Degenerate parabolic equations appearing in atmospheric dispersion of pollutants

Degenerate parabolic equations appearing in atmospheric dispersion of pollutants

Veli Shakhmurov

and Aida Sahmurova

1Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959, Istanbul, Turkey

2Khazar University, Baku Azerbaijan

3Okan University, Faculty of Health Sciences, Akfirat, Tuzla 34959, Istanbul,Turkey

Received 22 September 2016, appeared 18 December 2016 Communicated by Maria Alessandra Ragusa

Abstract. Linear and nonlinear degenerate abstract parabolic equations with variable coefficients are studied. Here the equation and boundary conditions are degenerated on all boundary and contain some parameters. The linear problem is considered on the moving domain. The separability properties of elliptic and parabolic problems in mixed Lp spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Keywords: differential-operator equations, degenerate PDE, semigroups of opera- tors, nonlinear problems, separable differential operators, positive operators in Banach spaces.

2010 Mathematics Subject Classification: 35A01, 35J56, 35Dxx, 35K51, 47G40.

1 Introduction

In this work, the boundary value problems (BVPs) for parameter dependent degenerate differential-operator equations (DOEs) are considered. Namely, equations and boundary conditions contain small parameters. These problems have numerous applications in PDE, pseudo DE, mechanics and environmental engineering. The BVP for DOEs have been studied extensively by many researchers (see e.g. [1,3,4,7–10,12–17,19–26,28,29] and the references therein). A comprehensive introduction to the DOEs and historical references may be found in [3,10,14–16,29]. The maximal regularity properties for DOEs have been studied e.g. in [1,4,11,19–22,24,25,28,29]. DOEs in Banach space valued function class are investigated e.g.

in [2,4,13,14,20,23,25,28,29]. Nonlinear DOEs are studied e.g. in [3,20,24,25]. The Fredholm property of BVP for elliptic equations are studied e.g. in [2,3,7].

The main objective of the present paper is to discusse the initial and BVP for the following nonlinear degenerate parabolic equation

∂u

∂t +

∑

a_k(x_k)^∂

[2]u

∂x²_k +B

t,x,u,D^[1]u

u=F

t,x,u,D^[1]u

, (1.1)

BCorresponding author. Email: veli.sahmurov@okan.edu.tr

wherea_k(x)are complex valued functions,BandFare nonlinear operators in a Banach space Eand

D^[1]u= ^∂

[1]u

∂x₁ ,∂^[1]u

∂x₂ , . . . ,∂^[1]u

∂x_n

!

, x= (x₁,x₂, . . . ,x_n)∈G=

∏

(0,b_k), D_k^[i]u= ^∂

[i]u

∂xⁱ_k =

x^αk(b_k−x_k)^{βk ∂}

∂x_k i

u(x), 0≤ α_k,β_k <1.

First, we consider the BVP for the degenerate elliptic DOE with small parameters

∑

ε_ka_k(x_k)^∂

[2]u

∂x²_k +A(x)u+λu+

∑

ε

1 2

kA_k(x)^∂

[1]u

∂x_k = f(x), (1.2) where a_k are complex-valued functions, ε_k are small parameters, A(x) and A_k(x) are linear operators,λis a complex parameter.

Namely we prove that, for f ∈ L_p(G;E), |argλ| ≤ ϕ, 0< ϕ≤ πand sufficiently large|λ|, problem (1.2) has a unique solutionu∈ W_p^[2](G;E(A),E)and the following coercive uniform estimate holds

∑

∑

|λ|^1−k2ε

i 2

k

∂^[i]u

∂xⁱ_k Lp(G;E)

+kAuk_L

p(G;E) ≤Ckfk_L

p(G;E).

Especially, it is shown that the corresponding differential operator is positive and also is a generator of an analytic semigroup. Then by using this result, we prove the well-posedeness in L_p(G;E) to initial and BVP for the following degenerate abstract parabolic equation with parameters

∂u

∂t +

∑

ε_ka_k(x_k)^∂

[2]u

∂x_k² +A(x)u= f(x,t), t ∈(0,T),x ∈G. (1.3) Finally, via maximal regularity properties of(0.3)and contaction mapping argument we de- rive the existence and uniqueness of solution of the problem (1.1).

