Global estimates for solutions of singular parabolic and elliptic equations with variable nonlinearity

Nonlinear Analysis 195 (2020) 111724

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Nonlinear Analysis www.elsevier.com/locate/na

Global estimates for solutions of singular parabolic and elliptic equations with variable nonlinearity Stanislav Antontsev a,b ,1 , Sergey Shmarev c ,∗,2 a

CMAF-CIO, University of Lisbon, Portugal Novosibirsk State University and Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia c Mathematics Department, University of Oviedo, c/Federico García Lorca 18, 33007, Oviedo, Spain b

article

info

abstract

Article history: Received 6 March 2019 Accepted 2 December 2019 Communicated by Vicentiu D Radulescu

We consider the homogeneous Dirichlet problem for the equation

MSC: 35K67 35B65 35K55 35K99

in the cylinder QT = Ω × (0, T ), Ω ⊂ Rd , d ≥ 2, with the variable exponent 2d < p− ≤ p(x, t) ≤ p+ ≤ 2, p± = const. We find sufficient conditions on p, ∂Ω , d+2 f and u(x, 0) which provide the existence of solutions with the following global regularity properties:

(

ut = div (ϵ2 + |∇u|2 )

Keywords: Singular parabolic equation Variable nonlinearity Higher regularity Strong solutions

p(x,t)−2 2

)

∇u + f (x, t),

ϵ ≥ 0,

(0.1)

ut ∈ L∞ (0, T ; L2 (Ω )), |∇u| ∈ C 0 ([0, T ]; L2 (Ω )), p

|ut | p−1 , |uxi xj |p , |∇ut |p , (ϵ2 + |∇u|2 ) p = p(x, t), i, j = 1, 2, . . . , d.

p−2 2

|uxi xj |2 ∈ L1 (QT ),

For the solutions of the stationary counterpart of Eq. (0.1),

(

div (ϵ2 + |∇v|2 )

p0 (x)−2 2

∇v

the inclusions |vxi xj |p0 , (ϵ2 + |∇v|2 )

)

= Φ(x)

p0 −2 2

in Ω ,

v = 0 on ∂Ω ,

|vxi xj |2 ∈ L1 (Ω ) are established. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Let us consider the homogeneous Dirichlet problem { p(z)−2 ut = div(|∇u| ∇u) + f (z) in QT = Ω × (0, T ), u = 0 on ∂Ω × (0, T ), u(x, 0) = u0 (x) in Ω ,

(1.1)

∗ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Antontsev), [email protected] (S. Shmarev). 1 The first author was supported by the Russian Federation government, Grant No. 14.W03.31.0002, Russia, and by the Portuguese Foundation for Science and Technology, Portugal, under the project: UID/MAT/04561/2019. 2 The second author acknowledges the support of the Research Grant MTM2017-87162-P, Spain.

https://doi.org/10.1016/j.na.2019.111724 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

2

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

where Ω is a domain in Rd with the boundary ∂Ω , z = (x, t) denotes a point in the cylinder QT . The variable exponent p(z) is a given measurable function. It is assumed that p(z) satisfies the inequalities 2d < p− ≤ p(z) ≤ p+ ≤ 2 d+2

in QT

with some constants p± and meets certain regularity assumptions which will be specified later. The primary goal of this work is the derivation of conditions of higher regularity for the solutions of problem (1.1). The method employed yields the same properties for the solutions of the regularized equation (1.1) (Eq. (2.9)) and its stationary counterpart (2.13). Equations of the type (1.1) fall into the class of parabolic equations with nonstandard growth, intensively studied in the last decades. These equations degenerate or become singular at the points where |∇u| = 0 or |∇u| = ∞, for this reason one may not expect the existence of classical solutions and the solutions of problem (1.1) are understood in a weak sense (see Definition 2.1). If u0 ∈ L2 (Ω ) and f ∈ L2 (QT ), the weak p(z) solution is a function u(z) ∈ C 0 ([0, T ]; L2 (Ω )) with |∇u| ∈ L1 (QT ). The time derivative ut of the weak solution is a distribution which need not belong to a Lebesgue space Lq (QT ) with any q ≥ 1. The natural questions arise: (a) what is the intrinsic regularity of local solutions of Eq. (1.1) in the interior of the cylinder QT ? (b) how does the global regularity of weak solutions of problem (1.1) change if the problem data possess better smoothness? Both questions have been addressed in a number of researches. Let Ω ′ ⊂ Ω be a boxed domain, dist(Ω , Ω ′ ) > 0, and Q′ = Ω ′ × (τ, T ), τ > 0, and u(z) be a local solution of Eq. (1.1). It is known that p(z) besides the natural integrability of the gradient, |∇u| ∈ L1 (QT ), the property of higher integrability p(z)+δ 1 ′ holds: there is a constant δ > 0 such that |∇u| ∈ L (Q ) — see, e.g., [9,32], and [31] for global estimates in QT . For Eq. (1.1) with p(z) continuous with the logarithmic module of continuity (condition (1.6)) the local H¨ older-continuity is proven in [4,12,29]. Moreover, if the exponent p(z) is H¨older-continuous, then the 1, 1

gradient of the local solution u(z) is locally H¨ older-continuous [12,30] and u ∈ Cx,t2 (Q′ ) - [12]. For Eq. (1.1) 2d with f = 0 and p ≡ p(t) ∈ ( d+2 , p+ ] the gradient of the solution is locally H¨older-continuous, provided that p(t) is continuous on [0, T ] with a logarithmic modulus of continuity — [25]. The proofs of the local regularity properties are based on the intrinsic geometry of the nonlinear equations — see [3,12,17] and references therein for the details of the method. The global regularity of solutions was studied in [5,8,26,28]. It is shown in [28] that equations of the type (1.1) with d = 1 and p ≡ p(x) ∈ C 1 (Ω ) admit Lipschitz-continuous solutions. In case of arbitrary space 1, 1

dimension, the solutions of problem (1.1) belong to Cx,t2 (QT ∩ {t > τ }), provided that p(z) is Lipschitzcontinuous in QT and pt (z) ≤ 0 a.e. in QT . Furthermore, in the special case p ≡ p(x) the solutions belong to C 1,1 (QT ∩ {t > τ }) - [26]. The results of [26,28] are obtained by comparison with suitable barrier functions. 2 In [8], the inclusion Dij u ∈ L2 (QT ∩ {t > τ }), τ > 0, is established by means of the study of finite-difference approximations for the weak derivatives. This method, as well as the method of [26], requires continuation of the solution across the lateral boundary and application of the local results of [30] on the H¨older continuity of the gradient. The results of [8] hold for the strong solutions of problem (1.1) with p(z) ≤ 2, which are the weak solutions with ut ∈ L2 (QT ). The existence of such solutions is proved under the assumptions |∇p| ≤ C and pt ≤ 0 a.e. in QT — [8,26]. Another approach to the study of global regularity of solutions to problem (1.1) with p(z) < 2 was employed in [5]. The solution was constructed with the method of Galerkin and the conclusion about the higher regularity followed from the uniform estimates on the derivatives of finite-dimensional approximations. Such an approach allows one to remove the restriction pt ≤ 0 a.e. in QT and to show the p(z) 2 existence of solutions with ut ∈ L2 (QT ), |∇u| ∈ L2 (Ω ) for a.e. t ∈ (0, T ) and |Dij u| ∈ L1 (QT ) in the case when pt (z) and |∇p(z)| are a.e. bounded in QT .

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

3

Existence and local integrability of the time derivative for the evolution p-Laplace equations with constant p is studied in [23,24] in the degenerate (p ≥ 2) and singular (p < 2) cases. It is shown in [24] that in the singular case the time derivative exists in the weak sense and belongs to the Lebesgue space Lθloc (QT ) with 1 3 3 if 1 < p < . Furthermore, in the latter case the solution is assumed θ = 2 if p ≥ , and 1 < θ < 2 2−p 2 bounded. The higher regularity of solutions of the system of singular equations of the type (1.1) with constant 2 p ∈ (1, 2) is studied in [15]. It is proved that for certain ranges of p the derivatives ut and Dij u belong to the Lebesgue spaces in QT ∩ {t > δ}, δ > 0. Global higher regularity of solutions of problem (1.1) (and the non-singular problem (2.9)) with constant p ∈ (1, 2] is studied in [10]. It is shown that for f ∈ L2 (QT ) and u0 ∈ W01,p (Ω ) the inclusion u ∈ L2(p−1) (0, T ; W 2,q (Ω )) holds with some q ∈ (1, 2] depending on d and p. This global in t result is proven in [10] under the assumption (2 − p)C(r) < 1, where C(r) is the constant from the inequality ⎛ ⎞ r1 d ∑ 2d(p − 1) 2 ⎝ ∈ (1, 2]. (1.2) v ∈ W 2,r (Ω ) ∩ W01,r (Ω ), r = ∥Dij v∥rr,Ω ⎠ ≤ C(r)∥∆v∥r,Ω , d − 2(2 − p) i,j=1 The present work concerns the question of existence of the time derivative and the higher-order derivatives in x of solutions of problem (1.1) with variable nonlinearity in the singular case p(z) ≤ p+ ≤ 2. We derive conditions on the problem data, ∂Ω , p(z), f (z), u0 , which ensure the existence of a strong solution with the following properties: ut ∈ L∞ (0, T ; L2 (Ω )), p(z)

|ut | p(z)−1 , |∇u|

p(z)−2

|∇u| ∈ C 0 ([0, T ]; L2 (Ω )), 2

2 2 |Dij u| , |Dij u|

p(z)

, |∇ut |

p(z)

∈ L1 (QT ). p(z)

The same properties are proven for the solutions of the regularized evolution problem with |∇u| 2 p(z)−2 substituted by (ϵ2 + |∇u| ) 2 , ϵ > 0, and its stationary counterpart (2.13). For a rigorous formulation of the results we need to introduce the variable Lebesgue and Sobolev spaces the solutions of problem (1.1) belong to. 1.1. The function spaces In this subsection we collect known facts on the variable Lebesgue and Sobolev spaces. A detailed exposition of the theory of these spaces can be found in the monograph [18], see also [7, Ch.1]. Let Ω ⊂ Rd be a domain with the Lipschitz-continuous boundary ∂Ω . Given a measurable function p(x) : Ω ↦→ [p− , p+ ] ⊂ (1, ∞), p± = const, we define the set { } ∫ p(x) p(·) L (Ω ) = f : Ω ↦→ R: f is measurable on Ω , |f | dx < ∞ . Ω p(·)

The set L

(Ω ) equipped with the Luxemburg norm { ∥f ∥p(·),Ω := inf

} ∫ ⏐ ⏐p(x) ⏐f ⏐ ⏐ ⏐ α>0: dx ≤ 1 ⏐ ⏐ Ω α

becomes a Banach space. It follows directly from the definition of the norm that ( ) ∫ ( ) p− p+ p(x) p− p+ min ∥f ∥p(·) , ∥f ∥p(·) ≤ |f | dx ≤ max ∥f ∥p(·) , ∥f ∥p(·) . Ω

′

For all f ∈ Lp(·) (Ω ), g ∈ Lp (·) (Ω ) with p(x) ∈ (1, ∞),

p′ (x) =

p(x) p(x) − 1

(1.3)

4

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

the generalized H¨ older inequality holds: ) ( ∫ 1 1 + ′ − ∥f ∥p(·) ∥g∥p′ (·) ≤ 2 ∥f ∥p(·) ∥g∥p′ (·) . |f g| dx ≤ p− (p ) Ω If p(x) is measurable and 1 < p− ≤ p(x) ≤ p+ < ∞ in Ω , then Lp(·) (Ω ) is a reflexive and separable Banach space, and C0∞ (Ω ) is dense in Lp(·) (Ω ). Let p1 (x), p2 (x) be measurable on Ω functions such that pi (x) ∈ [p− , p+ ] ⊂ (1, ∞) a.e. in Ω . If p1 (x) ≥ p2 (x) a.e. in Ω , then the inclusion Lp1 (·) (Ω ) ⊂ Lp2 (·) (Ω ) is continuous and C = C(|Ω |, p± ). (1.4) ∀ u ∈ Lp1 (·) (Ω ) ∥u∥p2 (·),Ω ≤ C∥u∥p1 (·),Ω , 1,p(·)

Let us define the variable Sobolev space V (Ω ) ≡ W0 (Ω ) as the set of functions { } 1,p(·) W0 (Ω ) = u ∈ Lp(·) (Ω ) ∩ W01,1 (Ω ) : |∇u| ∈ Lp(·) (Ω ) equipped with the norm ∥u∥W 1,p(·) (Ω) = ∥∇u∥p(·),Ω + ∥u∥p(·),Ω .

