Next Article in Journal
Euclidean Submanifolds via Tangential Components of Their Position Vector Fields
Previous Article in Journal
An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors
Article Menu

Export Article

Mathematics 2017, 5(4), 50; doi:10.3390/math5040050

Article
The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth
André H. ErhardtOrcid
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, Oslo 0316, Norway
Academic Editor: Michel Chipot
Received: 17 September 2017 / Accepted: 8 October 2017 / Published: 12 October 2017

Abstract

:
In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation t u div a ( x , t , u ) + λ ( | u | p ( x , t ) 2 u ) = 0 in Ω T , where λ 0 and t u denote the partial derivative of u with respect to the time variable t, while u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.
Keywords:
nonlinear parabolic problems; existence theory; variable exponents; stability
MSC:
35K55; 35A01; 35B35

1. Introduction

The aim of this paper is to establish the existence theory to nonlinear parabolic equations with nonstandard p ( x , t ) -growth of the following form
t u div a ( x , t , u ) + λ ( | u | p ( x , t ) 2 u ) = 0 in Ω T ,
where λ 0 and the vector-field a ( x , t , · ) satisfy certain p ( x , t ) -growth and monotonicity conditions. More precisely, we will prove that there exists a unique weak solution to the following Dirichlet problem:
t u div a ( x , t , u ) = λ ( | u | p ( x , t ) 2 u ) in Ω T , u = 0 , on Ω × ( 0 , T ) , u ( · , 0 ) = u 0 , on Ω × 0 .
Moreover, we will show that two unique weak solutions u and v of (2) with different initial values u 0 ( x ) , v 0 ( x ) L 2 ( Ω ) satisfy the following stability estimate:
Ω | u ( x , t ) v ( x , t ) | 2 d x Ω | u 0 ( x ) v 0 ( x ) | 2 d x
for a.e. t [ 0 , T ) . More precisely, we prove the stability of the unique weak solution to the Dirichlet problem (2) in the sense that the solutions are controlled by the initial value completely, cf. [1,2,3].
The motivation of this paper contains several aspects. The first one is that in general parabolic problems are important for the modelling of space- and time-dependent problems, e.g., problems from physics or biology. In particular, evolutionary equations and systems can be used to model physical processes, e.g., heat conduction or diffusion processes. One example is the Navier–Stokes equation, the basic equation in fluid mechanics. Furthermore, we want to refer to [4], where fluids in motion are studied. Applications also include climate modelling and climatology (see [5,6]).
The second interesting aspect of this paper is the nonstandard growth setting. Such setting arises for example by studying certain classes of non-Newtonian fluids such as electro-rheological fluids or fluids with viscosity depending on the temperature. Some properties of solutions to systems of such modified Navier–Stokes equation are studied in [7]. In general, electro-rheological fluids are of high technological interest because of their ability to change their mechanical properties under the influence of an exterior electro-magnetic field (see [8,9,10]). Many electro-rheological fluids are suspensions consisting of solid particles and a carrier oil. These suspensions change their material properties dramatically if they are exposed to an electric field (see [11]). Most of the known results concern the stationary case with p ( x ) -growth condition (see, e.g., [8,12,13]). Furthermore, for the restoration in image processing, one also uses some diffusion models with nonstandard growth conditions (please see [14,15,16,17]). Moreover, we want to refer to [18,19,20,21] for some numerical aspects regarding the numerical approximation of problems related to the parabolic p-Laplacian, the p ( x ) -Laplacian or electro-rheological fluids, respectively. Finally, we would like to mention the papers [22,23], where the stability of solutions with respect to continuous perturbations in the growth exponent p ( x ) is studied.
In the context of parabolic problems with p ( x , t ) -growth applications are models for flows in porous media [24,25] or nonlinear parabolic obstacle problems [26,27,28,29,30]. Moreover, in the last few years, parabolic problems with p ( x , t ) -growth arouse more and more interest in mathematics (see, e.g., [26,27,28,29,30,31,32,33]). Furthermore, we want to highlight that, in the case of certain parabolic equations with nonstandard growth conditions, several existence results are available (please see [32,34,35,36,37]).
The third interesting aspect of the investigation of problems related to (2) is motivated amongst others by the following observation: In [38] (for the case p ( x , t ) = const . ), the authors explained where they studied the asymptotic behaviour of the solution u to the homogeneous case of the following evolutionary p-Laplace equation
t u div ( | u | p 2 u ) = λ ( | u | p 2 u ) in Ω T
that v ( x ) = lim t u ( x , t ) should be a solution to the stationary problem
div ( | D v | p 2 D v ) = λ ( | v | p 2 v ) in Ω .
For this equation, the first eigenvalue is the minimum of the Rayleigh quotient
λ 1 = min v Ω | D v | p d x Ω | v | p d x ,
cf. [39] and see also for further details [40]. Similarly, the stationary solution of the appropriate nonstandard p ( x , t ) -problem should be the solution of the corresponding eigenvalue problem of the p ( x ) -Laplacian (please see [41,42]). Therefore, the study of problems related to (1) are also of interest, since these problems are associated with the study of long-term behaviour of solutions and the corresponding eigenvalue problems.

