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ANISOTROPIC PARABOLIC EQUATIONS WITH VARIABLE NONLINEARITY S. ANTONTSEV AND S. SHMAREV

Abstract. We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)–Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz–Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

1. Introduction 1.1. Statement of the problem and assumptions. Let Ω ⊂ Rn be a bounded simple-connected domain and 0 < T < ∞. We consider the Dirichlet problem for the parabolic equation  i X d h   ut − ai (z, u)|Di u|pi (z)−2 Di u + bi (z, u) + d(z, u) = 0 dxi (1.1) i   u = 0 on Γ , u(x, 0) = u (x) in Ω, T

in QT ,

0

where z = (x, t) ∈ QT ≡ Ω × (0, T ], ΓT is the lateral boundary of the cylinder QT , Di denotes the partial derivative with respect to xi and ∂f (z, v) df (z, v) = Di f (z, v) + Di v. dxi ∂v The coefficients ai (z, u), bi (z, u) and d(z, u) may depend on z = (x, t), u(z) and obey the following conditions: (1.2)

ai (z, r), bi (z, r), d(z, r) are Carathéodory functions

(defined for (z, r) ∈ QT × R, measurable in z for every r ∈ R, continuous in r for a.a. z ∈ QT ), (1.3)

∀ (z, r) ∈ QT × R

0 < a0 ≤ ai (z, r) ≤ a1 < ∞,

a0 , a1 = const,

1991 Mathematics Subject Classification. 35K55; 35K65. Key words and phrases. Nonlinear parabolic equation, nonstandard growth conditions, anisotropic nonlinearity. The first author was partially supported by the research project POCI/MAT/61576/2004, FCT/MCES (Portugal). The second author acknowledges the support of the research grant MTM–2004-05417 (Spain). 1

2

(1.4)

S. Antontsev, S. Shmarev

∀ (z, r) ∈ QT × R X p0 (z) |bi (z, r)| i ≤ b0 |r|λ + hb (z),

p0i =

i

pi (z) , pi (z) − 1

|d(z, r)| ≤ d0 |r|λ−1 + hd (z), with positive constants b, d0 , d1 , d2 , λ > 1, and λ . λ−1 The exponents pi (z) are given continuous in QT functions such that (1.5)

(1.6)

hb (z) ∈ L1 (QT ),

0

hd (z) ∈ Lλ (QT ),

λ0 =

+ − + pi (z) ⊂ (p− i , pi ) ⊆ (p , p ) ⊂ (1, ∞),

with finite constants p± , p± i > 1. Moreover, it will be assumed throughout the paper that the exponents pi (z) are continuous in QT with logarithmic module of continuity: (1.7)

∀ z, ζ ∈ QT , |z − ζ| < 1,

X

|pi (z) − pi (ζ)| ≤ ω(|z − ζ|),

i

where lim+ ω(τ ) ln

τ →0

1 = C < +∞. τ

1.2. Physical motivation and previous work. The paper addresses the questions of existence and uniqueness of weak solutions to problem (1.1). The main feature of equation (1.1) is the variable character of nonlinearity which causes a gap between the monotonicity and coercivity conditions. Because of this gap, equations of the type (1.1) are usually termed equations with nonstandard growth conditions. Equation (1.1) can be viewed as a generalization of the evolutional p–Laplacian equation (1.8)

ut = div (|∇u|p−2 ∇u)

with the constant exponent of nonlinearity p ∈ (1, ∞). During the last decades equation (1.8) was intensively studied and was casted for the role of a touchstone in the theory of nonlinear PDEs. There is extensive literature devoted to equation (1.8). We limit ourselves by referring here to monographs [23, 35], parers [5, 19, 28] and the review paper [29] which provide an excellent insight to the theory of evolutional p–laplacian equations. PDEs with variable nonlinearity are very interesting from the purely mathematical point of view. On the other hand, their study is motivated by various applications where such equations appear in the most natural way. Equations of the type (1.1) and their elliptic counterparts appear in the mathematical descriptions of motions of the non–newtonian fluids [17], in particular, electro–rheological fluids which are characterized by their ability to change the mechanical properties under the influence of the exterior electro–magnetic field [26, 38, 39]. Most of the known results concern the stationary models, see, e.g., [1, 2, 3]. Some properties of

Anisotropic parabolic equations with variable nonlinearity

3

solutions of the system of modified nonstationary Navier–Stokes equations describing electro–rheological fluids are studied in [4]. Another important application is the image processing where the anisotropy and nonlinearity of the diffusion operator and convection terms are used to underline the borders of the distorted image and to eliminate the noise [6, 8, 20]. Many of the frequently discussed schemes of image restoration lead to nonlinear elliptic and parabolic equations with linear growth in the diffusion operator; this situation corresponds to the case p− = 1 and is not discussed in the present paper. To the best of our knowledge, the reported results on the solvability of parabolic equations of the type (1.1) concern the equations with linear growth at infinity whose solutions are understood as elements of the space L2 0, T ; BV (Ω) ∩ L2 (Ω) , see, e.g., [7, 8, 20]. In our assumptions on the structure of the equation, the weak solutions possess better regularity and belong to Orlicz–Sobolev spaces W 1,p(x,t) (QT ) (the rigorous definition is given in Section 2 below). Moreover, it is proved in [16] that the gradient of the solution to the evolutional p(x, t)–laplacian satisfy the Meyer–type estimate: the gradient is integrable with the exponent p(z)(1 + δ), δ > 0, instead of p(z) as is prompted by the equation. It is known also that the solutions of equation (1.1) may extinct in a finite time [10, 15], a property typical for the solutions of the fast diffusion equation. In contrast to the case of the fast diffusion equation with constant exponents of nonlinearity, the variable nonlinearity makes that this property may persist even if the equation eventually transforms into the linear one. It is worth mentioning here the papers [30, 31, 32] devoted to the study of similar effects in solutions of equations with singularly perturbed coefficients and exponents of nonlinearity. Parabolic equations with variable nonlinearity of the type ut = div |u|γ(x,t) ∇u + F (x, t, u, ∇u) are studied in papers [11, 14]. This class of equations generalizes the famous porous media equation (PME) to the case of variable exponents of nonlinearity. It is shown in [11] that the weak solutions of this equation display many of the properties intrinsic to the solutions of PME. However, the methods used in the study of solvability of such equations are specific for the generalized PME and can not be directly applied to equations of the type (1.1) which are nonlinear with respect to Di u. Stationary counterparts of equation (1.1) and the generalized PME were studied by many authors. We refer here to [12, 13, 36] for a review of the relevant results. 1.3. Organization of the paper and description of results. The paper is organized as follows. In Section 2 we introduce the function spaces of Orlicz–Sobolev type and present a brief description of their main properties. In our conditions on the regularity of the data, the smooth functions are dense in these spaces, which allows us to construct a solution using the sequence of Galerkin’s approximations. The main existence result for problem (1.1) is stated in Theorem 3.1. We prove that problem (1.1) has at least one global weak solution if the growth conditions (1.4) and (1.6) are fulfilled with 2 ≤ λ = max{2, p− − δ} for some δ > 0. The assertion remains true if λ = max{2, p− }, but under the additional condition of smallness of the data u0 , hd and hb in the corresponding norms. The case λ > max{2, p− } is studied in Theorem 3.2. We show that in this range of exponents,

4


and with the functions hb , hd satisfying (1.5), problem (1.1) has a local in time solution if the parameters λ, p− and n are subject to the conditions 2 2n max{2, p− } < λ < p− 1 + , max 1, < p− , n 2+n 2 np− p− 1 + if n > p− . < n n − p−

(1.9)

The proofs of these assertions do not require monotonicity of the term d(z, u). The monotonicity of the diffusion part of the equation is used to prove the convergence of Galerkin’s approximations. In Section 7 we briefly discuss the possibility of extension of the existence results to the case of homogeneous Neumann boundary condition. Section 4 is devoted to derivation of L∞ bounds for the solutions of problem (1.1). We assume that the functions hd , hb are subject to the stronger restrictions (1.10)

|d(z, r)| ≤ d0 |r| + hd , |bi (z, r)| ≤ b0 |r| + hb ,

hb , hd ∈ L1 (0, T ; L∞ (Ω)).