(1.5)

0

By Clog (Ω ) we denote the set of functions continuous on Ω with the logarithmic modulus of continuity: |p(x2 ) − p(x1 )| ≤ ω(|x2 − x1 |),

(1.6)

where ω ≥ 0 satisfies the condition lim ω(τ ) ln

τ →0+

1 = C < ∞, τ

C = const. 1,p(·)

1,p(·)

It is known that for p(x) ∈ Clog (Ω ) the set C0∞ (Ω ) is dense in W0 (Ω ) and the space W0 (Ω ) coincides ∞ with the closure of C0 (Ω ) with respect to the norm (1.5). We will use the notation p(x, 0) ∈ Clog (Ω ) and p(z) ∈ Clog (QT ) for the functions p of the arguments z = (x, t) satisfying condition (1.6) in the domain Ω and the cylinder QT = Ω × (0, T ). 1,p(·) For the elements of W0 (Ω ) with p(x) ∈ C 0 (Ω ) the Poincar´e inequality holds: ∥u∥p(·),Ω ≤ C(d, Ω )∥∇u∥p(·),Ω . An immediate consequence of the Poincar´e inequality is that an equivalent norm of V (Ω ) can be defined by ∥u∥V (Ω) = ∥∇u∥p(·),Ω . Let p(x), q(x) ∈ C 0 (Ω ), 1 < p− ≤ p(x) ≤ p+ < ∞, d ≥ 2. If p+ ≤ d and q(x) < embedding V (Ω ) ⊂ Lq(·) (Ω ) is continuous, compact, and ∥v∥q(·),Ω ≤ C∥v∥V (Ω) −

1,p(·)

dp(x) d−p(x)

in Ω , then the

∀v ∈ V (Ω ). −

2d (Ω ) ⊂ W01,p (Ω ). If p− > d+2 , then the embedding W01,p (Ω ) ⊂ L2 (Ω ) is According to (1.4) W0 compact. 1,p (·) Let p(z) ∈ C 0 (QT ), p0 (x) = p(x, 0) ∈ [p− , p+ ], and V (Ω ) = W0 0 (Ω ). By V ′ (Ω ) we denote the dual space of V (Ω ), i.e., the set of bounded linear functionals over V (Ω ): ( ′ )d+1 p0 ϕ ∈ V ′ (Ω ) ⇔ ϕ = (ϕ0 , ϕ1 , . . . , ϕd ) ∈ Lp0 (·) (Ω ) , p′0 = , p0 − 1 ( ) ∫ d ∑ ⟨ψ, ϕ⟩V,V ′ = ϕ0 ψ + ϕi Dxi ψ dx ∀ ψ ∈ V (Ω ). Ω

i=1

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

5

Let us introduce the spaces of functions defined on the cylinder QT p(x,t)

Vt (Ω ) = {u : Ω ↦→ R|u ∈ L2 (Ω ) ∩ W01,1 (Ω ), |∇u|

∈ L1 (Ω )},

p(x,t)

2

W(QT ) = {u : (0, T ) ↦→ Vt (Ω )| u ∈ L (QT ), |∇u|

t ∈ (0, T ),

1

∈ L (QT )}.

By W′ (QT ) we denote the dual of W(QT ): Φ ∈ W′ (QT ) if and only if ′

∃ Φ0 ∈ L2 (QT ), Φ1 ∈ (Lp (x,t) (QT ))d , ∫ ∀ u ∈ W(QT ) ⟨u, Φ⟩W,W′ = (uΦ0 + ∇u · Φ1 ) dz. QT

2. Assumptions and main results By agreement, throughout the text we use the notation ∂v , Di v = ∂xi

2 Dij v

∂2v = , ∂xi ∂xj

p(z)

|vxx |

=

d ∑

2 |Dij v|

p(z)

.

i,j=1

Definition 2.1. A function u is called weak solution of problem (1.1) if 1. u ∈ C 0 ([0, T ]; L2 (Ω )) ∩ W(QT ), ut ∈ W′ (QT ); 2. for every test-function ϕ ∈ W(QT ), ϕt ∈ W′ (QT ) ∫ ∫ p(x,t)−2 ⟨ut , ϕ⟩W′ ,W + |∇u| ∇u · ∇ϕ dz = QT

3. for every ϕ(x) ∈ C01 (Ω )

f ϕ dz;

QT

∫ (u(x, t) − u0 (x))ϕ(x) dx → 0

as t → 0;

Ω

4. a weak solution is called strong solution if ut ∈ L2 (QT ),

p(z)

|∇u|

∈ L∞ (0, T ; L1 (Ω ))

and for every test-function ϕ ∈ W(QT ) ∫ ∫ ( ) p(x,t)−2 ut ϕ + |∇u| ∇u · ∇ϕ dz = QT

f ϕ dz.

QT

Let us recall the known results on the existence of weak and strong solution of problem (1.1). Proposition 2.1 ([6,19]). If ⎧ Ω ⊂ Rd , d ≥ 2, with the Lipschitz-continuous boundary ∂Ω , ⎪ ⎪ ⎨ 2d < p− ≤ p(z) ≤ p+ < ∞, p(z) ∈ Clog (QT ), ⎪ d +2 ⎪ ⎩ u0 ∈ L2 (Ω ), f ∈ L2 (QT ), then problem (1.1) has a unique weak solution which satisfies the estimate ∫ p(z) ess sup ∥u(t)∥22,Ω + |∇u| dz ≤ CM0 (0,T )

QT

(2.1)

(2.2)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

6

with a constant C = C(T, |Ω |, p± ) and M0 = 1 + ∥u0 ∥22,Ω + ∥f ∥22,QT . Conditions (2.1) guarantee the existence of a unique weak solution in the sense of Definition 2.1, i.e., a solution u ∈ W(QT ) with ut ∈ W′ (QT ). Notice that in [19] the existence of a unique weak solution is proven for p(z) ∈ (1, ∞) without any restriction on p− from below. However, the counterexample in [17, Ch.XII] 2d shows that Eq. (1.1) with constant p ∈ (1, d+2 ] admits unbounded solutions and, thus, in this range of p higher regularity of the solution cannot be expected. Under further restrictions on the data the solutions possess better regularity and convert into strong solutions in the sense of Definition 2.1. Proposition 2.2 (Th.1.1, [5]). Assume that in the conditions of Proposition 2.1 (a)

Ω is a convex domain with the boundary ∂Ω ∈ C 2 ,

(b)

u0 ∈ W01,2 (Ω ),

|f | p(z) ∈ L1 (QT ),

(c)

∥∇p∥∞,QT ≤ C ∗ ,

∥pt ∥∞,QT ≤ C∗ ,

4

(2.3) C∗ , C ∗ = const.

If p+ < 2, then the weak solution of problem (1.1) is a strong solution in the sense of Definition 2.1 and the following estimate holds: ∫ ( ) p(z)−2 2 p(z) 2 2 sup ∥∇u(·, t)∥2,Ω + ∥ut ∥2,QT + |∇u| |uxx | + |∇u| dz (0,T )

QT

(

≤ C 1 + ∥∇u0 ∥22,Ω +

∫

4

|f | p(z)

(2.4)

) dz.

QT

The proof of Proposition 2.2 given in [5] relied on the assumption p+ < 2, which was used to ensure p(z) global integrability of the term |∇u| | ln |∇u ∥. We will show that Proposition 2.2 continues to be true if the condition p+ < 2 is removed and substituted by p+ = 2. The further regularity of strong solutions is studied for u0 and f subject to the conditions |f |

p′ (z)

∈ L1 (QT ),

p0 −2

u0 ∈ W01,2 (Ω ), Φ := div(|∇u0 |

ft ∈ L2 (QT ), |∇u0 |

p0 −2

Di u0 ∈ W 1,2 (Ω ),

2

∇u0 ) ∈ L (Ω ),

i = 1, . . . , d,

(2.5)

p0 = p(x, 0) = lim p(x, t). t→0+

The main result for the singular parabolic equation is given in the following theorem. Theorem 2.1. (i) Assume that conditions (2.1), (2.3) are fulfilled and p+ ≤ 2. Then every weak solution of problem (1.1) is a strong solution which satisfies estimate (2.4). ′ (ii) If conditions (2.1), (2.3), (2.5) are fulfilled and Φ ∈ Lp0 (·) (Ω ), then the solution of problem (1.1) satisfies the estimate ess sup ∥ut (t)∥22,Ω + ess sup ∥∇u(t)∥22,Ω ≤ CM1 (2.6) (0,T )

(0,T )

( ) with constants C = C (T, |Ω |, p± , C∗ ) and M1 = M1 ∥Φ∥p′ (Ω) , ∥u0 ∥22,Ω , ∥f ∥22,QT , ∥ft ∥22,QT . 0

(iii) Let the conditions of item (ii) be fulfilled and p− ≥

2(d + 1) . d+2

(2.7)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

7

( )d then the solution of problem (1.1) satisfies estimates (2.4), (2.6), ∇u ∈ C 0 ([0, T ]; L2 (Ω )) , and the following estimate holds: ∫ ( ) p′ (z) p(z) p(z)−2 2 p(z) sup ∥∇u(t)∥22,Ω + |ut | + |∇ut | + |∇u| |∇ut | + |uxx | dz (0,T )

QT

( ∫ ≤ C 1 + ∥∇u0 ∥22,Ω +

p′ (z)

|f |

)

(2.8)

dz

QT

with constants C = C(T, |Ω |, p± , C ∗ , C∗ , ∥∇u0 ∥2,Ω , ∥f ∥p′ (·),QT ) and M1 . The existence of weak and strong solutions of problem (1.1) stated in Propositions 2.1, 2.2 is proved in [5,6,19] with the method of Galerkin. Following [5], we obtain the strong solution of problem (1.1) as the limit when ϵ → 0 of the family of solutions of the non-singular problems ⎧ ( ) 2 p(z)−2 2 ⎪ 2 ⎪ u = div (ϵ + |∇u | ) ∇u ϵt ϵ ϵ + f (z) in QT , ⎨ (2.9) uϵ = 0 on ∂Ω × (0, T ), ⎪ ⎪ ⎩ u (x, 0) = v (x) in Ω , ϵ > 0, ϵ

ϵ

where vϵ approximate the initial datum u0 . The solution of problem (2.9) with ϵ > 0, u ≡ uϵ , is understood in the sense of Definition 2.1. It is sought as the limit of the sequence u(m) (x, t) =

m ∑

ui,m (t)ψi (x),

(2.10)

i=1

where {ψi } are the eigenfunctions of the Dirichlet problem for the Laplace operator. Let us denote (

2

γϵ (z, s) = ϵ + |s|

2

) p(z)−2 2

,

s ∈ Rd , z ∈ QT , ϵ > 0.

(2.11)

The coefficients ui,m (t) are defined as the solutions of the Cauchy problem for the system of m ordinary nonlinear differential equations ∫ ∫ u′i,m (t) = − γϵ (z, ∇u(m) )∇u(m) · ∇ψi dx + f (z)ψi dx, (2.12) Ω Ω (m) ui,m (0) = vi , i = 1, 2, . . . , m, (m)

where the constants vi

are chosen so that v (m) =

m ∑

(m)

vi

ψi (x) → vϵ (x)

in L2 (Ω ).

i=1

By the Carath´eodory theorem for every finite m system (2.12) has a continuous solution on an interval (0, Tm ). In our approach, the study of the existence and regularity properties of solutions to problem (1.1) reduces (m) to deriving uniform in m and ϵ a priori estimates for the functions uϵ and their derivatives. These estimates allow one to justify passing to the limit as m → ∞ and then as ϵ → 0, and provide the same regularity properties of the solutions of problems (2.9) and (1.1). We omit the rigorous formulation of the results on the regularity of solutions of the regularized problem (2.9). These results follow immediately from the uniform estimates (5.1), (5.10) in Lemmas 5.1, 5.2, estimate

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

8

(5.11) and Lemma 5.3: under the conditions of Theorem 2.1 the Galerkin approximations of the solutions of problem (2.9) satisfy the estimates ∫ (m) (m) 2 2 sup ∥uϵt (t)∥22,Ω + sup ∥∇u(m) (t)∥ + γϵ (z, ∇u(m) )|∇uϵt | dz ≤ L1 , ϵ 2,Ω ϵ (0,T )

∫

(0,T )

(

QT 2

(m) γϵ (z, ∇u(m) )|u(m) ϵ ϵxx | + |uϵxx |

p(z)

(m) p(z)

+ |∇uϵt |

′ (m) p (z)

)

+ |uϵt |

dz ≤ L2

QT

with constants Li depending on the data but independent of m and ϵ. In order to estimate the higher-order derivatives of the solution of problem (2.9) up to the initial moment one has to require that the data satisfy the necessary compatibility conditions at the instant t = 0. To this end we consider the equation for uϵt obtained from (2.9) by means of formal differentiation in t and endowed with the natural initial condition given by Eq. (2.9) at the instant t = 0. The next step is to construct a sequence of approximations for the initial function u0 which simultaneously provide the initial conditions for both uϵ and uϵt . This problem leads to an elliptic problem which is interesting on its own. We construct vϵ (x) as the weak solutions of the homogeneous Dirichlet problem for the equations ) ( 2 p0 (x)−2

div (ϵ2 + |∇vϵ | )

2

∇vϵ

p0 (x)−2

= div(|∇u0 |

∇u0 ),

ϵ > 0,

with p0 = lim p(x, t) as t → 0+ . The functions vϵ are obtained as the limits of Galerkin’s approximations in the same basis {ψi } as the solutions of the parabolic problems (2.9). Let us consider the elliptic problem div(γϵ (x, 0, ∇v)∇v) = Φ

in Ω ,

v = 0 on ∂Ω

(2.13)

with a parameter ϵ > 0 and the function γϵ defined in (2.11). Definition 2.2. A function v ∈ V (Ω ) is called weak solution of problem (2.13) if ∫ (γϵ (x, 0, ∇v)∇v · ∇η + Φη) dx = 0 ∀η ∈ V (Ω ).