1.1. General Assumptions

In this paper, we consider a bounded domain Ω R n of dimension n 2 and we write Ω T : = Ω × ( 0 , T ) for the space-time cylinder over Ω of height T > 0 . Here, u t or t u , respectively, denote the partial derivative with respect to the time variable t and u denotes the one with respect to the space variable x. Moreover, we denote by P Ω T = ( Ω ¯ × 0 ) ( Ω × ( 0 , T ) ) the parabolic boundary of Ω T and we write z = ( x , t ) for points in R n + 1 . Furthermore, we consider vector-fields a: Ω T × R n R n that are assumed to be Carathéodory functions—i.e., a ( z , w ) is measurable in the first argument for every w R n and continuous in the second one for a.e. z Ω T —and satisfy the following nonstandard growth and monotonicity properties, for some growth exponent p: Ω T ( 2 n n + 2 , ) and structure constants 0 < ν 1 L and μ [ 0 , 1 ] :
| a ( z , w ) | L ( 1 + | w | ) p ( z ) 1 ,
( a ( z , w ) a ( z , w 0 ) ) · ( w w 0 ) ν ( μ 2 + | w | 2 + | w 0 | 2 ) p ( z ) 2 2 | w w 0 | 2 ,
for all z Ω T and w , w 0 R n . Furthermore, the growth exponent function p: Ω T ( 2 n n + 2 , ) satisfies the following conditions: there exist constants γ 1 < and γ 2 < , such that
2 n n + 2 < γ 1 p ( z ) γ 2 and | p ( z 1 ) p ( z 2 ) | ω ( d P ( z 1 , z 2 ) )
hold for any choice of z 1 , z 2 Ω T , where ω : [ 0 , ) [ 0 , 1 ] denotes a modulus of continuity. More precisely, we assume that ω ( · ) is a concave, non-decreasing function with
lim ρ 0 ω ( ρ ) = 0 = ω ( 0 ) .
Moreover, the parabolic distance is given by d P ( z 1 , z 2 ) : = max { | x 1 x 2 | , | t 1 t 2 | } for z 1 = ( x 1 , t 1 ) , z 2 = ( x 2 , t 2 ) R n + 1 . In addition, for the modulus of continuity ω ( · ) , we assume the following weak logarithmic continuity condition
lim sup ρ 0 ω ( ρ ) log 1 ρ < + .
Finally, we point out that the monotonicity condition (5) implies, by using the growth condition (4) and Young’s inequality, the coercivity property
a ( z , w ) · w ν c ( γ 1 , γ 2 ) | w | p ( z ) c ( γ 1 , γ 2 , ν , L )
for all z Ω T and w R n .