Under these assumptions we prove in Theorem 4.1 that the weak solutions of problem (1.1) are globally bounded. The growth restriction can be relaxed for the terms d(z, u) of special form. Namely, if we assume that in the foregoing assumptions d(z, u) = d1 (z, u)|u|σ(z)−2 u + d2 (z, u)|u|λ−2 u + hd with 1 < λ ≤ inf σ(z) < M, QT

d1 ≥ d01 = const > 0,

|d2 | ≤ d2 = const < ∞,

and that the inequality d01 Rσ(z)−1 − d02 Rλ−1 − b0 R − sup hd (z) − sup |hb (z)| ≥ 0 QT

QT

holds in QT for some R > 0, then the solutions of (1.1) are globally bounded. Moreover, once such a bound is established, we use it to prove the existence of a global weak solution applying Theorem 3.1. We finally drop conditions (1.9) and show the under assumptions (1.10) problem (1.1) admit a local bounded solution for every λ ≥ 1. Uniqueness of weak solutions is studied in Section 5. It is shown that the weak solution of problem (1.1) is unique if the function u 7→ d(z, u) is monotone increasing and |ai (z, u) − ai (z, v)| ≤ ω(|u − v|) with the module of continuity ω satisfying the condition Z ds p+ + 0 → ∞ as → 0 for some 1 < α < (p ) = . α p+ − 1 ω (s) If the omit the condition of monotonicity of d(z, u), the uniqueness of weak solutions still can be proved but under stronger continuity and growth assumptions: d(z, u) is


5

Lipschitz–continuous with respect to u and ω α (s) = Cs2 . In the proof of uniqueness we follow ideas of [9, 13, 21, 22] were similar arguments were applied to the study of elliptic equations with nonstandard growth conditions. In Section 6 we study the dependence of the regularity of solutions to problem (1.1) on the regularity properties of the exponents pi , ai and σ in the partial case when ai ≡ ai (z),

d(z, u) = c(z)|u|σ(z)−2 u − f (z),

σ(x,0)

c(z) ≤ 0.

pi (x,0)

We show that if u0 ∈ L (Ω), Di u0 ∈ L (Ω), and if the exponents pi and σ are nonincreasing functions of t, then the solutions of problem (1.1) possess better regularity properties: |u|σ(z) , |Di u|pi (z) ∈ L∞ (0, T ; L1 (Ω)),

ut ∈ L2 (QT ),

|Di u|pi | ln |Di u||pit |, |u|σ | ln |u||σt | ∈ L1 (QT ). In the concluding Section 7 we give certain extensions of the results to other classes of equations close to (1.1).

2. The function spaces 1,p(x)

2.1. Spaces Lp(x) (Ω) and W0 (Ω). The definitions of the function spaces used throughout the paper and a brief description of their properties follow [24, 25, 33, 37]. The further references can be found in the review papers [27, 40]. Let ( (2.1)

Ω ⊂ Rn be a bounded domain, ∂Ω be Lipschitz-continuous, p(x) satisfy condition (1.7) of log–continuity.

By Lp(x) (Ω) we denote the space of measurable functions f (x) on Ω such that Z Ap(·) (f ) = |f (x)|p(x) dx < ∞. Ω

The space Lp(x) (Ω) equipped with the norm kf kp(·),Ω ≡ kf kLp(x) (Ω) = inf λ > 0 : Ap(·) (f /λ) ≤ 1 1, p(x)

becomes a Banach space. The Banach space W0 (1, ∞) is defined by

(2.2)

(Ω) with p(x) ∈ [p− , p+ ] ⊂

 n o 1, p(x)  (Ω) = f ∈ Lp(x) (Ω) : |∇ f | ∈ Lp(x) (Ω), u = 0 on ∂Ω ,  W0 X kuk = kDi ukp(·),Ω + kukp(·),Ω . 1,p(x)  W0 (Ω)  i

An equivalent norm of

1, p(x) W0

is given by

kukW 1,p(·) (Ω) = 0

X i

kDi ukp(·),Ω .

6

S. Antontsev, S. Shmarev 1, p(x)

• If condition (2.1) is fulfilled, then C0∞ (Ω) is dense in W0 (Ω). The space ∞ can be defined then as the closure of C0 (Ω) with respect to the norm (2.2) – see [41, 44].

1, p(x) W0 (Ω)

• The space W 1,p(x) (Ω) is separable and reflexive provided that p(x) ∈ C 0 (Ω). • Let

1 < q(x) ≤ sup q(x) < inf p∗ (x) Ω

Ω

1,p(x)

Then the embedding W0

  p(x)n with p∗ (x) = n − p(x)  ∞

if p(x) < n, if p(x) > n.

(Ω) ,→ Lq(x) (Ω) is continuous and compact.

• It follows directly from the definition that p− p+ p− p+ min kf kp(·) , kf kp(·) ≤ Ap(·) (f ) ≤ max kf kp(·) , kf kp(·) .

(2.3)

0

• H¨ older’s inequality. For all f ∈ Lp(x) (Ω), g ∈ Lp (x) (Ω) with p(x) ∈ (1, ∞),

p0 =

p p−1

the following inequality holds:

Z |f g| dx ≤

(2.4)

1 1 + 0 − p− (p )

kf kp(·) kgkp0 (·) ≤ 2 kf kp(·) kgkp0 (·) .

Ω

In particular, for every constant q ∈ (1, p− ) kf kq,Ω ≤ C kf kp(·),Ω

with the constant C = 2 k1k

p(·) ,Ω p(·)−q

.

• If conditions (2.1) are fulfilled, then there exists a constant C > 0 such that (2.5)

1,p(x)

∀ f ∈ W0

(Ω)

kf kp(·),Ω ≤ C k∇f kp(·),Ω

(Poincaré inequality).

2.2. Spaces Lp(x,t) (QT ) and anisotropic spaces W(QT ). Let pi (z) satisfy conditions (1.6) and (1.7). For every fixed t ∈ [0, T ] we introduce the Banach space n Vt (Ω) = u(x) : u(x) ∈ L2 (Ω) ∩ W01,1 (Ω), X kukVt (Ω) = kuk2,Ω + kDi ukpi (·,t),Ω ,

o Di u(x) ∈ Lpi (x,t) (Ω) ,

i

and denote by

Vt0 (Ω)

its dual. For every t ∈ [0, T ] the inclusion −

Vt (Ω) ⊂ X = W01,p (Ω) ∩ L2 (Ω) holds, which is why Vt (Ω) is reflexive and separable as a closed subspace of X. By W(QT ) we denote the Banach space


7

n o W(QT ) = u : [0, T ] 7→ Vt (Ω)| u ∈ L2 (QT ), Di u ∈ Lpi (z) (QT ), u = 0 on ΓT , X kukW(QT ) = kDi ukpi (·),QT + kuk2,QT . i 0

W (QT ) is the dual of W(QT ) (the space of linear functionals over W(QT )):

w ∈ W0 (QT ) ⇐⇒

 n X    w = w + Di wi ,  0 

0

w0 ∈ L2 (QT ),

i=1

     ∀φ ∈ W(QT ) hhw, φii =

Z

wi ∈ Lpi (z) (QT ), ! X wφ + wi Di φ dz.

QT

i

0

The norm in W (QT ) is defined by kvkW0 (QT ) = sup hhv, φii| φ ∈ W(QT ), kφkW(QT ) ≤ 1 . Let v = (v1 , . . . , vn ), p(z) = (p1 (z), . . . , pn (z)), and Ap(·),QT (v) =

n Z X

pi (z)

|vi |

dz.

i=1Q

T

The following counterpart of (2.3) holds: min

nX

+

kDi ukppi (·),QT ,

i

(2.6)

X

−

kDi ukppi (·),QT

o

i

≤ Ap(·),QT (∇ u) ≤ max

nX i

−

kDi ukppi (·),QT ,

X

o + kDi ukppi (·),QT .

i

Set n o + V+ (Ω) = u(x)| u ∈ L2 (Ω) ∩ W01,1 (Ω), |∇u| ∈ Lp (Ω) . Since V+ (Ω) is separable, it is a span of a countable set of linearly independent functions {ψk (x)} ⊂ V+ (Ω). Without loss of generality, we may assume that this system forms an orthonormal basis of L2 (Ω). Proposition 2.1. Let conditions (2.1) hold. Then the set {ψk } is dense in Vt (Ω) for every t ∈ [0, T ]. Proof. In our conditions on ∂Ω and pi , for every u ∈ Vt (Ω) there is a sequence uδ (·, t) ∈ C ∞ (Ω) such that supp u (·, t) ⊂⊂ Ω and ku − uδ kVt (Ω) → 0 as δ → 0. Such a sequence is obtained via convolution of u with the Friedrics’s mollifiers. Since uδ ∈ C0∞ (Ω) ⊂ V+ (Ω) and {ψm } is dense in V+ (Ω), one may choose constants cm such that (k)

uδ

≡

k X m=1

cm ψm (x) → uδ

strongly in V+ (Ω) as δ → 0.