Note that, the equation and boundary conditions are degenerated on all edges of bound- aryG. Moreover, it happened with the different rate at both boundary edges.

In application, the system of degenerate nonlinear parabolic equations is presented. Par- ticularly, we consider the system that serves as a model of systems used to describe photo- chemical generation and atmospheric dispersion of ozone and other pollutants. The model of the process is given by initial and BVP for the atmospheric reaction–advection–diffusion system having the form

∂u_i

∂t =

∑

"

a_ki(x)^∂

[2]u_i

∂x²_k +b_ki(x)^∂

[1]

∂x_k(u_iω_k)

# +

∑

d_ku_k+ f_i(u) +g_i, (1.4) where

x∈ G₃ ={x= (x₁,x₂,x₃), 0< x_k < b_k},

u_i =u_i(x,t), i,k =1, 2, 3, u=u(x,t) = (u₁,u₂,u₃), t∈ (0,T)

and the state variablesu_i represent concentration densities of the chemical species involved in the photochemical reaction. The relevant chemistry of the chemical species involved in the photochemical reaction and appears in the nonlinear functions f_i(u), with the terms g_i, rep- resenting elevated point sources,a_ki(x),b_ki(x)are real-valued functions. The advection terms

ω = ω(x) = (ω₁(x),ω₂(x),ω₃(x)), describe transport from the velocity vector field of at- mospheric currents or wind. In this direction the work [11] and references there can be mentioned. The existence and uniqueness of solution of the problem (1.4) is established by the theoretic-operator method, i.e., this problem reduced to degenerate differential-operator equation.

Modern analysis methods, particularly abstract harmonic analysis, the operator theory, in- terpolation of Banach spaces, semigroups of linear operators, microlocal analysis, embedding and trace theorems in vector-valued Sobolev–Lions spaces are the main tools implemented to carry out the analysis.

2 Notations, definitions and background

Let γ = γ(x) be a positive measurable function on Ω ⊂ Rⁿ and E be a Banach space. Let L_p,γ(_Ω;E)denote the space of strongly measurableE-valued functions defined onΩwith the norm

kfk_L

p,γ =kfk_L

p,γ(_Ω;E) = Z

kf(x)k^p_Eγ(x)dx ¹_p

, 1≤ p<_∞.

Let p=(p₁,p₂, . . . ,p_n). L_p,γ(G;E), G = _∏ⁿ_k=1(0,b_k) will denote the space of all E-valued p-summable functions with mixed norm, i.e., the space of all measurable functions f defined on Gequipped with norm

kfk_L

p,γ(G;E)=























bn

Z

0





. . .

b2

Z

0





b₁

Z

0

kf(x)k^p_E¹γ(x)dx₁





p2 p1

dx₂







p3 p2

. . .











pn pn−1

dxn













1 pn

<_∞.

For γ(x) ≡ 1 we will denote these spaces by L_p(_Ω;E) and L_p(G;E), respectively (see e.g. [5] forE=_C).

The Banach spaceEis called anUMD-space if the Hilbert operator (H f)(x) =lim

ε→0

Z

|x−y|>ε

f(y) x−ydy

is bounded in L_p(R,E), p ∈ (1,∞) (see e.g. [6] ). UMD spaces include e.g. L_p, l_p spaces, Lorentz spaces L_pqand Lorentz–Morrey spacesR^p,q,λ, when p,q∈ (1,∞),λ∈[0,n) [18].

Let E₁ and E2 be two Banach spaces continuously embedding in a locally convex space.

By (E₁,E₂)_θ,p, 0 < θ < 1, 1 ≤ p ≤ _∞ we will denote the interpolation spaces obtained from {E₁,E₂}by theK-method [27, §1.3.2].

Let E0 and E be two Banach spaces and E0 is continuously and densely embeds into E. Let us consider the Sobolev–Lions-type space W_p,γ^m (a,b;E₀,E), consisting of all functions u∈L_p,γ(a,b;E₀)that have generalized derivativesu^(m) ∈ L_p,γ(a,b;E)with the norm

kuk_Wm

p,γ =kuk_Wm

p,γ(a,b;E0,E)=kuk_L

p,γ(a,b;E0)+u^(m)

Lp,γ(a,b;E)< _∞.