1.2. The Function Spaces

The spaces L p ( Ω ) , W 1 , p ( Ω ) and W 0 1 , p ( Ω ) denote the usual Lebesgue and Sobolev spaces, while the nonstandard p ( z ) -Lebesgue space L p ( z ) ( Ω T , R k ) is defined as the set of those measurable functions v: Ω T R k for k N , which satisfy | v | p ( z ) L 1 ( Ω T , R k ) , i.e.,
L p ( z ) ( Ω T , R k ) : = v : Ω T R k is measurable in Ω T : Ω T | v | p ( z ) d z < + .
The set L p ( z ) ( Ω T , R k ) equipped with the Luxemburg norm
v L p ( z ) ( Ω T ) : = inf δ > 0 : Ω T v δ p ( z ) d z 1
becomes a Banach space. This space is separable and reflexive (see [34,35]). For elements of L p ( z ) ( Ω T , R k ) , the generalized Hölder’s inequality holds in the following form: if f L p ( z ) ( Ω T , R k ) and g L p ( z ) ( Ω T , R k ) , where p ( z ) = p ( z ) p ( z ) 1 , we have
Ω T f g d z 1 γ 1 + γ 2 1 γ 2 f L p ( z ) ( Ω T ) g L p ( z ) ( Ω T )
(see also [35]). Moreover, the norm · L p ( z ) ( Ω T ) can be estimated as follows
1 + v L p ( z ) ( Ω T ) γ 1 Ω T | v | p ( z ) d z v L p ( z ) ( Ω T ) γ 2 + 1 .
Notice that we will use also the abbreviation p ( · ) for the exponent p ( z ) . Next, we introduce nonstandard Sobolev spaces for fixed t ( 0 , T ) . From assumption (6), we know that p ( · , t ) satisfies
| p ( x 1 , t ) p ( x 2 , t ) | ω ( | x 1 x 2 | )
for any choice of x 1 , x 2 Ω and for every t ( 0 , T ) . Then, we define for every fixed t ( 0 , T ) the Banach space W 1 , p ( · , t ) ( Ω ) as
W 1 , p ( · , t ) ( Ω ) : = { u L p ( · , t ) ( Ω , R ) | u L p ( · , t ) ( Ω , R n ) }
equipped with the norm
u W 1 , p ( · , t ) ( Ω ) : = u L p ( · , t ) ( Ω ) + u L p ( · , t ) ( Ω ) .
In addition, we define W 0 1 , p ( · , t ) ( Ω ) as the closure of C 0 ( Ω ) in W 1 , p ( · , t ) ( Ω ) and we denote by W 1 , p ( · , t ) ( Ω ) its dual. For every t ( 0 , T ) , the inclusion W 0 1 , p ( · , t ) ( Ω ) W 0 1 , γ 1 ( Ω ) holds true.
Furthermore, we consider more general nonstandard Sobolev spaces without fixed t. By W g p ( · ) ( Ω T ) , we denote the Banach space
W g p ( · ) ( Ω T ) : = u [ g + L 1 ( 0 , T ; W 0 1 , 1 ( Ω ) ) ] L p ( · ) ( Ω T ) | u L p ( · ) ( Ω T , R n )
equipped by the norm
u W p ( · ) ( Ω T ) : = u L p ( · ) ( Ω T ) + u L p ( · ) ( Ω T ) .
If g = 0 , we write W 0 p ( · ) ( Ω T ) instead of W g p ( · ) ( Ω T ) . Here, it is worth mentioning that the notion ( u g ) W 0 p ( · ) ( Ω T ) or u g + W 0 p ( · ) ( Ω T ) , respectively, indicate that u agrees with g on the lateral boundary of the cylinder Ω T , i.e., u W g p ( · ) ( Ω T ) .
Our next aim is to introduce the dual space of W 0 p ( · ) ( Ω T ) . Therefore, we denote by W p ( · ) ( Ω T ) the dual of the space W 0 p ( · ) ( Ω T ) . Assume that v W p ( · ) ( Ω T ) . Then, there exist functions v i L p ( · ) ( Ω T ) , i = 0 , 1 , , n , such that
v , w Ω T = Ω T v 0 w + i = 1 n v i D i w d z
for all w W 0 p ( · ) ( Ω T ) . Furthermore, if v W p ( · ) ( Ω T ) , we define the norm
v W p ( · ) ( Ω T ) = sup { v , w Ω T | w W 0 p ( · ) ( Ω T ) , w W 0 p ( · ) ( Ω T ) 1 } .
Notice that whenever (11) holds, we can write v = v 0 i = 1 n D i v i , where D i v i has to be interpreted as a distributional derivate. By
w W ( Ω T ) : = w W p ( · ) ( Ω T ) | w t W p ( · ) , ( Ω T )
we mean that there exists w t W p ( · ) ( Ω T ) , such that
w t , φ Ω T = Ω T w · φ t d z for all φ C 0 ( Ω T )
(see also [34]). The previous equality makes sense due to the inclusions
W p ( · ) ( Ω T ) L 2 ( Ω T ) ( L 2 ( Ω T ) ) W p ( · ) ( Ω T ) ,
which allow us to identify w as an element of W p ( · ) ( Ω T ) .
Finally, we are in the situation to give the definition of a weak solution to the parabolic nonstandard growth equation (1):
Definition 1.
We identify a function u L 1 ( Ω T ) as a weak solution of the parabolic equation (1), if and only if u C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W p ( · ) ( Ω T ) and
Ω T u · φ t a ( z , u ) · φ d z = λ Ω T | u | p ( x , t ) 2 u · φ d z
holds, whenever φ C 0 ( Ω T ) .
Remark 1.
In this paper, we consider certain initial value problems. Therefore, we should also mention the meaning when referring to an initial condition of the type u ( · , 0 ) = u 0 a.e. on Ω. Here, we shall always mean
1 h 0 h Ω | u u 0 | 2 d x d t 0 a s h 0 .
In particular, when u C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) , then (13) is obviously equivalent with saying u ( · , 0 ) = u 0 .