8

S. Antontsev, S. Shmarev (k)

Given an arbitrary > 0, kuδ − uδ kV+ (Ω) < for all k ∈ N from some k() on. By (2.4) (k)

(k)

kuδ − uδ kVt (Ω) ≤ C kuδ − uδ kV+ (Ω) ≤ C with a constant C = C(n, |Ω|, p± , σ ± ) independent of . It follows now that for all sufficiently large k and small δ (k)

(k)

ku − uδ kVt (Ω) ≤ ku − uδ kVt (Ω) + kuδ − uδ kVt (Ω) < 2 ∀ t ∈ [0, T ]. Proposition 2.2. For every u ∈ W(QT ) there is a sequence {dk (t)}, dk (t) ∈ C 1 [0, T ], such that

m

X

dk (t)ψk (x)

u −

k=1

→0

as m → ∞.

W(QT )

Proof. In view of Proposition 2.1, the assertion immediately follows because the Pm + + functions k=1 dk (t)ψk (x) are dense in Lp (0, T ; W 1,p (Ω)) ∩ L2 (0, T ; L2 (Ω)).

3. Existence theorems In this section we prove the existence of weak solutions to problem (1.1) under the general growth conditions (1.4). The solution of problem (1.1) is understood in the following sense. Definition 3.1. A function u(x, t) ∈ W(QT ) ∩ L∞ (0, T ; L2 (Ω)) is called weak solution of problem (1.1) if for every test-function ζ ∈ Z ≡ {η(z) : η ∈ W(QT ) ∩ L∞ 0, T ; L2 (Ω) , ηt ∈ W0 (QT )}, and every t1 , t2 ∈ [0, T ] the following identity holds:

(3.1)

Zt2 Z

uζt −

t1 Ω

Z t2 X ai |Di u|pi −2 Di u + bi (z, u) Di ζ − d(z, u)ζ dz = uζdx . t1

i

Ω

The following are the main results of this section. Theorem 3.1.

a) Let us assume that

1) the coefficients ai (z, r), bi (z, r), d(z, r) satisfy conditions (1.2), (1.3), (1.4), 2) the exponents pi (z) satisfy (1.6) and (1.7), 3) the constant λ satisfies the condition (3.2)

λ = max{2, p− − δ}

with some δ > 0.


9

Then for every u0 ∈ L2 (Ω) problem (1.1) has at least one weak solution u ∈ W(QT ) satisfying the estimate (3.3)

kuk2L∞ (0,T ;L2 (Ω))

Z +

a0

X

h i |Di u|pi dz ≤ M ku0 k2L2 (Ω) + K + 1

i

QT

with a constant M independent of u and K = khb k1,QT + khd kλ0 ,QT . Moreover, ut ∈ W0 (QT ). b) The assertion remains true if (3.2) is substituted by the condition λ = max{2, p− } and the constant b0 + d0 in (1.4) is appropriately small in comparison with a0 . Theorem 3.2. Let us assume that in the conditions of Theorem 3.1 condition (3.2) is substituted by the following one: (3.4)

np− 2 < , max{2, p− } < λ < p− 1 + n n − p−

2n max 1, < p− . 2+n

Then there exists T0 > 0, defined through ku0 k2L2 Ω + K, such that problem (1.1) has at least one weak solution u ∈ W(QT0 ) satisfying estimate (3.3) in QT0 . The weak solution exists globally in time if ku0 k2L2 (Ω) + K is sufficiently small.

3.1. Proof of Theorems 3.1, 3.2. 3.1.1. Galerkin’s approximations. A solution of problem (1.1) is constructed as the limit of the sequence of Galerkin’s approximations. Let us define the operator hLv, φiΩ Z n X ai (z, v)|Di v|pi −2 Di v + bi (z, v) Di φ + d(z, v)φ dx, = vt φ + Ω

φ ∈ Vt (Ω).

i=1

The approximate solutions to problem (1.1) are sought in the form u(m) (z) =

m X

(m)

ck (t)ψk (x),

i=1

where the coefficients (3.5)

(m) ck (t)

D

are defined from the relations

Lu(m) , ψk

E

= 0,

k = 1, . . . , m.

Ω

Equalities (3.5) generate the system of m ordinary differential equations for the (m) coefficients ck (t):

(3.6)

 0 (m) (m) (m)   ck = Fk t, c1 (t), . . . , cm (t) , Z (m)  u0 (x)ψk dx k = 1, . . . , m.  ck (0) = Ω

10


If the coefficients ai , bi , d and the exponents pi , σ satisfy the conditions of Theorem 3.1 a), the functions Fk are continuous in all their arguments. 3.1.2. A priory estimates. Lemma 3.1. Let the conditions of Theorem 3.1 fulfilled. Then for every n a) beom (m) T < ∞ and m ∈ N system (3.6) has a solution ck (t) on the interval (0, T ) k=1

and the corresponding function u(m) satisfies the estimate (3.7) ku(m) (·, t)k2L∞ (0,T ;L2 (Ω)) +

Z a0

QT

X

h i |Di u(m) |pi dz ≤ M ku0 k2L2 (Ω) + K + 1 ,

i

with the constants M , K defined in the conditions of Theorem 3.1. (m)

Proof. By Peano’s Theorem, for every finite m system (3.6) has a solution ci (t), (m) i = 1, . . . , m, on an interval (0, Tm ). Multiplying each of equalities (3.5) by ck (t) and summing over k = 1, . . . , m, we arrive at the relation

(3.8)

1 2

τ ku(m) k22,Ω t=0 +

Z

Xh

ai (z, u(m) )|Di u(m) |pi dz i i +bi (z, u(m) )Di u(m) + d(z, u(m) )u(m) dz = 0, τ ∈ [0, Tm ]. Qτ

Using (1.3), (1.4) and applying Young’s inequality, we estimate: ∀ > 0 0

(3.9)

(3.10)

|bi (z, u(m) )Di u(m) | ≤ a0 |Di u(m) |pi + C |bi (z, u(m) )|pi ≤ a0 |Di u(m) |pi + C (b0 |u(m) |λ + |hb |), 0

|d(z, u(m) )u(m) | ≤ (d0 + d )|u(m) |λ + C|hd |λ , −

d ∈ (0, 1),

+

with a constant C depending on , d , a0 , p , p . Plugging (3.9)–(3.10) into (3.8), choosing sufficiently small and simplifying, we get the estimate

(3.11)

Z X 1 (m) 2 t=τ ku k2,Ω + a0 |Di u(m) |pi dz 2 t=0 Qτ i Z 0 ≤C (d0 + b0 + d )|u(m) |λ + |hb | + |hd |λ dz Qτ Z ≤ C(d0 + b0 + d ) ku(m) kλ dz + CK. Qτ

2 Let λ = 2. Using Gronwall’s inequality to estimate the function u(m) (·, t) 2,Ω and then reverting to (3.8), we obtain the required estimate (3.7). − −δ) Let 2 < λ = p− − δ. This assumption yields the inequality λ < n(p n−p− +δ , which allows one to make use of the embedding theorem in Sobolev spaces: (3.12)

ku(m) (·, t)kλλ,Ω ≤ C(λ, p− , n)k∇u(m) kλp− ,Ω .

Applying now (2.3) and Young’s inequality, we arrive at the inequality


Z

|u(m) |λ dx ≤ C

λ p− − |∇u(m) |p dx

Z

Ω

11

Ω

 (3.13)

≤C

λ p−

|Di u(m) |pi dx

 + 1

Ω

i

≤ a0

!

XZ XZ

|Di u(m) |pi dx + C(, δ, Ω, a0 , p± ).

Ω

i

Gathering these estimate with (3.8) and choosing appropriately small, we obtain the inequality X 1 (m) 2 t=τ + a0 ku k2,Ω 2 t=0 i

Z

|Di u(m) |pi dz ≤ C (K + 1) .

Qτ

The right–hand side of the obtained estimate does not depend on m, which is why the solution of system (3.6) can be continued to the maximal interval [0, T ]. Lemma 3.2. The assertion of Lemma 3.1 remains true for λ = max{2, p− }, provided that the constant b0 + d0 is sufficiently small in comparison with a0 . Proof. We only have to study the case λ = p− . Then the Poincaré inequality yields Z

|u(m) |λ dx ≤ C

Ω

Z

−

|∇u(m) |p dx,

C = C(n, λ).