Let

W_p,γ^[m] =W_p,γ^[m](0, 1;E₀,E)

=

u: u∈L_p(0, 1;E₀), u^[m] ∈ L_p(0, 1;E),kuk

Wp,γ^[m] =kuk_L

p(0,1;E0)+ u^[m]

Lp(0,1;E)<_∞

.

Now, let we defineE-valued Sobolev–Lions-type spaces with mixed L_p andL_p,γnorms. Let α_k(x) =x^α1k(b_k−x_k)^α2k, α= (α₁,α2,), p=(p₁,p2, . . . ,pn).

ConsiderE-valued weighted space defined by W_p,α^[m](G,E(A),E)

= (

u;u∈ L_p(G;E₀), ^∂_∂x^[mm^]u

k ∈ L_p(G;E), kuk

Wp,α^[m]

=kuk_L

p(G;E0)+

∑

∂^[m]u

∂x^m_k

Lp(G;E)<_∞ )

. Let ε_k be small parameters andε = (ε_1,ε₂, . . . ,ε_n). We denote byW_p,γ^m (_Ω;E₀,E)the space of all functions u ∈ L_p,γ(_Ω;E₀)possessing generalized derivatives ^∂_∂x^mm^u

k ∈ L_p,γ(_Ω;E) with the parametrized norm

kuk_Wm

p,γ,ε(_Ω;E0,E)=kuk_L

p,γ(_Ω;E0)+

∑

ε_k

∂^mu

∂x^m_k

L_p,γ(_Ω;E)

< _∞.

For definition ofR-sectorial operator see e.g. [7, p. 39]

In a similar way as in [21, Theorems 2.3, 2.4] we have the following result.

Theorem 2.1. Assume the following conditions be satisfied:

(1) γ=γ(x)is a weight function defined on domainΩ⊂ Rⁿsatisfying A_p condition;

(2) E is a UMD space, A is a R-sectorial operator in E and p_k ∈(1,∞); β= (β₁,β₂, . . . ,β_n); (3) there exists a bounded linear extension operator from W_p,γ^m (_Ω;E(A),E)to W_p,γ^m (Rⁿ;E(A),E).

Then, the embedding

D^βW_p,γ^m (_Ω;E(A),E)⊂ L_p,γ Ω;E

A^{1−|mβ|−µ}

is continuous and for0≤ µ≤1−^|_m^β|,0<h≤ h₀ <_∞ the following uniform estimate holds

∏

ε

|β| m

k kD^αuk_L

p,γ(Ω;E(A^1−κ−µ))≤ h^µkuk_Wm

p,γ,ε(_Ω;E(A),E)+h^−(1−µ)kuk_L

p,γ(_Ω;E)

for all u∈W_p,γ^m (_Ω;E(A),E).

Consider the BVP for the degenerate ordinary DOE with parameter

Lu= εa(x)u^[2](x) + (A(x) +λ)u(x) = f, (2.1) L₁u=

m1

i

∑

ε^σiδ_iu^[i](0) =0, L₂u=

m2

∑

ε^σiβ_iu^[i](1) =0, x∈(0, 1), (2.2) where u^[i] = x^γ1(1−x)^γ2_dx^diu(x), 0 ≤ γ_k < _1, σ_i = ₂ⁱ + ¹

2p(1−γ0), γ₀ = _min{γ₁,γ₂}, m_k ∈ {0, 1},δ_i,β_iare complex numbers; A(x)is a linear operator in a Banach spaceEforx ∈(0, 1), εis a small positive andλis a complex parameter.

We supposeδm₁ 6=0, βm₁ 6=0 and Z _x

0 z^−γ1(1−z)^−γ2dz<_∞.

Consider the operatorB_ε generated by problem (1.1)–(1.2) forλ=0, i.e., D(B_ε) =W_p,γ^[2](0, 1;E(A),E,L_k)

=ⁿu: u∈W_p,γ^[2](0, 1;E(A),E),L_ku=0, k=1, 2o , B_εu=−εa(x)u^[2]+A(x)u.

Condition 2.2. Assume the following conditions are satisfied:

(1) E is a UMD space and γ(x) = x^γ1(1−x)^γ2, 0≤ γ_k < 1− ¹_p, 1 < p < _{∞, a} ∈ C([0, 1])and a(x)<0for x ∈(0, 1);

(2) A is a R positive operator in E and A(x)A⁻¹(x0)∈C([0, 1];B(E))for x, x0∈ (0, 1).