1.3. Statement of the Result and Plan of the Paper

In the following, we mention our main result and we briefly describe the strategy of the proof to these results and the novelties of the paper. We start with some useful and important preliminary results (see Section 2). In Section 3, we prove the existence of a unique weak solution to (2) and we investigate its stability. The approach to prove the existence of weak solutions to the Dirichlet problem is to construct a solution, which solves the problem (2). We start by constructing a sequence of the Galerkin’s approximations, where the limit of this sequence is equal to the solution in (2). Then, we show that this approximate solution converges to a general solution. Finally, we will use this existence result to derive the desired stability estimate (3). This yields the following.
Theorem 1.
Let λ 0 , Ω R n be an open, bounded Lipschitz domain and p : Ω T [ γ 1 , γ 2 ] satisfies (6) and (7). Then, suppose that the vector-field a: Ω T × R n R n is a Carathéodory function and satisfies the growth condition (4) and the monotonicity condition (5). Moreover, let u 0 L 2 ( Ω ) . Then, there exists a unique weak solution u C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W 0 p ( · ) ( Ω T ) with t u W p ( · ) ( Ω T ) of (2) and this solution satisfies the following estimate:
sup 0 t T Ω | u ( · , t ) | 2 d x + Ω T | u | p ( · ) d z + Ω T | u | p ( · ) d z c u 0 L 2 ( Ω ) 2 + | Ω T | ,
with u ( · , 0 ) = u 0 and a constant c = c ( γ 1 , γ 2 , ν , L ) if λ 1 or λ = 0 and c = c ( γ 1 , γ 2 , ν , L , 1 λ ) if λ ( 0 , 1 ) . Furthermore, for two weak solutions u , v C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W 0 p ( · ) ( Ω T ) with t u , t v W p ( · ) ( Ω T ) and different initial values u 0 , v 0 L 2 ( Ω ) (i.e., u 0 v 0 ) of (2) the stability estimate (3), i.e.,
u ( x , t ) v ( x , t ) L 2 ( Ω ) 2 u 0 v 0 L 2 ( Ω ) 2
holds true for a.e. t [ 0 , T ) .
Remark 2.
Please note that we can conclude from (3) and Hölder’s inequality that
Ω | u ( x , t ) v ( x , t ) | d x ( Ω | u ( x , t ) v ( x , t ) | 2 d x ) 1 2
for a.e. t [ 0 , T ) —where Ω f d x : = 1 | Ω | Ω f d x —or
u ( x , t ) v ( x , t ) L 1 ( Ω ) | Ω | 1 2 u 0 ( x ) v 0 ( x ) L 2 ( Ω )
for a.e. t [ 0 , T ) , respectively.
Remark 3.
Moreover, we want to emphasise that we can also prove the existence of a unique weak solution to (2), if we assume that a ( · ) satisfies the growth condition (4), coercivity condition (8) and the monotonicity condition ( a ( z , w ) a ( z , w 0 ) ) · ( w w 0 ) 0 for all z Ω T and w , w 0 R n . Furthermore, the existence of solutions to the initial value problem (2) can be extended to general boundary value problems and, moreover, we are also able to prove the statement of Theorem 1 if we consider further inhomogeneities on the right-hand side of (1), i.e., f div ( | F | p ( x , t ) 2 F ) satisfying f L γ 1 ( Ω T ) and F L p ( x , t ) ( Ω T ) (please see the approach in [32]).