(2.14)

Ω

Theorem 2.2. Let us assume that ⎧ Ω ⊂ Rd , d ≥ 2, with the boundary ∂Ω ∈ Lip, ⎪ ⎪ ⎨ ( 2d ] ,2 , p0 = p(x, 0) ∈ Clog (Ω ), p0 ∈ [p− , p+ ] ⊂ ⎪ d+2 ⎪ ⎩ Φ ∈ L2 (Ω ).

(2.15)

(i) For every ϵ ∈ (0, 1) problem (2.13) has a unique weak solution vϵ ∈ V (Ω ) and ( ) 1 p− −1 ∥vϵ ∥2,Ω + ∥vϵ ∥V (Ω) ≤ C 1 + ∥Φ∥2,Ω with an independent of vϵ constant C. (ii) If Φ = ∆p0 u0 and u0 satisfies conditions (2.5), then limϵ→0 ∥vϵ − u0 ∥q,Ω = 0 with 2 ≤ q <

(2.16)

dp− . d−p−

Theorem 2.3. Let conditions (2.15) be fulfilled. Assume that Ω is a convex domain with the boundary ∂Ω ∈ C 2 , ′ p0 Φ ∈ Lp0 (Ω ), p′0 = ≥ 2, ∥∇p0 ∥∞,Ω ≤ C ∗ . p0 − 1

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

(i) |vϵxx |

p0

9

2 p0 −2

∈ L1 (Ω ), (ϵ2 + |∇vϵ | ) 4 |vϵxx | ∈ L2 (Ω ) and there holds the estimate ( ) ∫ ( ∫ ) 2 p0 −2 2 p p′ ∥vϵ ∥22,Ω + (ϵ2 + |∇vϵ | ) 2 |vϵxx | + |vϵxx | 0 dx ≤ C 1 + |Φ| 0 dx Ω

(2.17)

Ω

with an independent of vϵ , ϵ and Φ constant C. (ii) If Φ = ∆p0 u0 with u0 satisfying conditions (2.5), then ∥vϵ − u0 ∥W 1,2 (Ω) → 0

as ϵ → 0.

0

2,2 The Wloc (Ω )-regularity of solutions of the singular elliptic problems of the type (2.13) was studied in [13]. The global W 2,2 (Ω ) ∩ W01,1 (Ω )-regularity of solutions of the Dirichlet problem for Eq. (2.13) with ϵ ≥ 0 is studied in [16] in the case d = 2. If Φ ∈ Lr (Ω ) with a sufficiently large r > 2, then the solutions of problem (2.13) with constant exponent p0 ∈ (p, 2] belong to W 2,q (Ω ) ∩ W01,p0 (Ω ) with q ≥ 2, [11]. The threshold value p depends on d, r, p0 and the constant C(q) in (1.2). An analogous result is true for the solutions of problem (2.13) with variable exponent p ≤ p0 (x) ≤ p+ < 2, [14]. We also refer to [1,2] for the study of the regularity issues for elliptic equations with nonstandard growth and a review of the pertinent literature. The proofs of Theorems 2.2, 2.3 are given in Section 3. In Section 4 we collect the basic a priori estimates for the solutions of the system of ODEs (2.12) and sketch the proof of existence of strong solutions to problems (2.9) and (1.1). Item (iii) of Theorem 2.1 is proved in Section 5. We derive new a priori estimates (m) for the finite-dimensional approximations uϵ of the solutions of problem (2.9) and their derivatives in x and t. These uniform with respect to m and ϵ estimates provide the regularity properties of uϵ and u = lim uϵ .

Notation. Throughout the text the symbol C denotes the constants which can be calculated or estimated through the known quantities, but whose exact value is unimportant for the argument and may change from line to line even inside the same formula. 3. The nonlinear elliptic problem. the choice of the sequence {vϵ } 3.1. Existence of weak solutions: proof of Theorem 2.2 Let {ψi } be the orthonormal basis of L2 (Ω ) composed of solutions of the problem (ψi , ϕ)W 1,2 (Ω) = λi (ψi , ϕ) ∀ϕ ∈ W01,2 (Ω ).

(3.1)

0

−1

The system {λi 2 ψi } forms an orthogonal basis of W01,2 (Ω ). Let us denote Pm ≡ span{ψ1 , . . . , ψm } and ∑∞ (m) represent Φ = i=1 Φi ψi , Φ (m) (x) =

m ∑

(m)

Φi

ψi (x) → Φ(x)

in L2 (Ω ).

i=1

The solution v of problem (2.13) will be obtained as the limit of the sequence of finite-dimensional Galerkin’s approximations in the system {ψi } v = lim v (m) , m→∞

v (m) =

m ∑

(m)

vi

ψi (x) ∈ Pm .

i=1

(m)

The coefficients vi , i = 1, . . . , m, are defined as the solutions of the system of m nonlinear algebraic equations ∫ ∫ γϵ (x, 0, ∇v (m) )∇v (m) · ∇ψi dx = − Φ (m) ψi dx, i = 1, . . . , m. (3.2) Ω

Ω

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

10

3.1.1. Solvability of the finite-dimensional problems Lemma 3.1. Let Φ ∈ L2 (Ω ). For every m ∈ N the nonlinear algebraic system (3.2) has a solution v = (v1 , . . . , vm ) ∈ Rm . The assertion of Lemma 3.1 immediately follows from the Leray–Schauder principle. The applicability of this principle to the nonlinear finite-dimensional mapping defined by system (3.2) stems from several auxiliary lemmas. The m-dimensional subspace Pm of V (Ω ) can be endowed with the norm ∥v∥Pm = ∥v∥V (Ω) . Pm is isomorphic to the space v = (v1 , . . . , vm ) ∈ Rm , which is equipped with the usual scalar product and the norm m ∑ 2 (u, v)m = ui vi , |u|m = (u, u)m . i=1

A solution of problem (3.2) is sought as a fixed point of the mapping G : Pm × [0, 1] ↦→ Pm defined as follows: for every w ∈ Pm and τ ∈ [0, 1] v = G(w, τ ) ∈ Pm is the solution of the linear problem ∫ ∫ γϵ (x, 0, τ ∇w)∇v · ∇ψi dx = −τ Φ (m) ψi dx, i = 1, . . . , m. (3.3) Ω

Ω

Lemma 3.2. For every w ∈ Pm the problem v = G(w, 0) only has the trivial solution. ˜ the constant from the embedding inequality Proof . Let v = G(w, 0). Denote by C u ∈ W01,2 (Ω ).

˜ ∥u∥2,Ω ≤ C∥∇u∥ 2,Ω ,

(3.4)

By virtue of (3.3) m ∑

∫

p0 −2

vi

i=1

ϵ Ω

∫ ∇v · ∇ψi dx =

2

ϵp0 −2 |∇v| dx = 0

Ω

and for ϵ ∈ (0, 1) p− −2

ϵ

2 |v|m

=ϵ

p− −2

∥v∥22,Ω

˜ 2 ϵp− −2 ∥∇v∥22,Ω ≤ C ˜2 ≤C

∫

2

ϵp0 −2 |∇v| dx = 0. □

Ω

Lemma 3.3. The mapping G(τ, w) is continuous with respect to τ on the set {|w|m ≤ R}. Proof . Let ui = G(w, τi ), i = 1, 2, be the solutions of problem (3.3) corresponding to different τ . By the mean value theorem ⏐∫ 1 ⏐ ⏐ ⏐ d 2 2 p0 −2 |γϵ (x, 0, τ1 ∇w) − γϵ (x, 0, τ2 ∇w)| = ⏐⏐ (ϵ + (θτ1 + (1 − θ)τ2 )|∇w| ) 2 dθ⏐⏐ 0 dθ ∫ 1 2 − p0 2 p0 −4 2 (3.5) |τ1 − τ2 | (ϵ2 + (θτ1 + (1 − θ)τ2 )|∇w| ) 2 dθ|∇w| = 2 0 2 − p− p− −4 2 2 ≤ ϵ |τ1 − τ2 |Km |w|m , 2 where Km = sup{|∇ψi | : x ∈ Ω , i = 1, m}.

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

11

Combining (3.3) for ui and using (3.5) we find that ∫ 2 γϵ (x, 0, τ2 ∇w)|∇(u1 − u2 )| dx ≤ |τ1 − τ2 |∥Φ (m) ∥2,Ω ∥u1 − u2 ∥2,Ω Ω ∫ − 2 2 + |τ1 − τ2 |ϵp −4 Km |w|m |∇u1 ||∇(u1 − u2 )| dx Ω ( ) 2 2 ˜ (m) ∥2,Ω + ϵp− −4 Km ≤ |τ1 − τ2 | C∥Φ |w|m ∥∇u1 ∥2,Ω ∥∇(u1 − u2 )∥2,Ω , ˜ from (3.4). For every |w| ≤ R, ϵ ∈ (0, 1) and τ2 ∈ [0, 1] with the constant C m γϵ (x, 0, τ2 ∇w) = ≥

1 2 2−p0

(ϵ2 + τ22 |∇w| ) 1 2

2 ) (1 + |w|m Km

2

2−p− 2

2 ≥ (1 + R2 Km )

p− −2 2

(3.6) := σm .

Using (3.6) and simplifying, we obtain the estimate ( ) σm ∥∇(u1 − u2 )∥2,Ω ≤ C|τ1 − τ2 | ∥Φ (m) ∥2,Ω + ∥∇u1 ∥2,Ω with an independent of τi constant C. The estimate on ∥∇u1 ∥2,Ω follows from (3.3): ∫ 2 2 σm ∥∇u1 ∥2,Ω ≤ γϵ (x, 0, τ1 ∇w)|∇u1 | dx Ω

˜ (m) ∥2,Ω ∥∇u1 ∥2,Ω . ≤ τ1 ∥Φ (m) ∥2,Ω ∥u1 ∥2,Ω ≤ C∥Φ

□

Lemma 3.4. All fixed points of the mapping G(w, τ ) belong to the ball {|u|m < ρ} of radius ρ=

p−

1 1 M + M p− −1 , −1

M = 21+

2−p− 2

2 ˜ 2 (1 + Km C )

2−p− 2

∥Φ (m) ∥2,Ω ,

˜ from (3.4). with the constants Km from (3.6) and C Proof . Let u ∈ Pm be a fixed point of the mapping G(w, τ ) with |u|m = R. By (3.3) ∫ 2 p0 −2 2 2 σm ∥∇u∥2,Ω ≤ (ϵ2 + |∇u| ) 2 |∇u| dx Ω ∫ 2 ≤ γϵ (x, 0, τ ∇u)|∇u| dx Ω ∫ ˜ (m) ∥ ∥∇u∥ ≤τ |Φ (m) ||u| dx ≤ 2C∥Φ 2,Ω 2,Ω Ω

with σm from (3.6), whence ∥∇u∥2,Ω ≤

˜ 2C ∥Φ (m) ∥2,Ω σm

and ˜2 2−p− 2C ˜ ˜ 2 (1 + R2 K 2 ) 2 ∥Φ (m) ∥2,Ω R = |u|m ≤ C∥∇u∥ ∥Φ (m) ∥2,Ω = 2C 2,Ω ≤ m σm −

≤ M (1 + R2−p ),

M = 21+

2−p− 2

2 ˜ 2 (1 + Km C )

2−p− 2

∥Φ (m) ∥2,Ω .

Applying Young’s inequality to the second term on the right-hand side of this inequality we arrive at the estimate 1 (p− − 1)|u|m ≤ M + (p− − 1)M p− −1 . □

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

12

Proof of Lemma 3.1. Since the finite rank operator G(u, τ ) : Pm ↦→ Pm is compact, the existence of a fixed point of the mapping G(v, 1) follows now from Lemmas 3.2, 3.3, 3.4 and the Leray–Schauder principle — see, e.g., [22, Ch.4, Sec.8]. □ 3.1.2. Uniform a priori estimates Lemma 3.5. For every u ∈ V (Ω ) ∫

∫

2

p0

γϵ (x, 0, ∇u)|∇u| dx ≥ |∇u| Ω Ω { } + − 3 ≥ 2− 2 min ∥u∥pV (Ω) , ∥u∥pV (Ω) − C

J(∇u) :=

with the constant C =

dx − C (3.7)

2 − p− p− −2 2 2 |Ω |. p−

Proof . For every u ∈ V (Ω ) |∇u|

∫ J(∇u) = Ω

2

∫

2 2−p0

(ϵ2 + |∇u| )

∫

dx =

. . . ≡ I+ + I− .