By reasoning as in [21, Theorem 5.1] and by using the method used in [22, Theorem 1] we get the following theorem.

Theorem 2.3. Assume that Condition 2.2 holds. Then problem (2.1) has a unique solution u ∈ W_p,γ^[2](0, 1;E(A),E) for f ∈ L_p(0, 1;E) and |argλ| ≤ ϕ with sufficiently large |λ|. Moreover, the following uniform coercive estimate holds

∑

|λ|¹⁻²ⁱε

i 2

u^[i]

Lp(0,1;E)+kAuk_L

p(0,1;E)≤Ckfk_L

p(0,1;E). In a similar way as in [25, Theorem 3.1] we obtain the following theorem.

Theorem 2.4. Suppose the Condition2.2is satisfied. Then, the operator B_ε is uniformly R-positive in L_p(0, 1;E).

3 Degenerate elliptic equations with parameters

Consider the BVP for the following degenerate partial DOE with parameters

∑

ε_ka_k(x_k)^∂

[2]u

∂x²_k +A(x)u+λu+

∑

ε

1 2

kA_k(x)^∂

[1]u

∂x_k = f(x), (3.1) L_k1u=

mk1

i

∑

ε^σ_k^ikδ_kiu^[_x^i]_k(G_k0) =0, L_k2u=

mk2

i

∑

ε^σ_k^ikβ_kiu^[_k^i](G_kb) =0,

for x^(k) ∈ G_k, where A(x)and A_k(x)are linear operators, u = u(x), ε_k are small parameters, δ_ki,β_kiare complex numbers,λis a complex parameter,m_kj ∈ {0, 1}and

∂^[i]u

∂xⁱ_k =

x^α_k^1k(b_k−x_k)^{α2k ∂}

∂x_k i

u(x), 0≤α_1k,α_2k <1,

σ_ik= ⁱ

2 + ¹

2p_k(1−α_0k)^{, α0k} =min{α_1k,α_2k},

a_k are complex-valued functions and

x= (x₁,x₂, . . . ,x_n)∈G=

∏

(0,b_k),

G_k0= (x₁,x₂, . . . ,x_k−1, 0,x_k+1, . . . ,xn), p_k ∈(1,∞), G_kb = (x₁,x2, . . . ,x_k−1,b_k,x_k+1, . . . ,xn),

x^(k)= (x₁,x₂, . . . ,x_k−1,x_k+1, . . . ,x_n)∈G_k =

∏

j6=k

0,b_j . Let

α=α(x) =

∏

x^α_k^1k(b_k−x_k)^α2k. Remark 3.1. Under the substitutions

τ_k =

Z _xk

0 x⁻_k^αk(b_k−x_k)^−αkdx_k, k =1, 2, . . . ,n

the spacesL_p(G;E)andW_p,α^[2](G;E(A),E)are mapped isomorphically onto the weighted spaces L_p,˜α G;˜ E

andW_p,˜²_α G;˜ E(A),E

, respectively, where G˜ =

∏

0, ˜b_k , ˜b_k =

Z _bk

0 x⁻_k^α1k(b_k−x_k)^−α2kdx_k, ˜α(τ) =α(x₁(τ₁),x2(τ2), . . . ,xn(τn)). Consider the principal part of (3.1), i.e., consider the problem

∑

ε_ka_k(x_k)^∂

[2]u

∂x²_k +A(x)u+λu= f(x), (3.2)

mk1

i

∑

ε^σ_k^ikδ_kiu^[_xⁱ_k^](G_k0) =0,

mk2

i

∑

ε^σ_k^ikβ_kiu^[_k^i](G_kb) =0.

Condition 3.2. Assume

(1) E is a UMD space, γ(x) = _∏ⁿ_k=1x^α_k^1k(b_k−x_k)^α2k, where0 ≤ α_1k, α_2k < ₁− _p¹

k, p_k ∈ (1,∞), δ_kmk1 6=0,β_kmk2 6=0;

(2) A(x)is a uniformly R-positive operator in E, A(x)A⁻¹(x¯)∈C(G;^¯ L(E)), x ∈G;

(3) a_k(x)∈C^(m)(G^¯)and a_k(x_k)<0for x_k ∈ (0,b_k).