Ω

Combining (3.11) with this inequality, we have that

(3.14)

XZ 1 (m) 2 t=τ ku k2,Ω + a0 |Di u(m) |pi dz 2 t=0 Q τ i XZ ≤ C(d0 + b0 + d ) |Di u(m) |pi dz + CK i

with d ∈ (0, 1).

Qτ

The conclusion follows if we claim that C(b0 + d0 ) < a0 and choose d sufficiently small. Lemma 3.3. Let condition (3.4) be fulfilled. Then there exists T0 , depending on ku0 k22,Ω + K, such that the assertion of Lemma 3.1 is true on every interval [0, T ] with T < T0 . Proof. Instead of (3.12), we will make use of the interpolation inequality (3.15)

(1−θ)λ

(m) ku(m) kλλ,Ω ≤ C(λ, p− , n)k∇u(m) kθλ k2,Ω p− ,Ω ku

with the exponent

(3.16)

θ=

λ−2 np− ∈ (0, 1). λ np− − 2(n − p− )

The inclusion θ ∈ (0, 1) follows from condition (3.4):

,

12

(3.17)


θλ n(λ − 2) = 1 [42]. This subsequence contains a subsequence which converges to u a.e. in QT (see, e.g., [34, Th.2.8.1]). These conclusions together with the uniform in m estimates (3.7) allow one to extract from the sequence u(m) a subsequence (for the sake of simplicity we assume that it merely coincides with the whole of the sequence) such that   u(m) → u weakly in W(QT ) and strongly in Lq (QT ),    (m)  → u a.e. in QT ,   u 0 d z, u(m) → d (z, u) strongly in Lλ (QT ),  0   bi z, u(m) → bi (z, u) strongly in in Lpi (z) (QT ),   pi (z)−2  0  ai z, u(m) Di u(m) Di u(m) → Ai (z) weakly in Lpi (z) (QT )

(3.21)

for some functions u ∈ W(QT ),

0

Ai (z) ∈ Lpi (z) (QT ).

By the method of construction, each of the functions u(m) satisfies identity (3.1) with the test–function η ∈ Zm . Let us fix an arbitrary m ∈ N. Then for every s ≤ m and η ∈ Zs Z

τ =T u(m) η dx τ =0 Ω Z h i X u(m) ηt − ai |Di u(m) |pi −2 Di u(m) + bi Di η + dη dxdt = 0. − QT

i

Letting m → ∞ and using (3.21) we conclude that ∀ η ∈ Zs Z − (3.22)

τ =T uη dx τ =0 Ω Z " uηt −

+ QT

# X

(Ai (z) + bi (z, u))Di η + d(z, u)η dxdty = 0

i

with an arbitrary s ∈ N. It follows then that identity (3.22) holds for every η ∈ W(QT ). It remains to identify the limit functions Ai . Lemma 3.5. For almost all z ∈ QT (3.23)

Ai (z) = ai (z, u)|Di u|pi (z)−2 Di u,

i = 1, . . . , n.

Proof. We rely on the monotonicity of the operator M(s) = |s|p−2 s: ∀ ξ, η ∈ Rn


(M(ξ) − M(η)) (ξ − η) ( p 2−p |ξ − η| p−2 ≥ 2 p p (p − 1) |ξ − η| (|ξ| + |η| ) p

(3.24)

15

if 2 ≤ p < ∞, if 1 < p < 2.

According to (3.24), for every ξ ∈ Zm Z

ai (z, u(m) ) |Di u(m) |pi −2 Di u(m) − |Di ξ|pi −2 Di ξ Di (u(m) − ξ) dxdt ≥ 0.

QT

Let ξ ∈ Zm . It follows from (3.5) with the test–function η = u(m) − ξ Z QT

n

f (z) − d(z, u(m) ) η i o Xh − ai (z, u(m) )|Di ξ|pi (z)−2 Di ξ + bi (z, u(m) ) Di η + u(m) ηt dxdt i

Z − Ω

t=T u(m) η dx ≥ 0. t=0

Gathering (3.22) with this inequality and then letting m → ∞, one has that XZ

∀ ξ ∈ W(QT )

Ai (z) − ai (z, u)|Di ξ|pi −2 Di ξ Di (u − ξ) dxdt ≥ 0.

QT

i

Choosing now ξ = u±ζ with > 0, simplifying and then letting → 0, we conclude that ∀ ζ ∈ W(QT )

XZ

Ai (z) − ai (z, u)|Di u|pi −2 Di u Di ζ dxdt = 0.

QT

i

This completes the proof of Theorems 3.1, 3.2. Corollary 3.1. Under the conditions of Theorems 3.1, 3.2 the solution u ∈ W(QT ) satisfies the identity Z " (3.25) ∀ ζ ∈ W(QT )

# ut ζ +

Q

X

ai |Di u|pi −2 Di u + bi Di ζ + d ζ dz = 0.

i

4. L∞ estimates 4.1. Global estimates. Theorem 4.1. Let the conditions of Theorem 3.1 be fulfilled and, additionally,

(4.1) ∀ k ∈ N

0 ∂bi (z, s) pi (z) 0 sup{|s|pi (z)−1 | : z ∈ QT , s ∈ [−k, k]} = Bk < ∞, ∂s

and ∀ s ∈ R, z ∈ QT ,

16


(a)

|d(z, s)| ≤ d0 |s| + hd (z),

(b)

∂bi (z, s) ∂xi ≤ b0 |s| + hb (z)

(4.2)

with finite nonnegative constants d0 , b0 . Then the weak solution of problem (1.1) is bounded and satisfies the estimate ku(·, t)k∞,Ω ≤ eC0 t ku0 k∞,Ω Z t + e C0 t e−C0 t (khb (·, t)k∞,Ω + khd (·, t)k∞,Ω ) dτ

(4.3)

0

with C0 = b0 + d0 . Proof. Let us fix k ∈ N and consider the auxiliary problem  X d   ut − ai |Di u|pi −2 Di u + bi + dk (z, u) = 0 dx i i   u = 0 on Γ , u(x, 0) = u (x) in Ω

(4.4)

T

in QT ,

0

with dk (z, u) ≡ d(z, min{|u|; k} sign u). Since for every finite k |d (z, min{|u|; k} sign u) | ≤ d0 k λ−1 + hd , it follows from Theorem 3.1 that problem (4.4) has a weak solution u(z) in the sense of Definition 3.1. Let us introduce the function   k u uk = min{|u|, k} sign u ≡  −k

if u > k, if |u| ≤ k, if u < −k.

The function u2m−1 with m ∈ N can be taken for the test–function in (3.25). Let k in (3.25) t2 = t + h, t1 = t, with t, t + h ∈ (0, T ). Observe that dk (z, u) = d(z, uk ). Then 1 2m

t+h

Z d u2m (x, t) dx dt k dt t Ω X Z t+h Z 2(m−1) + (2m − 1)ai uk |Di uk |pi + bi (z, u)Di u2m−1 dxdt k

Z

i

Z

t

t+h

Ω

Z

+ t

Ω

d(z, uk )uk2m−1 dxdt = 0.

Dividing the last equality by h and letting h → 0, we have that ∀ a.e. t ∈ (0, T )


1 d 2m dt

17

Z

u2m k (x, t) dx Ω XZ 2(m−1) + (2m − 1)ai uk |Di uk |pi + bi (z, u)Di u2m−1 dx k

(4.5)

Ω

Zi

d(z, uk )uk 2m−1 dx = 0.

+ Ω

Indeed: by Lebesgue’s dominated convergence theorem for every φ ∈ L1 (0, T ) and R t+h a.e. t ∈ (0, T ) there exists limh→0 t φ(τ ) dτ = φ(t). Let us write (4.5) in the form: ∀ a.e. t ∈ (0, T ) 1 d (4.6) 2m dt

Z

u2m k (x, t) dx + (2m − 1)

XZ

Ω

2(m−1)

Ω

i

ai uk

|Di uk |pi dx =

n X

Ji + I,

i=1

where Z

bi (z, u)Di u2m−1 dx k

Ji = Ω

Z

Z ≡ Ω

bi (z, uk )Di u2m−1 dx, k

d(z, uk )uk 2m−1 dx.