... + Ω∩{|∇u|>ϵ}

2

Ω∩{|∇u|≤ϵ}

The estimate on I+ is straightforward: |∇u|

∫ I+ > Ω∩{|∇u|>ϵ}

2

2 2−p0

(2|∇u| )

dx > 2

p− −2 2

|∇u|

I− ≥ Ω∩{|∇u|≤ϵ}

1 p− −2 ≥ +2 2 p Lemma 3.6. If p− >

2d d+2 ,

(2ϵ2 )

2

2−p0 2

dx ≥ 2

p− −2 2

p0 2 2 a

∫ Ω∩{|∇u|≤ϵ}

∫ |∇u|

p0

dx −

Ω∩{|∇u|≤ϵ}

p0

|∇u|

dx.

Ω∩{|∇u|>ϵ}

2

To estimate I− we apply Young’s inequality in the form ap0 ≤ ∫

∫

+

(

2−p0 p0

(a > 0):

2 2 p |∇u| 0 + 1 − p0 p0

) dx

2 − p− p− −2 2 2 |Ω |. □ p−

then for every m ∈ N the solution of problem (3.2) satisfies the estimate

( ) 1 1 p− −1 p+ −1 ∥v (m) ∥2,Ω + ∥v (m) ∥V (Ω) ≤ C 1 + ∥Φ∥2,Ω + ∥Φ∥2,Ω with independent of ϵ and m constant C. Proof . By virtue of (3.2) with η = v (m) and H¨ older’s inequality ∫ J(∇v (m) ) = v (m) Φ (m) dx ≤ ∥v (m) ∥2,Ω ∥Φ (m) ∥2,Ω .

(3.8)

Ω

Applying Lemma 3.5 and the embedding inequality ∥v (m) ∥2,Ω ≤ C ′ ∥∇v (m) ∥p0 ,Ω ≤ C ′ ∥v (m) ∥V (Ω) we obtain { } + − 3 2− 2 min ∥v (m) ∥pV (Ω) , ∥v (m) ∥pV (Ω) − C ≤ J(∇v (m) ) ≤ C ′ ∥v (m) ∥V (Ω) ∥Φ (m) ∥2,Ω

(3.9)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

13

with the constant C from (3.7). By Young’s inequality, this estimate can be written in the form 3

+

+

(p+ )′

3

−

−

(p− )′

2− 2 ∥v (m) ∥pV (Ω) ≤ C + δ∥v (m) ∥pV (Ω) + C(δ, p+ )∥Φ (m) ∥2,Ω

if ∥v (m) ∥V (Ω) < 1,

2− 2 ∥v (m) ∥pV (Ω) ≤ C + δ∥v (m) ∥pV (Ω) + C(δ, p− )∥Φ (m) ∥2,Ω

if ∥v (m) ∥V (Ω) ≥ 1 5

with any δ > 0. The needed estimate follows from these inequalities if we choose δ = 2− 2 , notice that ∥Φ (m) ∥2,Ω ≤ ∥Φ∥2,Ω , and add (3.9). □ ′

Corollary 3.1. The functions {|γϵ (x, 0, ∇v (m) )∇v (m) |} are uniformly bounded in Lp0 (·) (Ω ). Proof . For every ξ ∈ Rd and ϵ ∈ (0, 1) 2 p0 −2

|γϵ (x, 0, ξ)ξ| = (ϵ2 + |ξ| ) whence ∫ (γϵ (x, 0, ∇v

(m)

)|∇v

(m)

∫

p′0

|)

dx ≤

Ω

2

2

2 p0 −1

|ξ| ≤ (ϵ2 + |ξ| )

(ϵ +

2

≤2

2 p0 −1 p0 |∇v (m) | ) 2 p0 −1

p0 −1 2

(1 + |ξ|

(

p0 −1

),

∫

dx ≤ C 1 +

|∇v

(m) p0

)

|

dx.

Ω

Ω

The conclusion follows from Lemma 3.6 and (1.3). □ 3.1.3. Passing to the limit The uniform a priori estimates on v (m) allow one to extract from {v (m) } a subsequence, (for which we keep the same notation), such that v (m) → v in L2 (Ω ) and a.e. in Ω , ∇v (m) ⇀ ∇v in (Lp0 (Ω ))d ,

(3.10) p′0

γϵ (x, 0, ∇v (m) )∇v (m) ⇀ A in (L (Ω ))d ′

with some v ∈ V (Ω ), A ∈ (Lp0 (Ω ))d . Let us fix some k ∈ N. By virtue of (3.2), for every m ≥ k ∫ ∫ (m) (m) γϵ (x, 0, ∇v )∇v · ∇η dx = − Φ (m) η dx ∀η ∈ Pk . Ω

(3.11)

Ω

Letting m → ∞ we have

∫

∫ A · ∇η dx = −

Ω

Φη dx

η ∈ Pk .

Ω

Since {ψi } is dense in V (Ω ), we may now let k → ∞ to conclude that the same is true for every η ∈ V (Ω ). To identify the limit A we make use of the monotonicity of γϵ (x, 0, ξ)ξ. Proposition 3.1 ([8], Sec.3). For all x ∈ Ω , ξ, ζ ∈ Rd , (ξ ̸= ζ) and ϵ > 0 2

2 p0 −2

(γϵ (x, 0, ξ)ξ − γϵ (x, 0, ζ)ζ, ξ − ζ) ≥ C(1 + |ξ| + |ζ| )

2

2

|ξ − ζ|

with a constant C independent of ϵ, ξ, ζ. Let us take in (3.11) η = v (m) . For every ψ ∈ Pm ∫ ∫ (m) (m) (m) 0= γϵ (x, 0, ∇v )∇v · ∇v dx + Φ (m) v (m) dx Ω ∫Ω ∫ (m) (m) (m) = γϵ (x, 0, ∇v )∇v · ∇(v − ψ) dx + Φ (m) v (m) dx Ω ∫ Ω + γϵ (x, 0, ∇v (m) )∇v (m) · ∇ψ dx. Ω

(3.12)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

14

By Proposition 3.1 ∫ ∫ (m) (m) (m) γϵ (x, 0, ∇v )∇v · ∇(v − ψ) dx ≥ γϵ (x, 0, ∇ψ)∇ψ · ∇(v (m) − ψ) dx. Ω

Ω

Plugging this inequality into (3.12) we obtain ∫ ∫ ∫ 0≥ γϵ (x, 0, ∇ψ)∇ψ · ∇(v (m) − ψ) dx + Φ (m) v (m) dx + γϵ (x, 0, ∇v (m) )∇v (m) · ∇ψ dx. Ω

Ω

Ω

Since each term of the last relation has a limit as m → ∞, we arrive at the inequality ∫ ∫ ∫ 0≥ γϵ (x, 0, ∇ψ)∇ψ · ∇(v − ψ) dx + Φv dx + A · ∇ψ dx Ω Ω ∫Ω = (γϵ (x, 0, ∇ψ)∇ψ − A) · ∇(v − ψ) dx. Ω

Let us choose ψ in the special way: ψ = v + λζ with λ > 0 and an arbitrary ζ ∈ V (Ω ). The last inequality takes the form ∫ λ (γϵ (x, 0, ∇v + λ∇ζ)∇(v + λζ) − A) · ∇ζ dx ≤ 0. Ω

Simplifying and letting λ → 0 we arrive at the inequality ∫ (γϵ (x, 0, ∇v)∇v − A) · ∇ζ dx ≤ 0

∀ζ ∈ V (Ω ),

Ω

which is impossible unless ∫ (γϵ (x, 0, ∇v)∇v − A) · ∇ζ dx = 0

∀ζ ∈ V (Ω ).

Ω

The proof of existence of a weak solution is completed. Estimate (2.16) follows from Lemma 3.6. 3.1.4. Theorem 2.2(ii): convergence as ϵ → 0 Let {vϵ } be the family of solutions of problem (2.13). According to (2.16) ∥vϵ ∥V (Ω) are uniformly bounded, ′ by Corollary 3.1 γϵ (x, 0∇vϵ )∇vϵ are uniformly bounded in (Lp0 (Ω ))d . It follows that {vϵ } is precompact in − ′ dp q p0 d Lq (Ω ) with q < d−p − and there exist v ∈ L (Ω ), χ ∈ (L (Ω )) such that vϵ → v in Lq (Ω ) and a.e. in Ω , ∇vϵ ⇀ ∇v in (Lp0 (Ω ))d , ′

γϵ (x, 0, ∇vϵ )∇vϵ ⇀ χ in (Lp0 (Ω ))d . Letting ϵ → 0 in (2.14) we find that ∫ (χ · ∇ψ − Φψ) dx = 0

∀ψ ∈ V (Ω ).

Ω

Take ψ ∈ V (Ω ) and write (2.14) in the form ∫ ∫ (γϵ (x, 0, ∇vϵ )∇vϵ − γϵ (x, 0, ∇ψ)∇ψ)∇(vϵ − ψ) dx + Φvϵ dx Ω Ω ∫ ∫ p −2 + γϵ (x, 0, ∇vϵ )∇vϵ · ∇ψ dx + |∇ψ| 0 ∇ψ · ∇(vϵ − ψ) dx Ω Ω ∫ p −2 + (γϵ (x, 0, ∇ψ) − |∇ψ| 0 )∇ψ · ∇(vϵ − ψ) dx = 0. Ω

(3.13)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

15

The last term J of this equality tends to zero as ϵ → 0. Indeed: for the functions ′

ϕϵ = |γϵ (x, 0, ∇ψ) − |∇ψ|p0 −2 ∇ψ|p0 we have

′ p0 −1 p0

|ϕϵ | ≤ (2|∇ψ|

)

p0

≤ C|∇ψ|

∈ L1 (Ω ),

ϕϵ → 0 a.e. in Ω as ϵ → 0

whence, by the dominated convergence theorem, 1 p′ ϵ 0

J ≤ 2∥|ϕ |

p+ −1 p−

{

∥p′ (·),Ω ∥∇(vϵ − ψ)∥p0 (·),Ω ≤ C max ∥ϕϵ ∥1,Ω 0

p− −1 p+

}

, ∥ϕϵ ∥1,Ω

→0

as ϵ → 0.

Using monotonicity and the choice of v, in (3.13) we may pass to the limit as ϵ → 0, which leads to the inequality ∫ p −2 (χ − |∇ψ| 0 ∇ψ) · ∇(ψ − v) dx ≤ 0 ∀ψ ∈ V (Ω ). Ω

It follows that

∫ (χ − |∇ψ|

p0 −2

∀ψ ∈ V (Ω )

∇ψ) · ∇(ψ − v) dx = 0

Ω

(see Section 3.1.3). Thus, the limit function v is a weak solution of problem (2.13) with ϵ = 0. On the other hand, u0 is also a weak solution of the same problem. Combining relations (2.14) for u0 and v with the test function u0 − v we obtain ∫ p −2 p −2 (|∇v| 0 ∇v − |∇u0 | 0 ∇u0 ) · ∇(v − u0 ) dx = 0. Ω

Using the well-known inequality p0 −2

(|ξ|

p0 −2

ξ − |ζ|

p0

2

ζ, ξ − ζ) ≥ (p0 − 1)|ξ − ζ| (|ξ|

p

+ |ζ| 0 )

p0 −2 p0

∀ξ, ζ ∈ Rd ,

and noting that p− (2 − p0 ) ≤ p0 (2 − p− ), we have: ∫ ∫ p− p0 −2 p− 2−p0 p− p− p p p p |∇(v − u0 )| (|∇v| 0 + |∇u0 | 0 ) 2 p0 (|∇v| 0 + |∇u0 | 0 ) 2 p0 dx |∇(v − u0 )| dx = Ω

Ω

(∫ ≤

p0

2

|∇(v − u0 )| (|∇v|

p

+ |∇u0 | 0 )

p0 −2 p0

Ω

)1− p2− ) p2− (∫ p− (2−p0 ) p p dx (|∇v| 0 + |∇u0 | 0 ) p0 (2−p− ) dx Ω

p− 1− 2

( ) ∫ p p ≤ C 1 + (|∇v| 0 + |∇u0 | 0 ) dx Ω

(∫ ×

(|∇v|

p0 −2

p0 −2

∇v − |∇u0 |

) p2− ∇u0 ) · ∇(v − u0 ) dx = 0.