First, we prove the separability properties of the problem (3.2).

Theorem 3.3. Assume that Condition 3.2 holds. Then problem (3.2) has a unique solution u ∈ W_p,α^[2](G;E(A),E)for f ∈ Lp(G;E),|argλ| ≤ ϕwith sufficiently large|λ|and the following coercive uniform estimate holds

∑

∑

|λ|¹⁻²ⁱε

i 2

k

∂^[i]u

∂xⁱ_k Lp(G;E)

+kAuk_L

p(G;E)≤Ckfk_L

p(G;E). (3.3)

Proof. Consider the BVP

(L+λ)u =a₁(x₁)ε₁D_x^[2₁^]u(x₁) + (A(x₁) +λ)u(x₁) = f(x₁), (3.4) L_1ju=0, j=1, 2, x₁ ∈(0,b₁),

where L_1j are boundary conditions of type (3.2) on(0,b₁). By virtue of Theorem2.3, problem (3.4) has a unique solution u∈W_p^[2_1,α^]

1(0,b₁;E(A),E) for f ∈ L_p1(0,b₁;E), |argλ| ≤ ϕ with sufficiently large |λ|and the coercive uniform estimate holds

∑

|λ|^1−2jε

j 2

1

u^[j]

Lp1(0,b₁;E)+kAuk_L

p1(0,b₁;E) ≤Ckfk_L

p1(0,b₁;E). Now, let us consider the following BVP

∑

ε_ka_k(x_k)D^[_k^2]u(x₁,x₂) +A(x₁,x₂)u(x₁,x₂) +λu(x₁,x₂) = f(x₁,x₂), (3.5) L_k1u=0, L_k2u=0, k=1, 2, x₁,x₂ ∈G₂= (0,b₁)×(0,b₂).

Letp₂ = (p₁,p₂)andα(2) = (α₁,α₂). Since L_p2 0,b₂;L_p1(0,b₁); E

= L_p2(G₂;E), the BVP (3.5) can be expressed as

a2ε₂D^[₂^2]u(x2) + (Bε₁(x2) +λ)u(x2) = f(x2), L_2ju=0, j=1, 2,

for x₁ ∈ (0,b₁), where Bε₁ is a differential operator in Lp₁(0,b₁;E)for x₂ ∈ (0,b₂), generated by problem (3.4). By virtue of [3, Theorem 4.5.2], L_p1(0,b₁;E)∈UMDfor p₁∈ (1,∞). Hence, by [28, Corollary 4.1] the space L_p1(0,b₁;E) satisfies the multiplier condition. Moreover, the Theorem2.4implies the uniformR-positivity of operatorBε₁. Hence, by Theorem2.3, problem (3.5) has a unique solution u ∈ W_p^[2]

2,α(2)(G₂;E(A);E) for f ∈ L_p2(G₂;E), |argλ| ≤ ϕ with sufficiently large|λ|and (3.3) holds forn=2. By continuing this we obtain the assertion.

Theorem 3.4. Let the Condition3.2 hold and let A_k(x)A⁻(¹₂^−ν)(x) ∈ C(G;^¯ L(E))for0 < ν < ¹₂. Then, problem(2.1)has a unique solution u∈W_p,α^[2](G;E(A),E)for f ∈ L_p(G;E),|argλ| ≤ϕwith sufficiently large |λ|and the coercive uniform estimate holds

∑

∑

|λ|¹⁻²ⁱε

i 2

k

∂^[i]u

∂xⁱ_k Lp(G;E)

+kAuk_L

p(G;E) ≤Ckfk_L

p(G;E). (3.6) Proof. By assumption and by Theorem 2.1, for all h > 0 we have the following Ehrling–

Nirenberg–Gagliardo-type estimate kL₁uk_L

p(G;E)≤ h^µ kuk

Wp,α^[2](G;E(A),E)+h^−(1−µ)kuk_L

p(G;E). (3.7) LetO_ε denote the operator generated by the problem (3.2) and

L₁u=

∑

ε

1 2

kA_k(x)^∂

[1]u

∂x_k .