I=− Ω

Integrating by parts, we find that Z Z ∂bi (z, uk ) 2m−1 ∂bi (z, uk ) 2m−1 (1) (2) uk Di uk dx − uk dx = Ji + Ji . (4.7) Ji = − ∂u ∂x i Ω Ω Applying Young’s and H¨ older’s inequalities and plugging (4.1)-(4.2), we estimate (1) (2) Ji , Ji and I as follows: Z

(1)

|Ji | ≤ a0 (2m − 1)

|uk |2(m−1) |Di uk |pi dx

Ω

C(pi , )

Z

p0i −1

0! ∂bi (z, uk ) pi |uk |2m−1 dx ∂u

|uk | 1 (2m − 1) pi −1 Ω Z ≤ a0 (2m − 1) |uk |2(m−1) |Di uk |pi dx +

Ω 1

+ Bk

|Ji2 |

Z

C(pi , )|Ω| 2m 1

(2m − 1) pi −1

1 1− 2m |uk |2m dx ,

Ω

Z Z ∂bi (z, uk ) 2m−1 uk dx ≤ ≤ (b0 |uk | + hb ) |u|2m−1 dx k ∂xi Ω Ω Z 2m ≤C |u|2m dx k + hb Ω

Z ≤ b0

2m

|uk | Ω

Z dx + Ω

1 Z 2m

h2m b dx

2m

|uk | Ω

2m−1 2m dx ,

18


Z |I| ≤

Z

|d(z, uk )||uk |2m−1 dx ≤

Ω

(d0 |uk | + hd )|uk |2m−1 dx

Ω

Z ≤ d0

2m

|uk |

1 Z 2m

Z

h2m d dx

+

Ω

2m

|uk |

Ω

2m−1 2m dx .

Ω

Let us introduce the function yk (t) = kuk (·, t)k2m,Ω . Choosing sufficiently small and substituting the above estimates into (4.6), we arrive at the inequality for the function yk (t): yk2m−1

P 1 C(pi , )|Ω| 2m 2m−1 dyk yk (t) ≤ Bk i + (b0 + d0 ) yk2m (t) 1 dt pi −1 (2m − 1) + yk2m−1 (khb (·, t)k2m,Ω + khd (·, t)k2m,Ω ) ,

or P 1 2m dyk i C(pi , )|Ω| + (b0 + d0 ) yk (t) + (khb (·, t)k2m,Ω + khd (·, t)k2m,Ω ) . (t) ≤ Bk 1 dt (2m − 1) pi −1 Multiplying this inequality by e−C0 t , C0 = (b0 + d0 ), and integrating over the interval (0, t) we arrive at the estimate e−C0 t kuk (·, t)k2m,Ω ≤ ku0 k2m,Ω + t Bk Z +

P

i

1

C(pi , )|Ω| 2m 1

(2m − 1) pi −1

t

e−C0 τ (khb (·, t)k2m,Ω + khd (·, t)k2m,Ω ) dτ

0

which yields, as m → ∞, ∀k ∈ N (4.8)

kuk (·, t)k∞,Ω ≤ eC0 t ku0 k∞,Ω Z t + e C0 t e−C0 t (khb (·, t)k∞,Ω + khd (·, t)k∞,Ω ) dτ 0

≡ K. The right–hand side of this estimate does not depend on k. Let us choose now k ≥ K + 1. Under this choice of k uk ≡ min{|u|; k} sign u = u,

d (z, uk ) ≡ dk (z, u) ≡ d(z, u),

which means that the solution of problem (4.4) with k ≥ K +1 is, in fact, a solution of problem (1.1) which satisfies estimate (4.3). Remark 4.1. It is worth mentioning here paper [43] which addresses the question of local boundedness of solutions to equation (1.1) with anisotropic but constant growth conditions. The method of proof is based on application of suitable embedding theorems in the anisotropic Sobolev spaces.


19

4.2. Global existence via boundedness. Let us consider the case when in equation (1.1) the term d(z, u) is of the special form: (4.9)

d(z, u) = d1 (z, u)|u|σ(z)−2 u + d2 (z, u)|u|λ−2 u + hd (z),

with (4.10) 0 < d01 ≤ d1 (z, u) < ∞,

|d2 (z, u)| ≤ d02 < ∞,

d01 , d02 , λ = const > 0.

If σ(z) and λ satisfy conditions (1.3), (1.4), the existence of a weak solution follows from Theorems 3.1–3.2. If we additionally assume that the conditions of Theorem 4.1 are fulfilled, then this weak solution is bounded. We now turn to the study of the case 2 < λ < σ − ≤ σ(z) ≤ σ + < ∞,

(4.11)

which does not fall into the scope of Theorems 3.1, 3.2, 4.1. Let us take a positive number R0 < ∞ such that ∀ z ∈ QT σ(z)−1

(4.12) P(z, R0 ) ≡ d01 R0

− d02 R0λ−1 − b0 R0 − sup |hd (z)| − sup |hb (z)| ≥ 0. QT

QT

−

Because of condition σ > λ > 2, such a number always exists, provided that sup |hb | + sup |hd | < ∞.

(4.13)

QT

QT

Theorem 4.2. Let the coefficients ai , bi and the exponents pi , satisfy the conditions of Theorem 4.1, and let d(z, u) satisfy condition (4.9). Let us assume that σ(z) is measurable in QT and that conditions (4.9) − (4.12) are fulfilled. Then problem (1.1) has in QT at least one bounded weak solution satisfying the estimate kuk∞,QT ≤ max sup |u0 |; R0 . Ω

Remark 4.2. The conditions of Theorem 4.2 are surely fulfilled for the diffusion– absorption equation ut = ∆p(z) u − |u|σ(z)−2 u + hd (z), In this case kuk∞,QT ≤ supΩ |u0 | + khd k∞,QT .

σ(z) > 1.

Proof of Theorem 4.2. Fix an arbitrary finite number R > 0 and consider the regularized problem  i X d h   ut − ai |Di u|pi (z)−2 Di u + biR (z, u) + dR (z, u) = 0 dxi (4.14) i   u = 0 on Γ , u(x, 0) = u (x) in Ω, T

in QT ,

0

with dR (z, u) = d1 (z, u)|uR |σ(z)−2 uR + d2 (z, uR ) + hd (z),

biR (z, u) = bi (z, uR ),

20


and

0 if |u|> R, Di u if |u|≤ R. The regularized problem (4.14) has a global weak solution. Moreover, since bi satisfy the conditions of Theorem 4.1, this solution is globally in time bounded: kuk∞,QT ≤ C(R). The theorem will be proved if we show that the constant C(R) is in fact independent of R. Let us set R = max R0 , sup |u0 | Di uR ≡

uR = min{|u|, R} sign u,

Ω

with R0 satisfying the inequality P(z, R0 ) ≥ 0. Let us take for the test–function in (3.25) the function Di u if u > R, u+ = max{u − R, 0}, Di u + ≡ 0 if u ≤ R. Arguing like in the proof of Theorem 4.1 we arrive at the equality 1 d 2 dt

(4.15)

Z

u2+ (x, t) dx +

Ω

XZ Zi

(ai |Di u+ |pi + bi (z, uR )Di u+ ) dx

Ω

dR (z, u)u+ dx = 0 ∀ a.e. t ∈ (0, T ),

+ Ω

which can be written in the form 1 d 2 dt

∀ a.e. t ∈ (0, T )

Z

u2+ dx +

XZ

Ω

i

ai |Di u+ |pi dx + I ≡

Ω

n X

(1)

Ji

i=1

In the last relation Z

I=

(1)

Ji

Z Ω

The terms

σ(z)−1

d1 (z, u) (min{|u|, R}) sign u Ω λ−1 + d2 z, (min{|u|, R}) sign u + hd (z) u+ dx.

=−

∂bi (z, uR ) u+ Di uR dx = 0, ∂u

(2)

Ji

Z = Ω

∂bi (z, uR ) u+ dx ∂xi

(j) Ji

are estimated exactly like in the proof of Theorem 4.1: Z Z (2) ∂bi (z, uR ) u+ dx ≤ (b0 R + |hb |) u+ dx. Ji ≤ ∂xi Ω Ω

Further, Z I≥

d01 R Ω

σ(z)−1

− d02 R

λ−1

− sup |hd | u+ dx. QT

Gathering these estimates we find that Z Z XZ 1 d 2 pi u dx + ai |Di u+ | dx + P(z, R0 )u+ ≤ 0. 2 dt Ω + Ω Ω i

(2)

+ Ji

.