Ω

It follows that v = u0 a.e. in Ω . 3.2. Higher regularity of weak solutions. 3.2.1. Proof of Theorem 2.3(i) ′

Lemma 3.7. Let p0 (x) satisfy conditions (2.15). For every w ∈ Lp0 (Ω ) there exists a sequence {w(m) } such that w(m) ∈ Pm and ∥w(m) − w∥p′ (·),Ω → 0 as m → ∞. 0

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

16

′

2d Proof . The inequalities d+2 < p− ≤ p0 ≤ 2 yield continuous embedding W01,2 (Ω ) ⊂ Lp0 (Ω ). Fix ′ some δ > 0. Since p0 ∈ Clog (Ω ), the set C0∞ (Ω ) is dense in both W01,2 (Ω ) and Lp0 (Ω ), and there exists wδ ∈ C0∞ (Ω ) such that ∥w − wδ ∥p′ ,Ω < δ. For wδ ∈ C0∞ (Ω ) ⊂ W01,2 (Ω ) 0

∥wδ ∥2W 1,2 (Ω) = 0

(m) wδ

∞ ∑

λi (wδ , ψi )22,Ω < ∞,

i=1 m ∑

=

(wδ , ψi )2,Ω ψi (x) → wδ in W01,2 (Ω ).

i=1 (m)

Then there is m such that ∥wδ − wδ (m)

∥w − wδ

(m)

∥p′ ,Ω ≤ C(p± )∥wδ − wδ 0

(m)

∥p′ ,Ω ≤ ∥w − wδ ∥p′ ,Ω + ∥wδ − wδ 0

0

∥W 1,2 (Ω) < δ and 0

(m)

∥p′ ,Ω ≤ δ + C(p± )∥wδ − wδ 0

∥W 1,2 (Ω) < 2δ. 0

′

□ ′

Given Φ ∈ Lp0 (Ω ), one may choose a sequence {Φ (m) } such that Φ (m) ∈ Pm and Φ (m) → Φ in Lp0 (Ω ). By virtue of (3.10) the corresponding sequence {v (m) } converges then to the solution v ≡ vϵ of problem dp− (2.13), and by Theorem 2.2(ii) vϵ converges to u0 in Lq (Ω ) with q ∈ [2, d−p − ). Lemma 3.8. Under the conditions of Theorem 2.3 the functions v (m) satisfy the estimates ( ) ∫ ( ∫ ) 2 p0 −2 (m) 2 p′ (m) p0 (ϵ2 + |∇v (m) | ) 2 |vxx | + |vxx | dx ≤ C 1 + |Φ| 0 dx Ω

(3.14)

Ω

with a constant C which does not depend on m and ϵ. Proof . The proof of Lemma 3.8 is split into three steps. Since the parameters m, ϵ and the function v (m) do not change throughout the proof, by convention we shall use the notation v ≡ v (m) ,

2 p0 (x)−2

γϵ ≡ γϵ (x, 0, ∇v (m) ) = (ϵ2 + |∇v (m) | )

2

.

2

Step 1: estimate on γϵ |vxx | . We prove first the auxiliary estimate: for every m ∈ N ∫

(

2

γϵ |vxx | dz ≤ C 1 + Ω

∫ (

2

2

ϵ + |∇v|

) p20

ln

2

(

2

2

ϵ + |∇v|

)

) dx + |IΦ | ,

(3.15)

Ω

∫

Φ (m) (x)∆v dx,

IΦ = −

(3.16)

Ω

with an independent of v positive constant C = C(p± ). Recall that ∂Ω ∈ C 2 by assumption. It follows from the classical regularity theory for the linear elliptic equations [20] that ψi ∈ C 2 (Ω ) ∩ C ∞ (Ω ), whence (m) v ≡ v (m) ∈ C 2 (Ω ) ∩ C ∞ (Ω ) for every finite m. Let us multiply each of equations (3.2) by λi vi , integrate by parts in Ω using Green’s formula (see [27, pp. 69–70]), and sum up the results: −

m ∑

(m)

λi v i

∫ (γϵ ∇v, ∇ψi )2,Ω =

i=1

For every i, k = 1, 2, . . . , d ( Dk (γϵ Di v) = γϵ

2 Dik v

+ (p0 − 2)Di v

Φ (m) (x)∆v dx =

Ω

div (γϵ ∇v) ∆v dx.

(3.17)

Ω

∇v · ∇(Dk v) ϵ2 + |∇v|

∫

2

) + γϵ Di v

) Dk p0 ( 2 2 ln ϵ + |∇v| . 2

(3.18)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

17

Let n = (n1 , . . . , nd ) be the vector of the exterior unit normal to ∂Ω . Integrating two times by parts in Ω and using (3.18), we represent the right side of (3.17) in the form ∫ div (γϵ ∇v)∆v dx =

∫ ∑ d

Ω

2 Dkk v

Ω k=1

d ∑

Di (γϵ Di v) dx

i=1

∫

d ∑

∫ γϵ (∇v · n)∆v dS −

= ∂Ω

Ω k,i=1 d ∑

∫ γϵ

= ∂Ω

(

2 Dkk vDi vni

−

2 Dki vDi vnk

)

d ∑

∫ dS +

(3.19) 2 Dki vDk

(γϵ Di v) dx

Ω k,i=1

k,i=1 d ∑

∫ =

3 Dkki v (γϵ Di v) dx

2

2 |Dki v| γϵ dx + I1 + I2 + I∂Ω ,

Ω k,i=1

where ⎛ ∫ (p0 − 2)γϵ ⎝

I1 = Ω

2 Dki vDi v

( (p0 − 2)γϵ

Ω

d ∑

ϵ2 + |∇v|

(∇v · ∇(Dk v))

(

2

(p0 − 2) ϵ + |∇v|

2

) p−2 2 −1

(

Ω

dx

2

ϵ2 + |∇v|

d ∑

) dx )

(∇v · ∇(Dk v))

2

dx,

k=1 d ∑

∫ I2 =

2

∇v · ∇(Dk v)

k=1

∫ =

⎞ ∇v · ∇(Dk v) ⎠

k,i=1

∫ =

d ∑

2 γϵ Dki vDi v

Ω k,i=1

) Dk p0 ( 2 2 ln ϵ + |∇v| dx, 2

∫ γϵ (∆v(∇v · n) − ∇v · ∇(∇v · n)) dS.

I∂Ω = ∂Ω

It is known that if ∂Ω ∈ C 2 and Ω is convex, then I∂Ω ≥ 0 for every v ≡ v (m) ∈ C 2 (Ω ) — see [21, Ch.1, Sec 1.5] for the proof in the cases d = 2, 3, the case of an arbitrary space dimension can be considered in the same way [5, Appendix A]. Combining (3.17) with (3.19) we arrive at the inequality ∫ γϵ

d (∑

Ω

2 |Dki v|

2

)

dx = −I1 − I2 − I∂Ω + IΦ ≤ |I1 | + |I2 | + |IΦ |

k,i=1

with IΦ defined in (3.16). Let us estimate I1 , I2 . The estimate on I1 is straightforward: −

∫

|I1 | ≤ (2 − p )

γϵ Ω

d ( ∑

2 Dki v

)2

dx.

k,i=1

To estimate I2 , we use the Young inequality: for every δ ∈ (0, 1) ⎛ ⎞ ( ) ∫ d ∑ 1 1 1 2 2 ⎝ |Dki v|γϵ2 ⎠ |∇v|γϵ2 | ln(ϵ2 + |∇v| )| dx |I2 | ≤ ∥∇p0 ∥∞,Ω 2 Ω i,k=1

∫ ≤δ

γϵ Ω

d ∑ k,i=1

2

2 |Dki v| dx + C(δ, C ∗ )

∫ ( Ω

2

ϵ2 + |∇v|

) p20

( ) 2 ln2 ϵ2 + |∇v| dx.

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

18

Gathering these estimates, we arrive at (3.15): inequality ∫ γϵ Ω

d ∑

) ( ∫ ( ( ) ) p0 2 2 2 2 (m) 2 ln2 ϵ2 + |∇v| dx + |IΦ | |Dki v | dx ≤ C 1 + ϵ2 + |∇v|

(3.20)

Ω

k,i=1

with a constant C which does not depend on v ≡ v (m) and ϵ. (m) p0 (x)

Step 2: estimate on |vxx |

. By Young’s inequality, for i, k = 1, 2, . . . , d ∫ ( ) p0 ) ( 2 2 ) p20 − p20 p0 2 2 2 − p0 ( 2 p0 2 2 2 γϵ dx ≤ ϵ + |∇v| dx. |Dki v| dx = (Dki v) γϵ + (Dki v) γϵ 2 2 Ω Ω Ω

∫

∫

(3.21)

The first term on the right-hand side of (3.21) is estimated by virtue of (3.15). The second term is estimated by ( ) ∫ ( ∫ ) p0 2 2 p (3.22) ϵ2 + |∇v| dx ≤ C 1 + |∇v| 0 dx Ω

Ω

with a constant C depending only on p± , |Ω |. Gathering (3.20), (3.21), (3.22), we conclude that ∫

p0

|vxx |

dx ≡

Ω

d ∫ ∑

2 |Dki v|

p0

dx

Ω

i,k=1

(

∫

≤C 1+

p0

|∇v|

dx +

∫ (

Ω

2

2

ϵ + |∇v|

) p20

(3.23) ln

2

(

2

2

ϵ + |∇v|

)

) dx + |IΦ | .

Ω

By Young’s inequality, for every σ ∈ (0, 1) ∫ ∫ ∫ p′ p0 (m) |IΦ | ≤ |Φ ||∆v| dx ≤ σ |vxx | dx + C(σ) |Φ (m) | 0 dx, Ω

Ω

Ω

and from (3.23) we obtain the estimate ∫ |vxx |

C0

p0

∫ dx ≤ 1 +

Ω

|∇v|

p0

dx +

∫ (

Ω

2

2

ϵ + |∇v|

) p20

ln

2

(

2

ϵ + |∇v|

2

)

∫ dx +

|Φ (m) |

p′0

dx

(3.24)

Ω

Ω

with a constant C0 > 0 that does not depend on v. ) ( ( ) p0 2 2 2 2 2 ln ϵ2 + |∇v| . Let us consider first the case of small oscillation Step 3: estimate on ϵ + |∇v| of p0 (x) in Ω : 1 1 1 − + < . (3.25) p− p d+2 Since

2d d+2

< p− ≤ p0 (x) ≤ p+ ≤ 2 by hypothesis, the following inequalities hold: for every µ > 0 and y > 0

y

p0 2

⎧ + ( ) ⎨y p 2+µ y − µ2 ln2 y if y ≥ 1, ln2 y ≤ ⎩ p2− 2 y ln y ≤ C(p− ) if y ∈ (0, 1).

It follows that 2 p0

(ϵ2 + |∇v| )

2

p+ +µ

2

ln2 (ϵ2 + |∇v| ) ≤ C(|∇v|

+ 1)

with a constant C depending only on p± , µ and, thus, (∫ ) ∫ ( ) p0 ( ) 2 2 2 p+ +µ ln2 ϵ2 + |∇v| dx ≤ C |∇v| dx + 1 ϵ2 + |∇v| Ω

Ω

(3.26)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

19

with C depending on p± , µ. To estimate the integral on the right-hand side we apply the Gagliardo– Nirenberg inequality. By virtue of (3.25) there exists a constant µ > 0 so small that ) ( 1 1 − ∈ (0, 1). α=d p− p+ + µ −

+

Then for every v ∈ W 2,p (Ω ) ∩ W 1,p

+µ

(Ω )

1−α ∥∇v∥p+ +µ,Ω ≤ C1 ∥vxx ∥α p− ,Ω ∥∇v∥p− ,Ω + C2 ∥∇v∥p− ,Ω

(3.27)

with an independent of v constants C1 , C2 . Condition (3.25) implies the inequality p+ + µ < 1. p− Applying Young’s inequality we may rewrite (3.27) in the form ∫ ∫ + p+ +µ p− |∇v| dx ≤ λ |vxx | dx + C(λ)∥∇v∥βp− ,Ω + C2 ∥∇v∥pp− +µ ,Ω α

Ω

(3.28)

Ω

with an arbitrary λ > 0 and

(1 − α)(p+ + µ) > p− . p− − α(p+ + µ) 2 Let {Ωi }K i=1 be a finite cover of Ω by domains Ωi ⊆ Ω with the boundaries ∂Ωi ∈ C . Each of Ωi is contained in a ball Bi of radius R. Because of continuity of p0 the radius R can be taken so small that in each of Ωi the exponent pi = p0 |Ωi satisfies the oscillation condition (3.25): 1 1 1 1 1 − < . − − + ≡ minΩi p0 maxΩi p0 d+2 pi pi β ≡ β(p± ) = p−

+ + − Denote βi = β(p± and p− i ) and notice that p0,i ≤ p 0,i ≥ p . Gathering inequalities (3.28) for v in the domains Ωi and using Young’s inequality together with (1.4), (1.3) we find that ∫ ( K ∫ ) p0 ( ) p0 ( ) ( ) ∑ 2 2 2 2 2 2 2 2 2 ϵ + |∇v| ϵ2 + |∇v| ln ϵ + |∇v| dx ≤ ln2 ϵ2 + |∇v| dx Ω

i=1

≤

K ∫ ∑

p+ +µi i

|∇v|

i=1 Ωi K ∫ ∑

p− i

|vxx |

≤λ

Ωi

i=1

∫

dx + C0

dx +

K ( ∑ i=1

p0 (x)

≤λ

Ωi

|vxx | Ω

βi C1,i ∥∇v∥ − pi ,Ωi

p+ +µi i

+ C2,i ∥∇v∥

)

,Ωi p− i

+ C0

( ) + +ν dx + C ′ (λ, K) 1 + ∥∇v∥γp0 (·),Ω + ∥∇v∥pp0 (·),Ω

with γ = maxi βi , ν = maxi µi . Using the estimate on ∥∇v∥p0 (·),Ω from Lemma 3.6 and (1.3) we conclude that for every λ > 0 the function v satisfies the inequality ∫ ∫ ( ) p0 ( ) 2 2 2 p 2 2 2 ϵ + |∇v| ln ϵ + |∇v| dx ≤ λ |vxx | 0 dx + C (3.29) Ω

Ω

with C = C(|Ω |, p± , µ, d, λ, ∥Φ∥2,Ω ). By virtue of (3.24) and (3.20) we obtain ( ) ∫ ∫ ( ∫ ) ′ 2 p0 p0 (x) (m) p0 γϵ |vxx | + |vxx | dx ≤ C 1 + |∇v| + |Φ | dx . Ω

Ω (m)

Ω

p′0

Because of (1.3) and the convergence Φ → Φ in L (Ω ), for all sufficiently large m ( ) ∫ ∫ p′ p′ |Φ (m) | 0 dx ≤ C 1 + |Φ| 0 dx , Ω

whence (3.14).