By using the estimate (3.7) we obtain that there is aδ∈(0, 1)such that

L₁(Oε+λ)⁻¹

B(X)<δ.

Hence, from perturbation theory of linear operators we obtain the assertion.

4 Abstract Cauchy problem for degenerate parabolic equation with parameter

Consider the initial and BVP for degenerate parabolic equation with parameter:

∂u

∂t +

∑

ε_ka_k(x_k)^∂

[2]u

∂x²_k +A(x)u+du= f(x,t), t∈(0,T),x ∈G. (4.1)

m_k1 i

∑

ε^σ_k^ikδ_kiu^[_xⁱ_k^](G_k0,t) =0,

m_k2 i

∑

ε^σ_k^ikβ_kiu^[_k^i](G_kb,t) =0,

u(x, 0) =0, t∈(0,T), x^(k)∈ G_k, (4.2) whereu= u(x,t)is a solution,δ_ki,β_kiare complex numbers,ε_k are positive parameters,a_k are complex-valued functions onG,A(x)is a linear operator in a Banach spaceE, domainsG,G_k, G_k0,G_kb,σ_ik andx^(k)are defined in Section2and

∂^[i]u

∂xⁱ_k =

x^α1k(b_k−x_k)^{α2k ∂}

∂x_k i

u(x,t), d>0.

For ¯p=(p₀,p), p=(p₁,p₂, . . . ,p_n), G_T = (0,T)×G, L_˜p,fl(G_T;E)will denote the space of allE-valued weighted ˜p-summable functions with mixed norm.

Theorem 4.1. Suppose the Condition 3.2 holds for ϕ > ^π₂. Then, for f ∈ L_p(G_T;E) and suffi- ciently large d>0problem(4.1)–(4.2)has a unique solution belonging to W^1,_¯p,α^[2](GT;E(A),E)and the following coercive estimate holds

∂u

∂t

L¯p(G_T;E)

+

∑

ε_k

∂^[2]u

∂x²_k

L¯p(G_T;E)

+kAuk_L

¯p(GT;E) ≤Ckfk_L

¯p(G_T;E). Proof. The problem (4.1) can be expressed as the following abstract Cauchy problem

du

dt + (Oε+d)u(t) = f(t), u(0) =0. (4.3) From Theorems 2.4, 3.3 we get that Oε is R-sectorial in F = Lp(G;E). By [18, §1.14], Oε is a generator of an analytic semigroup in F. Then by virtue of [28, Theorem 4.2], problem (4.3) has a unique solutionu∈ W_p¹₀(0,T;D(Oε),F)for f ∈ L_p0(0,T;F)and sufficiently larged> 0.

Moreover, the following uniform estimate holds

du dt

Lp0(0,T;F)

+kO_εuk_L

p0(0,T;F)≤Ckfk_L

p0(0,T;F). SinceLp₀(G_T;F) =L_¯p(G_T;E), by Theorem3.3 we have

k(Oε+d)uk_L

p0((0,T);F) =D(Oε). Hence, the assertion follows from the above estimate.

5 Degenerate parabolic DOE on the moving domain

Consider the degenerate problem (4.1)–(4.2) on the moving domainG(s) =_∏ⁿ_k=1(0,b_k(s)):

∂u

∂t +

∑

a_k(x_k)^∂

[2]u

∂x²_k +A(x)u+du= f(x,t), (5.1) L_k1u=

mk1

i

∑

ε^σ_k^ikδ_kiu^[_xⁱ_k^](G_k0(s),t) =0, L_k2u=

mk2

i

∑

ε^σ_k^ikβ_kiu^[_k^i](G_kb(s),t) =0,

u(x, 0) =0, t ∈(0,T), x∈ G(s), (5.2) where the end pointsb_k(s)depend of a parameters,x_k ∈(0,b_k(s))andb_k(s)are positive con- tinues function, G_k0(s), G_kb(s)are domains defined in Section2, replacing(0,b_k)by(0,b_k(s)) and

σ_ik = ⁱ

2 + ¹

2p(1−α_0k)^{, α0k} =min{α_1k,α_2k},

∂^[i]u

∂xⁱ_k =

x^α1k(b_k−x_k)^{α2k ∂}

∂x_k i

u(x,t). Let

G_T = G_T(s) = (0,T)×G(s). Theorem4.1implies the following.