21

Since P(z, R0 ) ≥ 0 by the choice of R0 , and u+ (x, 0) = 0 by the choice of R, the last inequality yields ∀ a.e. z ∈ Q u+ (z) = 0, whence u(z) ≤ R a.e. in QT . The same argument shows that u− (z) = max{−u(z) − R, 0} = 0 and, finally, |u(z)| ≤ R = max sup |u0 (x)|, R0 .

(4.16)

Ω

This inequality means that biR (z, u) ≡ bi (z, u), which completes the proof.

dR (z, u) ≡ d(z, u),

4.3. Local existence via boundedness. Let us consider problem (1.1) with the term d(z, u) satisfying the growth condition |d(z, u)| ≤ d0 |u|λ−1 + hd (z),

(4.17)

λ = const > 2.

For 0 ≤ λ ≤ 2 the existence of a global bounded solution to problem (1.1) is proved in Theorem 3.1. The next theorem asserts the existence of local bounded solution in the case λ > 2. Theorem 4.3. Let us assume that in the conditions of Theorems 3.1 and 4.1 the growth condition on the function d(z, u) is substituted by (4.17). Then for every u0 ∈ L∞ (Ω) there exists θ ∈ (0, T ] depending on λ, b0 , d0 , ku0 kL∞ (Ω) , khd kL1 (0,θ;L∞ (Ω)) and khb kL1 (0,θ;L∞ (Ω)) such that in the cylinder Qθ problem (1.1) has at least one weak solution u ∈ W(Qθ ) such that ut ∈ W0 (Qθ ) and kuk∞,Qθ < ∞. The solution can be continued to the interval [0, T ∗ ] where T ∗ = sup {θ ∈ [0, T ] : kuk∞,Qθ < ∞}. Proof. Let us consider the auxiliary problem

(4.18)

 X  ut − Di ai |Di u|pi (z)−2 Di u + bi + dr (z, u) = 0

in QT

i



u = 0 on Γ,

u(x, 0) = u0 (x) in Ω

with the right–hand side (4.19)

dr (z, u) = d(z, min{|u|, r} sign u),

r = const > 1.

As in the proof of Theorem 4.1, we will make use of the fact that |dr (z, u)| ≤ d0 rλ−1 + hd (z),

dr (z, u) = d(z, u)

if r ≥ u.

22


By Theorems 3.1, 4.1, for every r > 1 the regularized problem (4.18) has a global bounded weak solution u(z). Let us show that the function w(t) = ku(·, t)k∞,Ω can be estimated by a constant which does not depend on r. Following the proof of Theorem 4.1 we find that the solution of (4.18) satisfies inequality (4.3) with C0 = b0 and hd substituted by hd + d0 rλ−1 : Z t e−b0 t khb (·, t)k∞,Ω dt ku(·, t)k∞,Ω ≤ eb0 t ku0 k∞,Ω + eb0 t 0 Z t e−b0 t khd (·, t)k∞,Ω dt + d0 rλ−1 teb0 t ≡ R(r, t). + eb0 t 0

For every fixed r > 1 R(r, t) → ku0 k∞,Ω

as t → 0,

whence for every r ≥ ku0 k∞,Ω there is t ≡ t(r) such that ∀ t ∈ [0, t(r)]

ku(·, t)k∞,Ω ≤ r.

It follows that for r and t(r) chosen in this way ku(·, t)k∞,Ω ≤ r for all t ≤ t(r), i.e. the constructed solution of the regularized problem (4.18) is a weak solution of problem (1.1) in the cylinder Qt(r) . The possibility of continuation of this solution to the maximal interval [0, T ∗ ] follows from the fact that the function u(x, t(r)) possesses the same properties as the initial function u0 .

5. Uniqueness theorems In this section we study the question of uniqueness of weak solutions to the problem

(5.1)

 i X d h   ut − ai (z, u)|Di u|pi (z)−2 Di u + d(z, u) = 0 dxi i   u = 0 on Γ, u(x, 0) = u0 (x) in Ω.

in Q,

The weak solution is understood in the sense of Definition 3.1. Let us assume that the functions ai are continuous with the module of continuity ω, |ai (z, u1 ) − a(z, u2 )| ≤ ω(|u1 − u2 |),

(5.2)

and claim that the function ω is nonnegative and satisfies the condition Z (5.3)

1

ds → ∞ as → 0+ α ω (s)

for some 1 < α
0 such that for some τ ∈ (0, T ] w = u2 − u1 > δ on the set Ωδ = Ω ∩ {x : w(x, t) > δ} and |Ωδ | = µ > 0. We will show that this assumption leads to a contradiction unless µ = 0. Not loosing generality we assume that t = T . Set di ≡ d(z, ui ), aij ≡ aj (z, ui ), i = 1, 2, j = 1, . . . , n. By the definition of weak solution, for every test–function ζ ∈ Z and τ ∈ [0, T ] Z

wt ζ +

Qτ

(5.5)

n X

a2i (|Di u2 |pi −2 Di u2 − |Di u1 |pi −2 Di u1 )Di ζ + (d2 − d1 )ζ dz

i=1

Z +

n X

(a2i − a1i )|Di u1 |pi −2 Di u1 ) Di ζ dz = 0.

Qτ i=1

Let us denote Z A(u2 , u1 ) = d2 − d1 ,

J(u2 , u1 , ζ) = −

n X

(a2i − a1i )|Di u1 |pi −2 Di u1 Di ζ dz,

Qτ i=1

and write (5.5) in the form Z (5.6)

wt ζ +

Qτ

n X

a2i (|Di u2 |pi −2 Di u2 − |Di u1 |pi −2 Di u1 )Di ζ + A(u2 , u1 )ζ dz

i=1

= J(u2 , u1 , ζ). Let us introduce the functions

(5.7)

F (ξ) =

ξ

 Z 



ds ω α (s)

0

ξ > , ξ ≤ ,

G (η) =

Z 

η

F (s) ds

η > ,

 0

η≤

depending on the parameters δ ≥ > 0, and with the function ω(·) defined in (5.3). The definition of F and (5.4) yield: (5.8)

∀ u, v ∈ R

A(u, v)F (u − v) ≥ 0.

Set Q,τ ≡ {z ∈ Qτ : w > }. By the definition of F   Di w in Q , Di F (w) = ω α (w)  0 in Q \ Q .

24


Letting in (5.6) ζ = F (w), we obtain: Z G (w(x, τ ))dx Ω n X

Z +

(5.9)

Q,τ

a2i (|Di u2 |pi −2 Di u2 − |Di u1 |pi −2 Di u1 )

i=1

Di w ω α (w)

+ A(u2 , u1 )F (w) dz ≡ J(u2 , u1 , F (w)). Notice that since δ ≥ , then Ωδ ⊆ Ω , |Ω | ≥ |Ωδ | > µ and, by virtue of (5.3), Z G (w(x, τ )) dx ≥ µF (δ) → ∞ as → 0+.

(5.10) Ω

Let us consider first the case pi ≥ 2. By virtue of (1.3) and the first inequality of (3.24)

(5.11)

a0

Di w |Di w|pi ≤ a2i (|Di u1 |pi −2 Di u1 − |Di u2 |pi −2 Di u2 ) α . ω α (w) ω (w)

According to (5.3) p+ pi ≥ + ≥ α > 1. pi − 1 p −1 Applying Young’s inequality, we may estimate the integrand of J in the following way:

(5.12)

|Di w| Di w ≤ ω(w)|Di u1 |pi −1 α (a21i − a1i )|Di u1 |pi −2 Di u1 α ω (w) ω (w) 0 a0 |Di w|pi ≤ + C(a0 , p+ )|Di u1 |pi ω pi −α (w) 2 ω α (w) a0 |Di w|pi + C(a0 , p+ ) |Di u1 |pi . ≤ 2 ω α (w)

Let now 1 < p− ≤ pi < 2. Applying (1.3) and the second inequality of (3.24) we have 2

a0 (p− − 1)(|Di u1 | + |Di u2 |)pi −2 (5.13)

|Di w| ω α (w)

≤ a2i (|Di u2 |pi −2 Di u2 − |Di u1 |pi −2 Di u1 ) and

Di w , ω α (w)


(5.14)

25

Di w (a2i − a1i )|Di u1 |pi −2 Di u1 α ω (w) Di w ≤ ω(w)(|Di u1 | + |Di u2 |)pi −1 α ω (w) Di w ≤ ω(w)(|Di u1 | + |Di u2 |)pi −1 α ω (w) a0 (p− − 1) |Di w|2 ≤ (|Di u1 | + |Di u2 |)pi −2 α 2 ω (w) + Cω 2−α (w)(|Di u2 | + |Di u1 |)pi a0 (p− − 1) |Di w|2 pi e (|Di u1 | + |Di u2 |)pi −2 α + C(|D i u2 | + |Di u1 |) 2 ω (w)

≤ with

p+ ≤ 2. p+ − 1 Plugging the pointwise estimates (5.11), (5.12) and (5.13), (5.14) into (5.9) and dropping the nonnegative terms, we arrive at the inequality 1 ,

(s G (s) =

otherwise,

0

− 1 − ln

s

Proposition 5.1. There exists a positive number µ > 2 such that

(5.18)

( 2G (s) sF (s) ≤ const

for s ≥ µ for ≤ s ≤ µ.

for s > , otherwise.

