Therefore,F(·_,t₀)∈ H_α⁰(_Ω)_, a.e.t₀ ∈R. By using [8, Thm. 2] again, we getu_th ∈ H_α²(_Ω)for a.e.t₀∈_Rand

ku_thk²_H2

α(_Ω) ≤CkFk²_H0

α(_Ω) ≤C

"

kf_thk²_H0

α(_Ω)+ku_th+2k²_L

2(_Ω)+

h−1 k

∑

ku_tkk²_L

2(_Ω)

#

. (3.3)

Multiplying both sides of (3.3) by e^−t0γ, then integrating with respect to t₀ on R and using estimates from Theorem2.4again, we obtain

ku_thk²

Hα^2,0(Q,γ) ≤C

h+2 k

∑

kf_tkk²_H0

α(Q,γ).

It means that the assertion of the theorem is valid fork= h. The proof is completed.

From now on, let the assumption of Theorem 2.4 be satisfied for a given positive integer h+1.

Theorem 3.2. Assume that f_tk ∈H_α^h(Q,γ₀), k≤2,and h+1−α< ^π

ω, α∈[0, 1].

Then the generalized solution u of problem(1.2)–(1.3)belongs to H_α^2+h(Q,γ). In addition, kuk²

H²α^+h(Q,γ) ≤C

∑

kf_tkk²_Hh

α(Q,γ), (3.4)

where C is a constant independent of u,f . Proof. We have

kuk²_H2

α(Q,γ)=

∑

|p|+k≤2

Z

Q

r^{2(α+|p|+k−2)}|D^pu_tk|²+|u|²e^−γtdx dt

=

∑

|p|≤2

Z

Q

r^{2(α+|p|−2)}|D^pu|²+|u|²e^−γtdx dt

+

∑

|p|≤1

Z

Q

r^{2(α+|p|−1)}|D^pu_t|²e^−γtdx dt+

Z

Q

r^2α|u_tt|²e^−γtdx dt

=kuk²

Hα^2,0(Q,γ)+kutk²

Hα^1,0(Q,γ)+kuttk²_H0 α(Q,γ)

=

∑

ku_tkk²

H_α^2−k,0(Q,γ).

Therefore,u∈ H_α²(Q,γ)by Theorem3.1and Theorem2.4. Moreover, we have kuk²_H2

α(Q,γ) =

∑

ku_tkk²

H²_α^−k,0(Q,γ) ≤C

∑

kf_tkk²_H0

α(Q,γ).

Thus, the theorem is valid forh=0. Suppose that the assertion of the theorem is true forh−1, we will prove that it also holds fork =h. It is easy to see that

kuk²

Hα^2+h(Q,γ)=

h+2 k

∑

ku_tkk²

H^h_α^+2−k,0(Q,γ). (3.5) Hence, we will prove that

u_tk ∈ H_α^h+2−k,0(Q,γ),k =0, . . . ,h (3.6) and

ku_tkk²

H^hα^+2−k,0(Q,γ)≤C

k+2 s

∑

kft^sk²

Hα^h−k,0(Q,γ), k≤ h. (3.7) By using Theorem3.1, this holds fork= h. Suppose that it holds fork= h,h−1, . . . ,j+1, we will prove that it holds fork= j. Returning one more time to (3.2)(h= j), we get

Lu_tj = f_tj−u_tj+2 −

j−1 k

∑

j k

Lu_tj−ku_tk := F₁.

Notice that f_tj ∈ H_α^h(_Ω) ⊂ H_α^h−j(_Ω)for a.e. t ∈ _R (by the assumptions of theorem), u_tj+₂ ∈ H_α^h−j(_Ω)for a.e.t ∈ _R(by (3.6) which holds fork = j+2),u_tk ∈ H_α^h+1−k(_Ω)⊂ H_α^h−j(_Ω),k = 0, . . . ,j−1, (by the valid inductive assumption fork= h−1).

It implies thatF₁(·,t)∈ H_α^h−j(_Ω), a.e.t ∈_R. From [8, Thm. 2], we obtain u_tj ∈ H^h_α^+2−j(_Ω), a.e.t ∈_R

and

ku_tjk²

Hα^h+2−j(_Ω)≤ CkF₁k

H^h_α^−j(_Ω)

≤ C

kf_tjk²

Hα^h−j(_Ω)+ku_tj+₂k²

H^hα^−j(_Ω)+

j−1 k

∑

ku_tkk²

Hα^h−j(_Ω)

. (3.8)

Multiplying both sides of (3.8) bye^−γt, then integrating onR, we arrive at ku_tjk²

H^hα^+2−j,0(Q,γ) ≤C

j+2 k

∑

kf_tkk²

H^hα^−j,0(Q,γ).

It means that (3.6) and (3.7) are true fork = j. Thus, (3.6) and (3.7) hold for allk = 0, 1, . . . ,h.

From (3.5), we get

kuk²

Hα^h+2(Q,γ) ≤C

∑

kf_tkk²_Hh α(Q,γ). The proof is completed.

4 Asymptotics of the solution in a neighbourhood of the edge

According to the previous section, ifk+1−α< ^π

ω,α∈[0, 1]and f,f_t,f_tt ∈ H_α^k(Q,γ), then the solutionu ∈ H_α^2+k(Q,γ). Now we study the solution in the case ^π_ω < k+1−α. In this case, we can find an asymptotic representation ofuin the neighbourhood ofl₀ :x₁ =x₂=0. In this section, we use the notations y₁ = x₁,y₂ = x₂, y = (y₁,y₂),z_i = x_i+2,z = (z₁, . . . ,z_n−2),r = x²₁+x²₂ and(r,ϕ)for the polar coordinates of the pointy = (y₁,y₂) ∈ _Ωz = _Ω∩ {z = _const}_. SetQz =_Ωz×R. To begin with, we present the following lemma.

Lemma 4.1. Suppose that the following hypotheses are satisfied:

(i) f_t^s ∈ H^k,0_α (Q,γ), s ≤h.

(ii) k−α< ^π

ω <k+1−α< ^2π

ω, α∈ [0, 1].

Let u be the solution of (1.2)–(1.3), u≡0outside some neighbourhood of l0. Then u(y,z,t) =c(z,t)r^πωΦ(z,ϕ,t) +u₁(y,z,t)

where ct^s ∈ L₂(Qz,γ), (u₁)_t^s ∈ H_α^k+2,0(Qz,γ), s ≤h andΦ∈C^∞.

Proof. Using (i), we get from Theorem 3.2 that u_t^s ∈ H_α^k+1,0(Q,γ)_, s ≤ h, particularly, u_z ∈ H_α^k(_Ω), uttz∈ H_α^k(_Ω)for a.e.t∈R. On the other hand, we have

Lu_z = f_z−u_ttz−L_zu=_: f₁,