Proposition 5.1. Assume the Condition3.2 hold for ϕ> ^π₂. Then, problem(5.1)–(5.2) has a unique solution u∈W^1,_˜p,α^[2]((G(s));E(A),E) for f ∈ L_p(G_T(s);E) and sufficiently d > 0. Moreover, the following coercive uniform estimate holds

∂u

∂t

L¯p(GT;E)

+

∑

ε_k

∂^[2]u

∂x²_k L¯p(GT;E)

+kAuk_L

¯p(GT;E)≤ Ckfk_L

¯p(GT;E). (5.3) Proof. Under the substitution τ_k = x_kb_k(s) the problem (5.1)–(5.2) reduced to the following BVP in fixed domain G:

∂u

∂t +

∑

b⁻_k²(s)a˜_k(τ_k)^∂

[2]u

∂τ_k²

+A^˜(τ)u= f^˜(τ,t), t ∈R+,τ∈ G. (5.4)

mk1

i

∑

b^σ_k^ik(s)δ_kiu^[_xⁱ_k^](G_k0,t) =0,

mk2

i

∑

b_k^σik(s)β_kiu^[_k^i](G_kb,t) =0,

u(x, 0) =0, t∈ (0,T), x ∈G=

∏

(0,b_k), (5.5)

where

˜

a_k(τ) =a_k(x(τ)), A˜(τ) =A((x(τ))), f˜(τ) = f((x(τ))), x(τ) = (x₁(τ₁),x2(τ₂), . . . ,xn(τ_n)).

The problem (5.4)–(5.5), is a particular case of (4.1)–(4.2). So, by virtue of Theorem 4.1 we obtain the required assertion.

6 Nonlinear degenerate abstract parabolic problem

In this section, we consider initial and BVP for the following nonlinear degenerate parabolic equation

∂u

∂t +

∑

a_k(x_k)^∂

[2]u

∂x_k² +B

t,x,u,D^[1]u

u= F

t,x,u,D^[1]u

, (6.1)

m_k1 i

∑

δ_kiu^[_xⁱ_k^](G_k0,t) =0,

m_k2 i

∑

β_kiu^[_k^i](G_kb,t) =0,

u(x, 0) =0, t ∈(0,T), x∈ G, x^(k) ∈G_k, (6.2) whereu=u(x,t)is a solution,δ_ki,β_kiare complex numbers, a_k are complex-valued functions onG; domainsG, G_k,G_k0,G_kb andσ_ik,x^(k) are defined in Section2and

D^[_k^i]u= ^∂

[i]u

∂xⁱ_k =

x^α_k^k(b_k−x_k)^{αk ∂}

∂x_k i

u(x,t), 0≤ α_k <1.

LetG_T = (0,T)×G, whereG=_∏ⁿ_k=1(0,b_k). Moreover, let b_k ∈ (0,b_0k),G₀=

∏

(0,b_0k),T∈ (0,T₀), B_ki = W^2,p(G_k,E(A),E),L^p(G_k;E)

η_ik,p, η_ik = ^mkj

+ _p(1−¹

α_k)

2 , B₀ =

∏

∏

B_ki.

Remark 6.1. By virtue of [27, § 1.8.] and the Remark 3.1, operators u→ ^∂[i]u

∂xⁱ_k

x_k=0 are con- tinuous fromW_p,α^[2](G;E(A),E) onto B_ki and there are the constants C₁ and C0 such that for w∈W_p,α^[2](G;E(A),E),W ={w_ki},w_ki = ^∂[i]w

∂xⁱ_k ,i=0, 1, k=1, 2, . . . ,n

∂^[i]w

∂xⁱ_k Bki,∞

=sup

x∈G

∂^[i]w

∂x_kⁱ Bki

≤C₁kwk

Wp,α^[2](G;E(A),E), kWk_0,∞ =sup

x∈G

∑

k,i

kw_kik_B

ki ≤C0kwk

Wp,α^[2](G;E(A),E). Condition 6.2. Suppose the following hold:

(1) E is an UMD space and0≤ α₁,α2<1− ¹_p, p∈(1,∞);

(2) a_k are continuous functions onG,¯ a_k(x)<0, for all x∈G,δ_kmk1 6=0,β_kmk1 6=0,k =1, 2, . . . ,n;