Ω

□

Estimate (2.17) follows from the uniform estimates (3.14) for the finite-dimensional approximations v (m) .

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

20

3.2.2. Proof of Theorem 2.3(ii) According to the uniform estimate (2.17) the family of solutions of problem (2.13) {vϵ } is precompact in W01,2 (Ω ) and converges, up to a subsequence, to a function v˜: ∥vϵ − v˜∥W 1,2 (Ω) → 0. On the other hand, by 0 Theorem 2.2(ii) vϵ → u0 a.e. in Ω and it is necessary that v˜ = u0 a.e. in Ω . 4. The basic a priori estimates and existence of strong solutions of the evolution problem The existence of weak and strong solutions to the regularized problem (2.9) (ϵ > 0) stated in Propositions 2.1, 2.2 is proved with the method of Galerkin. The solutions are the limits of sequences of the finite-dimensional approximations (2.10) with the coefficients defined from the system of ODEs (2.12). The solution of the singular problem (1.1) u = lim uϵ as ϵ → 0 is found similarly to the stationary case considered in Section 3. We omit the details of the proofs, which can be found in [5] (strong solutions) and [8,19] (weak solutions), and confine ourselves to the presentation of the basic a priori estimates on the functions u(m) that allow one to prove the existence of a strong solution and are used in the proof of Theorem 2.1(i). Given ϵ > 0, let u(m) denote the function defined by (2.10) where the coefficients ui,m (t) are the solutions of system (2.12). To prove item (i) of Theorem 2.1, for the initial functions in problems (2.9) we take vϵ = u0 ∈ W01,2 (Ω ) with the “dummy” index ϵ. In the proof of items (ii)–(iii), for the initial datum vϵ (x) we take the solution of the regularized elliptic problem (2.13) with the right-hand side Φ = ∆p0 u0 . The data of problem (2.9) satisfy conditions (2.3): under the conditions of Theorem 2.3(ii) ∥vϵ − u0 ∥W 1,2 (Ω) → 0 and 0

±

∥vϵ ∥W 1,2 (Ω) ≤ C,

∗

C = C(d, |Ω |, p , C , ∥Φ∥p′ (·),Ω ), 0

0

4

′

uniformly with respect to ϵ, for p ≤ 2 f ∈ Lp (QT ) ⊆ L p (QT ). (a) Under the conditions of Proposition 2.1 the functions u(m) satisfy the estimates ∫ ( ) 2 (m) 2 sup ∥u (·, t)∥2,Ω + γϵ (z, ∇u(m) )|∇u(m) | dz ≤ C ∥vϵ ∥22,Ω + ∥f ∥22,QT , (0,T )

(4.1)

(4.2)

QT

∫

p(z)

|∇u(m) |

dz ≤ 2

2−p− 2

∫

QT

2 p−2

(ϵ2 + |∇u(m) | )

2

2

|∇u(m) | dz + |Ω |T

(4.3)

QT (m)

with an absolute constant C. Estimates (4.2) follow from (2.12) upon multiplication of ith equation by ui and summation of the results ([5, Lemma 2.1]). (b) Under the conditions of Proposition 2.2 the functions u(m) satisfy the estimates ∫ ( ) 2 p(z)−2 2 (m) 2 (m) p(z) sup ∥∇u (·, t)∥2,Ω + (ϵ2 + |∇u(m) | ) 2 |u(m) dz xx | + |uxx | (0,T )

QT

( ≤C 1+

∥∇vϵ ∥22,Ω

∫ |f |

+

4 p(z)

)

(4.4)

dz

QT

with a constant C = C(T, C ∗ ) which does not depend on m and ϵ. Estimates (4.4) follow from (2.12) by means of multiplication of the equation for ui,m by λi ui,m , summation of the results and integration by parts in Ω ([5, Lemma 2.2]). (√ ) 2 2 2d Lemma 4.1. Let p− ≥ d+2 , θj (t) = sin µj t , µj = πT j2 , j = 1, 2, . . ., and {ψi }∞ i=1 be the set of ′ eigenfunctions of problem (3.1). Then the set S = span {θi (t)ψj (x), i, j ∈ N} is dense in Lp (·) (QT ). ′

Proof . The proof is an imitation of the proof of Lemma 3.7. Given f ∈ Lp (·) (QT ) and an arbitrary δ > 0, ∑∞ we have to find v = i,j=1 vi,j θi ψj , vi,j = const, such that ∥f − v∥p′ (·),QT < Cδ with an independent of f

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

21

and v constant C. Let us introduce the Hilbert space Σ = {v(z) : v, vtt , ∆v ∈ L2 (QT )} equipped with the norm ∥v∥2Σ = ∥∆v∥22,QT + ∥vtt ∥22,QT + ∥v∥22,QT . ′

The collection of smooth functions C ∞ (QT ) is dense in Lp (·) (QT ) and there is fδ ∈ C ∞ (QT ) such that ∥f − fδ ∥p′ (·),QT < δ. The set {θi (t)} is an orthogonal basis of L2 (0, T ) and {θi (t)} × {ψj (x)} is an orthogonal basis of L2 (Ω × (0, T )). For every w ∈ Σ (∫ ) ∫ T ∞ ∑ w= θi (t) ψj (x)w(x, t) dx dt, wi,j θi (t)ψj (x), wi,j = i,j=1 ∞ ∑

∥w∥2Σ =

Ω

0 2 (1 + λ2j + µ2i )wi,j < ∞,

w(m) =

i=1

m ∑

wi,j θi ψj → w in Σ .

i,j=1

By the embedding inequality, for every w ∈ Σ ∥w∥p′ (·),QT ≤ C1 ∥w∥(p− )′ ,QT ≤ C2 ∥w∥Σ (m)

with independent of w constants C1 , C2 . Since fδ ∈ C ∞ (QT )∩Σ , there is m ∈ N such that ∥fδ −fδ Then (m) 2 ∥p′ (·),QT

∥f − fδ

(m) 2 ∥p′ (·),QT

≤ ∥f − fδ ∥2p′ (·),QT + ∥fδ − fδ

(m) 2 ∥Σ

≤ δ 2 + C22 ∥fδ − fδ

∥Σ < δ.

< (1 + C22 )δ 2 . □

Lemma 4.2. Estimate (4.4) remains true if in the conditions of Proposition 2.2 p+ = 2. ′

Proof . Take a sequence {f (m) } ⊂ S, f (m) → f in Lp (·) (QT ). Multiplying each of equations (2.12) for ui,m by λi ui,m and following the proof of Lemma 3.8 we arrive at the inequality ∫ ) ∫ 2 p(z) 1 d ( dx + |u(m) dx ∥∇u(m) (·, t)∥22,Ω + γϵ (z, ∇u(m) )|u(m) | xx | xx 2 dt Ω Ω ) ( ∫ ∫ (4.5) ′ p(z) 2 2 (m) p (z) 2 (m) 2 2 (m) 2 |f | dx ≤C 1+ |∇p|(ϵ + |∇u | ) ln (ϵ + |∇u | ) dx + Ω

Ω

with a constant C which does not depend on m and ϵ. Let us fix some µ ∈ (0, 2d/3(2 + d)) and consider K u(m) in a cylinder Qδ = Ω × (0, δ), δ ∈ (0, T ). Take a finite cover of Ω , {Ωi }i=1 , such that ∂Ωi ∈ C 2 and each Ωi is contained in a ball of finite radius R > 0. Let us denote p+ i = max p(z),

p− i = min p(z).

Qδ

Qδ

(i)

(i)

By continuity of p(z), the numbers R and δ can be chosen so small that 1 1 2 − + . < 3(2 + d) p + µ p− i i

(4.6)

By virtue of (3.26), for every i = 1, . . . , K ( ∫ ∫ p(z) 2 2 2 (m) 2 2 (m) 2 |∇p|(ϵ + |∇u | ) ln (ϵ + |∇u | ) dx ≤ C 1 + Ωi

+ (m) pi +µ

|∇u

|

) dx

(4.7)

Ωi

+ + + ∗ with C = C(p± i , µ, C ). We consider separately the cases pi ≤ 2 − µ and pi > 2 − µ. If pi ≤ 2 − µ, then ∫ ( ) ( ) p+ +µ |∇u(m) | i dx ≤ C 1 + ∥∇u(m) ∥22,Ωi ≤ C 1 + ∥∇u(m) ∥22,Ω . (4.8) Ωi

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

22

In the case p+ i > 2 − µ we make use of the Gagliardo–Nirenberg inequality: if 1 2

α=

1 d

+ 1 2

+

1 p+ +µ i 2 1 d − p− i

−

[

] 1 ∈ ,1 , 2

−

then for every w ∈ W 2,pi (Ωi ) ∩ L2 (Ωi ) ∥w∥1−α ∥∇w∥p+ +µ,Ω ≤ C1 ∥wxx ∥α 2,Ωi + C2 ∥w∥2,Ωi p− ,Ω i

i

i

(4.9)

i

with independent of w constants C1 , C2 . The inequality α < 1 follows from condition (4.6) because ⇔

α<1 1 2

The inequality α >

1 1 1 − + < d p + µ p− i i

2 1 < . 3(2 + d) d

⇐

is equivalent to 1 1 2 , > + − + 2 pi + µ pi

− + + which is true for all p− i ≤ 2 and pi > 2 − µ. Also, (4.6) yields the inequality α(pi + µ) < pi : − α(p+ i + µ) < pi

⇔

1 1 2 − + < + p (p + µ)(2 + d) p− i i i

⇐

2 2 < + . 3(d + 2) (pi + µ)(2 + d)

The norm ∥u(m) ∥2,Ω is estimated in (4.2) through the data of problem (2.9). Substituting this estimate into (4.9) for u(m) and then applying Young’s inequality two times, we estimate the right-hand side of (4.7) by Young’s inequality: for any λ > 0 ∫ ∫ ∫ p+ +µ p− p(z) i dx ≤ λ |u(m) | dx + C ≤ λ |u(m) dx + C ′ |∇u(m) | i (4.10) xx xx | Ωi

Ωi

Ωi

p± i ,

′

with constants C, C depending on λ, d, |Ω |, ∥f ∥2,QT , ∥vϵ ∥2,QT . Gathering estimates (4.8) and (4.10) and substituting them into (4.5) we obtain the inequality ∫ ) ∫ p(z) d ( (m) 2 (m) (m) 2 dx ∥∇u (·, t)∥2,Ω + γϵ (z, ∇u )|uxx | dx + |u(m) xx | dt Ω( ) ∫ Ω (4.11) p′ (z) dx , t ∈ (0, δ), ≤ C 1 + ∥∇u(m) (·, t)∥22,Ω + |f (m) | Ω ±

with a constant C = C(d, p , |Ω |, d, ω(·), ∥f ∥2,QT , ∥vϵ ∥2,Ω ). Omitting the nonnegative terms on the left-hand side, for the function y(t) = ∥∇u(t)∥22,Ω we obtain the differential inequality ( ) ∫ p′ (z) y ′ (t) ≤ Cy(t) + F (t), F (t) = C 1 + |f (m) | dx , Ω

whence (m)

∥∇u

(·, t)∥22,Ω

≤

∥∇vϵ ∥22,Ω eCt

+e

Ct

(

∫ |f

1+C

′ (m) p (z)

|

) dx ,

t ∈ [0, δ].

Ω

Reverting to (4.11) and integrating in t, we obtain the inequality ∫ ( ) 2 (m) 2 (m) p(z) sup ∥∇u (·, t)∥2,Ω + γϵ (z, ∇u(m) )|u(m) dz xx | + |uxx | (0,δ)

Qδ

( ∫ 2 ≤ C 1 + ∥∇vϵ ∥2,Ω +

|f

′ (m) p (z)

|

)

(4.12)

dz .