26


Proof. Set z = s/ and introduce the function f (z) = 2G (s) − sF (s) ≡ z − 1 − 2 ln z. Obviously, 2 2 ≥ 0, f 00 (z) = 2 ≥ 0 if z ≥ 2. z z Since f (z) is monotone increasing for z > 2 and tends to infinity as z → ∞, there is µ ≥ 2 such that f (z) ≥ 0 for z ≥ µ. For z ∈ [1, µ] f 0 (z) = 1 −

f (z) → ∞ as z → ∞,

f (1) = 0,

sF (s) = z − 1 ≤ µ − 1. Theorem 5.2. Let in the conditions of Theorem 5.1 condition (5.4) is substituted by condition (5.17). The the weak solution of problem (5.1) is unique. Proof. We will adapt the proof of Theorem 5.1. Let u1 , u2 be two different solutions of problem (1.1). Set u = u1 − u2 . Following the proof of Theorem 5.1 we arrive at the relation Z G (u(x, τ ))dx Ω

Z

(5.19)

+ Q,τ

n X

pi −2

a1i (|Di u1 |

pi −2

Di u1 − |Di u2 |

i=1

Di u Di u2 ) 2 u

! dz

= I1 + I2 , with n X

Z I1 = −

(a1i − a2i )|Di u2 |pi −2 Di u2

Q i=1

Di u dz, u2

Z I2 = −

(d(z, u1 ) − d(z, u2 ))F (u) dz. Q

The difference between this case and the one studied in Theorem 5.1 is that now the term I2 is not sign–defined. By Proposition 5.1 Z |I2 | ≤ C

Z

t

Z

uF (u) dz = C Q

!

Z

. . . dt ≡ I21 + I22 ,

... + 0

Ω∩(≤u≤µ)

Ω∩(µ≤u)

!

G (u)dx dt,

whence Z

t

Z

I21 ≤ C

Z t Z

G (u)dx dt ≤ C 0

Ω∩(≤u≤µ)

0

Ω

Let us introduce the function Z Y (t) =

G (u) dx. Ω

I22 ≤ C|Ω|T.


27

Substituting the above inequalities into (5.19) and taking into account (3.24), we find that the function Y (t) satisfies the Gronwall type inequality Z Y (t) ≤ C

t

Y (s)ds + 0

n Z X i=1

! pi

(|Di u1 |

pi

+ |Di u2 | ) dz + 1 .

Q

It follows that Y (t) ≤ K, which contradicts condition (5.10).

Corollary 5.1 (Comparison principle). Let u, v ∈ W(QT ) be two weak solutions of problem (5.1) such that u(x, 0) ≤ v(x, 0) a.e. in Ω. If the coefficients and the nonlinearity exponents satisfy the conditions of Theorem 5.1 or Theorem 5.2, then u ≤ v a.e. in QT .

6. Regularity of solutions for a class of model equations Let us consider the following simplified version of problem (1.1):  n X d   u − ai (z)|Di u|pi (z)−2 Di u + c(z)|u|σ(z)−2 u = f (z) t dxi (6.1) i=1   u = 0 on Γ, u(x, 0) = u0 (x) in Ω.

in QT ,

We want to trace the dependence of the regularity of weak solutions on the regularity of the data, especially, on the properties of the exponents pi (z) and σ(z). Let us accept the notations

(6.2)

n X

λp (t) =

i=1

max |pit (x, t)| ,

λa (t) =

Ω

n X i=1

λσ (t) = max |σt (x, t)| ,

max |ait (x, t)| , Ω

λc (t) = max |ct (x, t)| .

Ω

Ω

Theorem 6.1. Let us assume that a) ai , pi satisfy the conditions of Theorem 3.1, b) pit (z) ≤ 0 for a.a. z ∈ QT , c) σ(z) and c(z) are bounded measurable in QT functions, σt (z) exists a.e. in QT , and σt (z) ≤ 0,

0 ≤ c0 ≤ c(z)

a.e. in QT ,

d) λp (t), λσ (t), λa (t), λc (t) ∈ L1 (0, T ) and Z

T

(λp (t) + λa (t) + λσ (t) + λc (t)) dt = K < ∞. 0

e) u0 ∈ Lσ(x,0) (Ω), Di u0 ∈ Lpi (x,0) (Ω). Then the weak solution of problem (6.1) satisfies the estimate

28


sup

Z "X n

t∈(0,T )

Ω

# pi

ai |Di u |

+ c|u|

Z

2

|ut | dz

dx + QT

i=1 n X

Z +

(6.3)

σ

QT

! pi

σ

ai |Di u| | ln |Di u||pit | + c|u| | ln |u||σt |

dz

i=1

Z ≤C Ω

n X

! pi (x,0)

|Di u0 |

σ(x,0)

+ |u0 |

! dx + 1 ,

i=1

with an absolute constant C = C(p± , σ ± , q, T, K). Proof. Let us recall that under the conditions of Theorem 6.1 a weak solution of problem (6.1) can be obtained as the limit of the sequence of Galerkin’s approximations (see the proof of Theorem 3.1) u(m) =

m X

(m)

uk (t)ψk (x),

+

ψk (x) ∈ W01,p (Ω),

p+ = max sup pi (z), i

k=1 +

QT (m)

where the system {ψk } is dense in W01,p (Ω), and the functions uk are solutions of problem (3.5). By this reason, to prove Theorem 6.1 it suffices to derive estimate (6.3) for the approximate solutions u(m) . For the sake of simplicity, throughout the proof we use the notation u for the (m) approximate solution u(m) . Fix some m, multiply relations (3.5) by uk,t and take the sum over k = 1, . . . , m. This gives the equality (6.4) kut k22,Ω +

n Z X i=1

ai |Di u |pi −2 Di u Di ut dx +

Z

Ω

c|u|σ−2 u ut dx =

Z

Ω

f ut dx. Ω

We will use the easily verified formulas |Di u |pi pi ln |Di u |pi |Di u |pi 1 − pit − ait , + ai |Di u |pi 2 pi pi pi 1 ∂ |u|σ ln |u| |u|σ c + c|u|σ − σ − c . c|u|σ−2 u ut = t t ∂t σ σ2 σ σ Using them in (6.4), we obtain the equality ai |Di u|pi −2 Di u Di ut =

∂ ∂t

ai

kut k22,Ω + Y 0 (t) = I1 +

(6.5)

n X

I2,i +

i=1

n X

I3,i ,

i=1

where Z (6.6)

Y (t) = Ω

n X

|Di u |pi |u|σ ai +c pi σ i=1

!

Z dx,

I1 = −

f ut dx, Ω


1 ln |Di u |pi |Di u |pi (6.7) I2,i = −ai |Di u | − pit + ait dx, p2i pi pi Ω Z ln |u |σ |u |σ 1 − (6.8) I3,i = σ + c dx. −c|u |σ t t σ2 σ σ Ω The following estimates hold: Z

pi

δ 1 kut k22,Ω + kf k22,Ω , δ ∈ (0, 1), 2 2δ Z Z Z 1 pi 1 ai |Di u |pi pit dx ≤ max |pit | ai |Di u | 2 dx = λp (t) ai |Di u |pi 2 dx, 2 pi pi pi Ω Ω Ω Ω Z ln |Di u|pi ai |Di u |pi pit dx ≡ A + B, pi Ω Z |pit |ai |Di u|pi ln |Di u|dx = − |A| ≤ 0 A=− |I1 | ≤

Ω∩(|Di u|>1)

(recall that pit ≤ 0, ln |Di u|pi > 0), Z

. . . ≤ a1 |Ω| max |pit | max |τ pi ln τ | ≤ Cλp (t).

|B| =

Ω

Ω∩{|Di u|∈[0,1]}

τ ∈[0,1]

Z Z pi |Di u|pi ait |Di u | dx ≤ λa (t) dx. pi pi Ω Ω Gathering these estimate we obtain the inequality I2 ≤ −

n Z X i=1

|pit |ai |Di u |pi | ln |Di u|| dx + Cλp (t)

Ω n Z X

+ λp (t)

i=1

≤−

n Z X i=1

n Z X |Di u |pi 1 ai |Di u | 2 dx + λa (t) dx pi pi Ω i=1 Ω pi

|pit |ai |Di u |pi | ln |Di u ||dx + C [λp (t) + λa (t)] Y (t) + Cλp (t).