(3) there exist Φki ∈ B_ki such that the operator B(t,x,Φ) for Φ = {_Φki} ∈ B₀ is R-sectorial in E uniformly with respect to x∈ G₀ and t∈ [0,T₀];moreover,

B(t,x,Φ)B⁻¹ t⁰,x⁰,Φ

∈C(G;^¯ L(E)), t⁰∈(0,T), x⁰∈ G;

(4) A = B t⁰,x⁰,Φ

: G_T ×B₀ → L(E(A),E) is continuous. Moreover, for each positive r there is a positive constant L(r)such thatk[B(t,x,U)−B(t,x, ¯U)]υk_E ≤ L(r)kU−U^¯k_B

0kAυk_E for t∈ (0,T),x∈ G, U, ¯U ∈B₀, ¯U={u¯_ki}, ¯u_ki∈ B_ki,kUk_B

0,kU^¯k_B

0 ≤ r,υ∈D(A);

(5) the function F : G_T×B₀ → E such that F(·,U)is measurable for each U ∈ B₀ and F(t,x,·)is continuous for a.a. t∈ (0,T), x ∈ G.Moreover, kF(t,x,U)−F(t,x, ¯U)k_E ≤ _Ψr(x)kU−U^¯k_B

0

for a.a. t∈(0,T), x ∈G, U, ¯U∈ B₀andkUk_B

0,kU^¯k_B

0 ≤r; f(·) =F(·, 0)∈ L_p(G_T;E). The main result of this section is the following.

Theorem 6.3. Let the Condition6.2be satisfied. Then there is T∈ (0,T0)and b_k ∈(0,b_0k)such that problem(6.1)–(6.2)has a unique solution belonging to W_p,α^1,[2](G_T;E(A),E).

Proof. Consider the following linear problem

∂w

∂t +

∑

a_k(x_k)^∂

[2]w

∂x²_k +du= f(x,t), x∈G, t ∈(0,T),

m_k1 i

∑

δ_kiw^[_xⁱ_k^](G_k0,t) =0,

m_k2 i

∑

β_kiw^[_k^i](G_kb,t) =0, (6.3) w(x, 0) =0, t∈ (0,T), x ∈G, x^(k)∈ G_k, d>0.

By Theorem 4.1 and in view of Proposition 5.1 there exists a unique solution w ∈ W_p,α^1,[2](G_T;E(A),E) of the problem (6.3) for f ∈ Lp(G_T;E) and sufficiently large d > 0 and it satisfies the following coercive estimate

kwk

Wp,α^1,[2](GT;E(A),E) ≤C₀kfk_L

p(G_T;E),

uniformly with respect tob∈(0 ,b₀], i.e., the constantC₀does not depends on f ∈L_p(G_T;E) andb∈ (0b₀]where

A(x) =B(x, 0), f(x) =F(x, 0), x∈ (0,b).

We want to solve the problem (6.1)–(6.2) locally by means of maximal regularity of the linear problem (6.3) via the contraction mapping theorem. For this purpose, let wbe a solution of the linear BVP (6.3). Consider a ball

Br ={υ∈Y,υ−w∈Y₁,kυ−wk_Y ≤r}. For givenυ∈Br, consider the following linearized problem

∂u

∂t +

∑

a_k(x_k)^∂

[2]u

∂x²_k +A(x) =F(x,V) + [B(x, 0)−B(x,V)]υ,

m_k1 i

∑

δ_kiw^[_xⁱ_k^](G_k0,t) =0,

m_k2 i

∑

β_kiw^[_k^i](G_kb,t) =0, (6.4) w(x, 0) =0, t ∈(0,T), x∈ G, x^(k) ∈G_k.

where V = {υ_ki}, υ_ki ∈ B_ki. Define a map Q on B_r by Qυ = u, where u is solution of (6.4).

We want to show thatQ(Br)⊂ Brand that Qis a contraction operator provided T andb_k are sufficiently small, andr is chosen properly. In view of separability properties of the problem (6.3) we have

kQυ−wk_Y =ku−wk_Y ≤C₀{kF(x,V)−F(x, 0)k_X+k[B(0,W)−B(x,V)]υk_X}.