QT

Choosing u(m) (x, δ) for the initial function and repeating the above arguments in the cylinder Q2δ \ Qδ , we extend (4.12) to the cylinder Q2δ . Iterating, in a final number of steps in t we obtain the estimate in the cylinder QT . Since ∥f (m) ∥p′ (·),QT ≤ ∥f − f (m) ∥p′ (·),QT + ∥f ∥p′ (·),QT ≤ 1 + ∥f ∥p′ (·),QT (for large m), the assertion follows. □

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

23

(c) Under the conditions of Lemma 4.2 the functions u(m) satisfy the estimates ∫ 2 p (m) ∥ut ∥22,QT + sup (ϵ2 + |∇u(m) | ) 2 dx ≤ C (0,T )

(4.13)

Ω

with an independent of m and ϵ constant C. Estimates (4.13) follow upon multiplication the ith equation of (2.12) by u′i,m (t) and summation of the results. Following the proof of [5, Lemma 2.4] we arrive at the relations (∫ ( ) )p 2 2 d (m) ∥ut (t)∥22,Ω + ϵ2 + |∇u(m) | dx dt Ω ( )p ∫ pt ϵ2 + |∇u(m) |2 2 ( ∫ )) p ( 2 (m) (m) 2 1 − ln ϵ + |∇u | =− dx + f (m) ut dx p2 2 Ω Ω ( ∫ ∫ ( ( ) ) )p 2 ∗ − 2 (m) 2 (m) p 2 (m) 2 2 ln ϵ + |∇u | dx ≤ C(C , p ) 1 + |∇u | dx + ϵ + |∇u | Ω

Ω

1 (m) 1 for every t ∈ [0, T ]. + ∥f (m) (t)∥22,Ω + ∥ut (t)∥22,Ω 2 2 Integrating in t and simplifying we obtain the inequality ∫ ∫ 2 p(z) p 1 1 (m) 2 ∥ut ∥2,QT + (ϵ2 + |∇u(m) | ) 2 dx ≤ C1 + ∥f (m) ∥22,QT + C2 |∇u(m) | dz 2 2 Ω QT ∫ ( p ) ( ) 2 2 2 ln2 ϵ2 + |∇u(m) | dz + ∥∇vϵ ∥22,Ω , t ∈ [0, T ], + C3 ϵ2 + |∇u(m) | QT

where all terms on the right-hand side are already estimated in (4.1), (4.3), (4.4) and (4.10). The uniform in m and ϵ estimates (4.2), (4.3), (4.4), (4.13) are sufficient for the proof of global in time existence of strong solutions of problems (2.9) and (1.1). For every ϵ > 0 the sequence u(m) contains a subsequence which converges to a strong solution of problem (2.9) uϵ , the family uϵ converges as ϵ → 0 to a strong solution of problem (1.1) — see [5, Proof of Theorem 1.1]. 5. Proof of Theorem 2.1(ii)–(iii) The proof of Theorem 2.1 is reduced to deriving uniform with respect to ϵ estimates (2.6) and (2.8) for the solutions of the regularized problems (2.9) with ϵ > 0. In turn, these estimates will follow from the (m) corresponding uniform estimates on the Galerkin approximations uϵ . In view of (4.2), (4.3), (4.4), (4.13) (m) (m) (m) we only have to estimate ∥ut ∥p′ (·),QT , ∥ut ∥L∞ (0,T ;L2 (Ω)) , ∥|∇ut |∥p(·),QT . (m)

5.1. Estimates on ut

in L∞ (0, T ; L2 (Ω ))

Lemma 5.1. Let the conditions of Theorem 2.1(ii) be fulfilled. Then u(m) satisfies the estimate ∫ (m) (m) 2 sup ∥ut (t)∥22,Ω + γϵ (z, ∇u(m) )|∇ut | dz (0,T )

QT

( ∫ ≤ C 1 + ∥Φ∥p′ (·),Ω + ∥vϵ ∥22,Ω + 0

QT

|f |

p′ (z)

dz + ∥ft ∥22,QT

(5.1)

) =: M

±

with an independent of m and ϵ constant C = C(T, |Ω |, p , C∗ ). Proof . Under the additional condition ft ∈ L2 (QT ) the function f has the trace f |t=0 := f0 ∈ L2 (Ω ) and ( ) ∥f0 ∥22,Ω ≤ C ∥ft ∥22,QT + ∥f ∥22,QT .

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

24

Since p(z) ∈ C 0 (QT ), the solutions ui,m (t) of system (2.12) are continuous on [0, T ], and the function γϵ (z, s)s is continuous with respect to s, then there exists lim u′i,m (t) as t → 0. Differentiating Eqs. (2.12) with respect to t we obtain the system of the second-order ODEs ∫ ∫ u′′i,m (t) = − (γϵ (z, ∇u(m) )∇u(m) )t · ∇ψi dx + ft ψi dx, i = 1, 2, . . . , m, (5.2) Ω

Ω

endowed with the initial conditions (m)

ui,m (0) = vi u′i,m (0)

=

,

(m) Φi

(5.3)

(m)

+ f0i ,

i = 1, 2, . . . , m.

Here {v (m) } is the sequence of finite-dimensional approximations for the solution vϵ of the nonlinear elliptic (m) problem (2.13). To pose the second boundary condition we use the algebraic system (3.2) for vi , i = 1, . . . , m. By the choice of the functions Φ (m) and the method of construction of v (m) v (m) → vϵ in W01,2 (Ω ),

Φ (m) → Φ in L2 (Ω ),

(5.4)

where Φ is defined in (2.5). For the sake of brevity, let us agree to use the notation γϵ ≡ γϵ (z, ∇u(m) ). Multiplying (5.2) by u′i,m (t) and summing the results for i = 1, 2, . . . , m, we arrive at the equalities ∫ ∫ ( ) 1 d (m) (m) (m) (m) 2 γϵ ∇u ∇ut dx = ft ut dx. ∥u (t)∥2,Ω + (5.5) 2 dt t t Ω Ω The straightforward computation gives (m)

(γϵ ∇u

(m) )t ∇ut

=

(m) 2 γϵ |∇ut |

[ +

(m) γϵ ∇ut

(m)

· ∇u

] (m) pt (p − 2)∇ut · ∇u(m) 2 (m) 2 . ln(ϵ + |∇u | ) + 2 2 ϵ2 + |∇u(m) |

Since p ≤ 2, this equality yields (m)

(γϵ ∇u(m) )t ∇ut

⏐ ⏐ 2 ⏐ pt ⏐ (m) | − γϵ ⏐∇ut ∇u(m) ln(ϵ2 + |∇u(m) | )⏐ . 2

(m) 2

≥ (p− − 1)γϵ |∇ut

By the Cauchy inequality ⏐ p− − 1 2 ⏐ 1 ⏐⏐ (m) (m) (m) 2 γϵ ⏐∇ut ∇u pt ln(ϵ2 + |∇u(m) | )⏐ ≤ γϵ |∇ut | 2 2 p−2 2 2 2 (m) 2 2 − 2 + C(p )∥pt ∥∞,QT (ϵ + |∇u | ) |∇u(m) | ln2 (ϵ2 + |∇u(m) | ) ≤

2 p 2 p− − 1 (m) 2 γϵ |∇ut | + C(p− )∥pt ∥2∞,QT (ϵ2 + |∇u(m) | ) 2 ln2 (ϵ2 + |∇u(m) | ). 2

Substituting these formulas into (5.5) we obtain the inequality ∫ d (m) (m) 2 ∥ut (t)∥22,Ω + (p− − 1) γϵ |∇ut | dx ≤ |I1 | + |I2 |, dt Ω ∫ ∫ 2 2 p (m) − 2 I1 = ft ut dx, I2 = C(p )∥pt ∥∞,QT (ϵ2 + |∇u(m) | ) 2 ln2 (ϵ2 + |∇u(m) | ) dx. Ω

(5.6)

Ω

The first term on the right-hand side of (5.6) is estimated by the Cauchy inequality: ∫ ∫ ( ) 1 (m) (m) |I1 | ≤ |ft ||ut | dx ≤ ft2 + (ut )2 dx. 2 Ω Ω

(5.7)

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

25

To estimate the second term we combine estimate (3.26) with either (3.29) and (4.4) if p+ < 2, or (4.10) and (4.4) if p+ = 2. It follows that there exists a positive constant µ such that ∫ 2 p 2 2 |I2 | ≤ C∥pt ∥∞,QT (ϵ2 + |∇u(m) | ) 2 ln2 (ϵ2 + |∇u(m) | )dx Ω ( ) ∫ p+ +µ ± ≤ C(p , µ, C∗ , d) 1 + |∇u(m) | dx (5.8) Ω ( ) ∫ p(z) ≤ C′ 1 + |u(m) dx , C ′ = C ′ (C∗ , p± , µ, d, |Ω |, ∥u0 ∥2,Ω , ∥f ∥2,QT ). xx | Ω

Plugging (5.8), (5.7) into (5.6) and dropping the nonnegative term on the left-hand side we obtain the differential inequality ( ) ∫ d (m) (m) 2 2 (m) p(z) 2 ∥u (t)∥2,Ω ≤ C 1 + ∥ut (t)∥2,Ω + |uxx | dx + ∥ft (t)∥2,Ω (5.9) dt t Ω with a constant C which depends on the same quantities as C ′ in (5.8) but is independent of m and ϵ. Set (m) y(t) = ∥ut (t)∥22,Ω . It follows from (5.9) that y(t) satisfies the inequality ) d ( y(t)e−Ct ≤ F (t)e−Ct dt ( ∫

p(z)

|u(m) xx |

F (t) = C 1 +

a.e. in (0, T ), dx + ∥ft (t)∥22,Ω

) .

Ω (m) p(z)

we find that Integrating it and using (4.12) to estimate |uxx | ( ) ∫ p(z) (m) 2 dz + ∥f ∥ y(t) ≤ eCt ∥ut (0)∥22,Ω + CeCt T + |u(m) | t 2,QT xx QT ( ) ∫ (m) p′ (z) ≤ eCt ∥ut (0)∥22,Ω + CeCt 1 + T + ∥∇vϵ ∥22,Ω + |f | dz + ∥ft ∥22,QT . QT

By the choice of the initial data, (5.3) and (5.4), for the sufficiently large m (m)

∥ut

(0)∥22,Ω =

m ∑ ( ) (m) (u′i,m (0))2 = ∥Φ (m) + f0 ∥22,Ω ≤ 2 ∥Φ∥22,Ω + ∥f0 ∥22,Ω . i=1

Reverting to inequality (5.9) for y(t) we obtain the estimate ∫ (m) (m) 2 2 ∥ut (t)∥2,Ω + γϵ (z, ∇u(m) )|∇ut | dz ≤ M,

t ∈ (0, T ),

QT

with the constant M from (5.1). (m) p(z)

5.2. Estimates on |uxx |

□

′ (m) p (z)

, |ut

|

(m) p(z)

and |∇ut

|

Lemma 5.2. Let Ω be a convex domain with the boundary ∂Ω ⊂ C 2 . If the conditions of Theorem 2.1(iii) are fulfilled, then the solutions of system (2.12) satisfy the estimates ) ∫ ( ′ 2 (m) p(z) (m) p (z) (m) p(z) γϵ (z, ∇u(m) )|u(m) | + |u | + |∇u | + |u | dz ≤ C, (5.10) t t xx xx QT

sup ∥∇u(m) (t)∥22,Ω ≤ C ′

(5.11)

(0,T )

with independent of ϵ and m constants C and C ′ = C ′ (C, |Ω |, T ). The constant C depends on the same quantities as the constant M in (5.1).

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

26

′

Proof . By Lemma 4.1 there exists a sequence {f (m) } such that f (m) ∈ S and f (m) → f in Lp (·) (QT ). By the definition of u(m) , for a.e. t ∈ (0, T ) the solution of system (2.12) can be regarded as the solution of (m) system (3.2) with the right-hand side f (m) (x, t) − ut (x, t). By virtue of (3.14) ( ) ∫ ∫ ( ∫ ) ′ p(z)−2 2 p(z) p′ (z) (m) p (z) (m) (m) (m) 2 (m) 2 dx ≤ C 1 + |f | dx + (ϵ + |∇u | ) 2 |uxx | + |uxx | |ut | dx . (5.12) Ω

Ω

Set α= and claim that 0≤

α(p− )′ <1 p−

1 − 12 p− 1 1 1 2 + d − p−

2 ≥ p− >

⇔

Ω

2(d + 1) d+2

(assumption (2.7)).