Ω

The terms I3,i are estimated likewise: Z c|u |σ σt dx ≤ Cλσ (t)Y (t), 2 σ Ω Z Z − c|u |σ ln |u |σ |σt | dx + c|u |σ ln |u |σ |σt |dx Ω∩(|uN |>1)

Ω∩(|uN |≤1)

≤−

Z X n

(c|u |σ | ln |u |σ ||σt |) dx + Cλσ (t),

Ω i=1

Z n X |u |σ ct dx ≤ Cλc (t)Y (t). Ω σ i=1

It follows that

29

30


I3 ≤ −

Z X n

(c|u |σ ln |u |σ |σt |) dx + C (λc (t) + λσ (t)) Y (t) + Cλσ (t).

Ω i=1

It follows that the function Y (t) satisfies the differential inequality Y 0 (t) + kut k22,Ω +

n Z X i=1

(6.9)

+

n Z X i=1

(c|u |σ ln |u |σ |σt |) dx

Ω

|pit |ai |Di u |pi | ln |Di u| dx

Ω

≤ C [Λ(t)Y (t) + λ(t) + 1] with Λ(t) = λp (t) + λa (t) + λc (t) + λσ (t),

λ(t) = λσ (t) + λp (t).

and the assertion follows from Gronwall’s lemma.

Remark 6.1. The assertion remains true for the solutions of the equation n X d pi (z)−2 σi (z)−2 ai (z)|Di u| Di u + ci (z)|u| u = f (z), ut − dxi i=1 provided that the functions ci (z), σi (z) satisfy the conditions of Theorem 6.1. Remark 6.2. Problem (6.1) includes, as a partial case, the problem ( (6.10)

ut = ∆p u + f u = 0 on ΓT ,

in QT , u(x, 0) = u0 (x) in Ω,

with constant p ∈ (1, ∞). It was proved in [18, Lemma 2.1] that the solutions of this problem satisfy the estimate (6.11)

kut k2L2 (QT ) + kukpL∞ (0,T ;W 1,p (Ω)) ≤ C kf k2L2 (QT ) + ku0 kpW 1,p (Ω) ,

which is contained in (6.3) if p = const. Moreover, if in the conditions of Theorem 6.1 the coefficients ai , c and the exponents pi and σ are variable but independent of t, then estimate (6.11) is true as well for the solutions of problem (6.1). 7. Extensions The results of this paper can be extended in various directions. Let us mention here several most obvious generalizations. 1. The class of equations (1.1) can be completed by the equations ut −

i X X d h ai (z, u)|Di u|pi (z)−2 Di u + bi (z, u) + di (z, u)Di u + d(z, u) = 0 dxi i i

which reduce to (1.1) by means of the substitution


Z ebi (z, u) ≡ bi (z, u) +

u

di (z, s) ds,

e u) ≡ d(z, u) − d(z,

0

XZ i

31

u

Di di (z, s) ds.

0

2. The main existence Theorem 3.1 remains true if in the growth conditions (1.4) hd (z) ∈ W0 (QT ), i.e., ( 0 hd (z) = h0d (z) + div H(z), h0d ∈ Lλ (QT ), 0 H(z) = (H1 , . . . , Hn ), Hi ∈ Lpi (z) (QT ). 3. The proofs of the main theorems can be easily adapted to the equations ut −

i X d h ai (z, u)|∇u|p(z)−2 Di u + bi (z, u) + d(z, u) = 0. dxi i

For the main function spaces we take n Vt (Ω) = u(x)| u(x) ∈ L2 (Ω) ∩ W01,1 (Ω), kukVt (Ω) = kuk2,Ω + k∇ukp(·,t),Ω ,

o |∇u(x)| ∈ Lp(x,t) (Ω) ,

and n o W(QT ) = u : [0, T ] 7→ Vt (Ω)| u ∈ L2 (QT ), |∇u| ∈ Lp(z) (QT ), u = 0 on Γ kukW(QT ) = k∇ukp(·),QT + kuk2,QT . The rest of the arguments does not need any change. 4. The proofs of the existence theorems can be adapted to the case of the Neumann boundary condition. For example, let us consider the problem

(7.1)

 X d pi (z)−2   a (z, u)|D u| D u + d(z, u) = 0 u − t i i i   dxi   i X ai (z, u)|Di u|pi (z)−2 Di u · νi = 0 on ΓT ,     i   u(x, 0) = u0 (x) in Ω,

in QT ,

where ν = (ν1 , ..., νn ) denotes the outer normal vector to ΓT . Let us introduce the function spaces n o Vt (Ω) = u(x) : u(x) ∈ L2 (Ω), Di u(x) ∈ Lpi (x,t) (Ω) , n o W(QT ) = u : [0, T ] 7→ Vt (Ω)| u ∈ L2 (QT ), Di u ∈ Lpi (z) (QT ) with the norms X kukVt (Ω) = kuk2,Ω + kDi ukpi (·,t),Ω , i P kukW(QT ) = i kDi ukpi (·),QT + kuk2,QT We say that function u(x, t) ∈ W(QT ) ∩ L∞ (0, T ; L2 (Ω)) is a weak solution of problem (7.1) if for every test-function

32


ζ ∈ Z ≡ {η(z) : η ∈ W(QT ) ∩ L∞ 0, T ; L2 (Ω) , ηt ∈ W0 (QT )}, and every t1 , t2 ∈ [0, T ] the following identity holds:

(7.2)

Zt2 Z t1 Ω

uζt −

X

Z t2 ai |Di u|pi −2 Di u Di ζ − d(z, u)ζ dz = uζdx . t1

i

Ω

Let us assume that the data of problem (7.1) satisfy the conditions of Theorem 3.1. Since the space W(QT ) is separable, a solution of problem (7.1) can be constructed as the limit of the sequence of Galerkin’s approximations (see the proof of Theorem 3.1). References [1] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), pp. 121–140. [2] E. Acerbi and G. Mingione, Regularity results for electrorheological fluids: the stationary case, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 817–822. [3] , Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), pp. 213–259. [4] E. Acerbi, G. Mingione, and 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 (2004), pp. 25–60. [5] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), pp. 311–341. [6] L. Alvarez, P.-L. Lions, and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal., 29 (1992), pp. 845–866. ´ n, The Cauchy problem for a strongly degenerate [7] F. Andreu, V. Caselles, and J. M. Mazo quasilinear equation, J. Eur. Math. Soc. (JEMS), 7 (2005), pp. 361–393. ´ n, Parabolic quasilinear equations min[8] F. Andreu-Vaillo, V. Caselles, and J. M. Mazo imizing linear growth functionals, vol. 223 of Progress in Mathematics, Birkh¨ auser Verlag, Basel, 2004. [9] S. Antontsev, M. Chipot, and Y. Xie, Uniqueness results for equations of the p(x)laplacian type, Advances in Mathematical Sciences and Applications, (17)1 (2007). To appear. [10] S. Antontsev and S. Shmarev, Extinction of solutions of parabolic equations with variable anisotropic nonlinearities, in Proceedings of the International Conference -Differential Equations and Dynamical Systems, Suzdal, 2006, Steklov Mathematics Institute. To appear. [11] , A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), pp. 515–545. [12] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 65 (2006), pp. 728–761. , Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, [13] in Handbook of Differential Equation, M. Chipot and P. Quittner, eds., vol. 3, Elsevier, 2006, ch. 1, pp. 1–100. [14] S. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, Fundamental and Applied Mathematics, 12 (4) (2006), pp. 3–19. [15] S. Antontsev and S. Shmarev, Parabolic equations with anisotropic nonstandard growth conditions, in Internat. Ser. Numer. Math., vol. 154, Birkh¨ auser, Verlag Basel/Switzerland, 2006, pp. 33–44. [16] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of p(x, t)Laplacian type, Adv. Differential Equations, 10 (2005), pp. 1053–1080. [17] S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), pp. 19–36.


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