(5.13)

−

By the Gagliardo–Nirenberg inequality, for every v ∈ W01,p (Ω ) ∩ L2 (Ω ) 1−α ∥v∥(p− )′ ,Ω ≤ C∥∇v∥α p− ,Ω ∥v∥2,Ω . (m)

Applying this inequality to ut and plugging (5.1) we estimate ∫ ∫ ′ − ′ (m) p (z) (m) (p ) |ut | dx ≤ |ut | dx + |Ω | Ω

Ω

) α(p−− )′ p (m) (p− )′ (1−α) + |Ω | ∥ut ∥2,Ω ≤C dx (∫Ω ) µ − α(p− )′ (m) p ≤ C′ |∇ut | dx + |Ω |, µ= . p− Ω (∫

− (m) p |∇ut |

(5.14)

Using Young’s inequality two times we obtain the inequality ∫ ∫ − (m) p (m) p(z) |∇ut | dx ≤ C + |∇ut | dx Ω

Ω

∫ (

(m) 2 )|∇ut |

) p(z) 2

−

p(z)

γϵ 2 (z, ∇u(m) ) dx Ω ∫ ∫ 2 p(z) (m) 2 ≤C+ γϵ (z, ∇u(m) )|∇ut | dx + (ϵ2 + |∇u(m) | ) 2 dx ∫Ω ∫Ω 2 p(z) (m) ≤C+ γϵ (z, ∇u(m) )|∇ut | dx + |∇u(m) | dx. =C+

(m)

γϵ (z, ∇u

Ω

(5.15)

Ω

Substituting (5.15) into (5.14), integrating the result over the interval (0, T ) and using H¨older’s inequality we obtain )µ ∫ ∫ T (∫ ′ (m) p (z) (m) 2 |ut | dz ≤ C1 + C2 γϵ (z, ∇u(m) )|∇ut | dx dt QT

∫

Ω

0 T

(∫

(m) p(z)

|∇u

+ C3

(5.16)

Ω

0

≤ C1′ + C2′

|

)µ dx dt

(∫ QT

(m) 2

γϵ (z, ∇u(m) )|∇ut

| dz

)µ

+ C3′

(∫

|∇u(m) |

p(z)

)µ dz

QT

with µ defined in (5.14). According to (5.13) µ ∈ (0, 1). The integrals on the right-hand side of (5.16) are estimated in (4.2), (4.3), (5.1). Returning to (5.12), integrating it over the interval (0, T ) and substituting (5.16), we arrive at (5.10).

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

27

To prove (5.11) it is sufficient to notice that for every t ∈ (0, T ) (∫ ) ∫ 1 1 1 t d ∥∇u(m) (t)∥22,Ω = ∥∇vϵ(m) ∥22,Ω + ∥∇u(m) (τ )∥22,Ω dx dτ 2 2 2 0 dτ Ω ∫ t∫ 1 (m) (m) 2 ut ∆u(m) dz. = ∥∇vϵ ∥2,Ω − 2 0 Ω (m)

Both terms on the right-hand side are uniformly bounded: ∥∇vϵ ∥2,Ω are bounded by virtue of (3.14), Lemma 3.6 and (3.27), the second term is bounded due to Young’s inequality and (5.10): ⏐ ∫ ⏐∫ t ∫ ∫ ′ ⏐ ⏐ (m) (m) p (z) (m) (m) p(z) ⏐ ⏐ ≤ u ∆u dz |u | dz + |u | dz ≤ C. □ t t xx ⏐ ⏐ 0

QT

Ω

QT

( )d To complete the proof of Theorem 2.1 we have to prove the inclusion ∇u ∈ C 0 ([0, T ]; L2 (Ω )) . Let us consider the function space ⏐ } { ⏐ |∇u| ∈ L2 (Q ), u ∈ Lp′ (·) (Q ), T t T ⏐ , Xp(·) (QT ) = u(z) ⏐ ⏐ ∆u ∈ Lp(·) (QT ), u = 0 on ∂Ω × (0, T ) ∥u∥Xp(·) (QT ) = ∥ut ∥p′ (·),QT + ∥∇u∥2,QT + ∥∆u∥p(·),QT . Lemma 5.3. Let p(z) satisfy conditions (2.1). If ∂Ω ∈ C 2 and u ∈ Xp(·) (QT ), then ∇u ∈ (C 0 ([0, T ]; L2 (Ω )))d after possible redefining on a set of zero measure in (0, T ). Proof . It follows from [18, Theorem 8.5.12 and Proposition 4.1.7] that if p(z) ∈ Clog (QT ) and ∂Ω ∈ C 2 , there exists the extension operator { } E : Xp(·) (QT ) ↦→ v ∈ Xq(·) (Q′ ), supp v is finite in Q′ , where Q′ = Ω ′ × (−T, 2T ), Ω ⊂ Ω ′ , q(z) ∈ Clog (Q′ ), q(z) = p(z) and Eu = u on QT . The extension operator is continuous: there exists a constant C such that ∥Eu∥Xq(·) (Q′ ) ≤ C∥u∥Xp(·) (QT )

∀ u ∈ Xp(·) (QT ).

The set C0∞ (Q′ ) is dense in Xq(·) (Q′ ) — see [18, Chapter 8]. Let {uµ } be a family of functions uµ ∈ C0∞ (Q′ ) such that ∥Eu − uµ ∥Xq (Q′ ) → 0 as µ → 0. For every µ, δ > 0 and τ, t ∈ [0, T ] ∥∇(uµ − uδ )∥22,Ω (t) ≤ ∥∇(uµ − uδ )∥22,Ω ′ (t) = ∥∇(uµ − uδ )∥22,Ω ′ (τ ) + 2

∫ t∫ τ

= ∥∇(uµ − uδ )∥22,Ω ′ (τ ) − 2

Ω′

∇(uµ − uδ )t · ∇(uµ − uδ ) dz

∫ t∫ τ

Ω′

(uµ − uδ )t ∆(uµ − uδ ) dz

≤ ∥∇(uµ − uδ )∥22,Ω ′ (τ ) + 4∥(uδ − uµ )t ∥q′ (·),Q′ ∥∆(uδ − uµ )∥q(·),Q′ ≤ ∥∇(uµ − uδ )∥22,Ω ′ (τ ) + 4C 2 ∥uδ − uµ ∥2Xp(·) (QT ) . Integrating this inequality with respect to τ over the interval (0, T ) and simplifying we obtain 1 ∥∇(uδ − uµ )∥22,Q′ + 4C 2 ∥uδ − uµ ∥2Xp(·) (QT ) T ( ) 1 2 ≤C 4+ ∥uµ − uδ ∥2Xp(·) (QT ) → 0 as µ, δ → 0. T ( )d It follows that {∇uµ } is a Cauchy sequence in C 0 ([0, T ]; L2 (Ω )) and ∥∇uµ − U ∥C 0 ([0,T ];L2 (Ω)) → 0 as ( )d µ → 0 with some U ∈ C 0 ([0, T ]; L2 (Ω )) . On the other hand, ∇uµ → ∇u a.e. in QT , which means that ∇u = U a.e. in QT . □ ∥∇(uµ − uδ )∥22,Ω (t) ≤

28

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

Let u = lim uϵ be the strong solution of problem (1.1). Since the uniform estimates (5.1) yield u ∈ Xp(·) (QT ), it follows from Lemma 5.3 that ∇u ∈ C 0 ([0, T ]; L2 (Ω )) and, hence, sup ∥∇u(t)∥2,Ω = ess sup ∥∇u(t)∥2,Ω . (0,T )

(0,T )

References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001) 121–140. [2] E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005) 117–148. [3] E. Acerbi, G. Mingione, G.A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 21 (1) (2004) 25–60. [4] Y.A. Alkhutov, V.V. Zhikov, H¨ older Continuity of solutions of parabolic equations with variable nonlinearity exponent, J. Math. Sci. (N.Y.) 179 (3) (2011) 347–389, http://dx.doi.org/10.1007/s10958-011-0599-9, translation of Tr. Semin. im. I. G. Petrovskogo No. 28 (2011), Part I, 8–74. [5] S. Antontsev, I. Kuznetsov, S. Shmarev, Global higher regularity of solutions to singular p(x, t)-parabolic equations, J. Math. Anal. Appl. 466 (1) (2018) 238–263, http://dx.doi.org/10.1016/j.jmaa.2018.05.075. [6] S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat. 53 (2) (2009) 355–399, http://dx.doi.org/10.5565/PUBLMAT_53209_04. [7] S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, in: Atlantis Studies in Differential Equations, Vol. 4, Atlantis Press, Paris, 2015, http://dx.doi.org/10.2991/97894-6239-112-3. [8] S. Antontsev, S. Shmarev, Higher regularity of solutions of singular parabolic equations with variable nonlinearity, Appl. Anal. (2017) 1–22, http://dx.doi.org/10.1080/00036811.2017.1382690. [9] S. Antontsev, V. Zhikov, Higher integrability for parabolic equations of p(x, t)-Laplacian type, Adv. Differential Equations 10 (9) (2005) 1053–1080. [10] H. Beir˜ ao da Veiga, On singular parabolic p-Laplacian systems under nonsmooth external forces, Regularity up to the boundary, in: Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in Mathematical Analysis of Partial Differential Equations, in: Amer. Math. Soc. Transl. Ser. 2, vol. 232, Amer. Math. Soc, Providence, RI, 2014, pp. 1–10, http://dx.doi.org/10.1090/trans2/232/01. [11] H. Beir˜ ao da Veiga, F. Crispo, On the global W 2,q regularity for nonlinear N -systems of the p-Laplacian type in n space variables, Nonlinear Anal. 75 (11) (2012) 4346–4354, URL https://doi.org/10.1016/j.na.2012.03.021. [12] V. B¨ ogelein, F. Duzaar, H¨ older Estimates for parabolic p(x, t)-Laplacian systems, Math. Ann. 354 (3) (2012) 907–938, http://dx.doi.org/10.1007/s00208-011-0750-4. [13] S. Challal, A. Lyaghfouri, Second order regularity for the p(x)-Laplace operator, Math. Nachr. 284 (10) (2011) 1270–1279, http://dx.doi.org/10.1002/mana.200810285. [14] F. Crispo, C.R. Grisanti, High regularity of the solution to the singular elliptic p(·)-Laplacian system, Nonlinear Anal. 190 (2020) 111603, http://dx.doi.org/10.1016/j.na.2019.111603. [15] F. Crispo, P. Maremonti, Higher regularity of solutions to the singular p-Laplacian parabolic system, Adv. Differential Equations 18 (9–10) (2013) 849–894. [16] L.M. DelPezzo, S. Mart´ınez, H 2 Regularity for the p(x)-Laplacian in two-dimensional convex domains, J. Math. Anal. Appl. 410 (2) (2014) 939–952, http://dx.doi.org/10.1016/j.jmaa.2013.09.016. [17] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [18] L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. Ruˇ ˙ zi˘ cka, Lebesgue and Sobolev Spaces with Variable Exponents, first ed., in: Lecture Notes in Mathematics, vol. 2017, Springer, Berlin, 2011. [19] L. Diening, P. N¨ agele, Ruˇ ˙ zi˘ cka, Monotone operator theory for unsteady problems in variable exponent spaces, Complex Var. Elliptic Equ. (0) (2011) 1–23, http://dx.doi.org/10.1080/17476933.2011.557157, URL http://www.tandfonline.com/ doi/abs/10.1080/17476933.2011.557157. [20] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer, Berlin, 1983. [21] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, second English edition, revised and enlarged, Gordon and Breach Science Publishers, New York, 1969, Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2. [22] O.A. Ladyzhenskaya, N.N. Ura´ltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. [23] P. Lindqvist, On the time derivative in a quasilinear equation, Skr. K. Nor. Vidensk. Selsk. (2) (2008) 1–7. [24] P. Lindqvist, The time derivative in a singular parabolic equation, Differential Integral Equations 30 (9–10) (2017) 795–808, URL http://projecteuclid.org/euclid.die/1495850427. [25] J. Ok, Regularity for parabolic equations with time dependent growth, J. Math. Pures Appl. (9) 120 (2018) 253–293, http: //dx.doi.org/10.1016/j.matpur.2018.04.003, URL http://www.sciencedirect.com/science/article/pii/S0021782418300473. [26] S. Shmarev, On the continuity of solutions of the nonhomogeneous evolution p(x, t)-Laplace equation, Nonlinear Anal. 167 (2018) 67–84, URL https://doi.org/10.1016/j.na.2017.11.002. [27] R. Temam, Navier–Stokes equations, in: Studies in Mathematics and its Applications, Vol. 2, revised ed., North-Holland Publishing Co., Amsterdam, New York, 1979, theory and numerical analysis, With an appendix by F. Thomasset.

S. Antontsev and S. Shmarev / Nonlinear Analysis 195 (2020) 111724

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[28] A.S. Tersenov, The one dimensional parabolic p(x)-Laplace equation, NoDEA Nonlinear Differential Equations Appl. 23 (3) (2016) 11, http://dx.doi.org/10.1007/s00030-016-0377-y, Art. 27. [29] M. Xu, Y.Z. Chen, H¨ older Continuity of weak solutions for parabolic equations with nonstandard growth conditions, Acta Math. Sin. (Engl. Ser.) 22 (3) (2006) 793–806, http://dx.doi.org/10.1007/s10114-005-0582-9. [30] F. Yao, H¨ older Regularity for the general parabolic p(x, t)-Laplacian equations, NoDEA Nonlinear Differential Equations Appl. 22 (1) (2015) 105–119, URL https://doi.org/10.1007/s00030-014-0277-y. [31] C. Zhang, S. Zhou, X. Xue, Global gradient estimates for the parabolic p(x, t)-laplacian equation, Nonlinear Anal. Theory Methods Appl. 105 (2014) 86–101, http://dx.doi.org/10.1016/j.na.2014.04.005, URL http://www.sciencedirect. com/science/article/pii/S0362546X14001333. [32] V.V. Zhikov, S.E. Pastukhova, On the property of higher integrability for parabolic systems of variable order of nonlinearity, Mat. Zametki 87 (2) (2010) 179–200, http://dx.doi.org/10.1134/S0001434610010256.