2. Preliminaries

In the following, we will refer to some useful tools, which we will need for our proof. First of all, we refer to two lemmas, which are useful tools when dealing with p-growth problems. To this aim, we define a function V μ , p : R k R k by
V μ , p ( A ) : = ( μ 2 + | A | 2 ) p 2 A
for A R k , p > 1 and μ 0 . Moreover, we cite the following lemma from ([43], Lemma 2.1), which is established for the case p 0 in [44] and in the case 0 > p > 1 in [43].
Lemma 1.
Suppose that μ 0 . Then, there exists a positive constant c, depending on p > 1 , such that for all A , B R k with A B , we have
c 1 ( μ 2 + | A | 2 + | B | 2 ) p 2 | A B | | V μ , p ( A ) V μ , p ( B ) | c ( μ 2 + | A | 2 + | B | 2 ) p 2 | A B | .
Since p ( · ) > 2 n n + 2 , we are able to choose p = p ( · ) 2 > 1 . Then, choosing μ = 0 and k = n 2 we consider V ( A ) = | A | p ( · ) 2 A . This allows us to conclude from Lemma 1 ((cf. [45], Lemma 2.2) in the case p ( · ) > 2 and ([46], Lemma 2) in the case 1 < p ( · ) < 2 ) the following lemma.
Lemma 2.
There exists a constant c: = c ( n , γ 1 , γ 2 ) , such that for any A , B R n , there holds
( | A | 2 + | B | 2 ) p ( · ) 2 2 | A B | 2 c V ( A ) V ( B ) · ( A B ) ,
where A B .
Finally, we need the following Theorem ([32], Theorem 1.3), since this Theorem implies the strong convergence in p ( z ) -Lebesgue spaces and therefore, it is important for our existence result.
Theorem 2.
Let Ω R n an open, bounded Lipschitz domain with n 2 and p ( · ) > 2 n n + 2 satisfying (6) and (7). Furthermore, define p ^ ( · ) : = max 2 , p ( · ) . Then, the inclusion W ( Ω T ) L p ^ ( · ) ( Ω T ) is compact.

3. Proof of the Main Result

First of all, we will prove the existence of a unique weak solution to the Dirichlet problem (2). Then, we are able to derive the desired stability estimate (3) immediately. The proof reads as follows:
Proof of Theorem 1.
We start by constructing a sequence of the Galerkin’s approximations, where the limit of this sequence is equal to the solution in (2). Therefore, we consider ϕ i ( x ) i = 1 W 0 1 , γ 2 ( Ω ) , which is an orthonormal basis in L 2 ( Ω ) . Since W 0 1 , γ 2 ( Ω ) is separable, it is a span of a countable set of linearly independent functions ϕ k W 0 1 , γ 2 ( Ω ) . Moreover, we have the dense embedding W 0 1 , γ 2 ( Ω ) L 2 ( Ω ) for any γ 2 > 2 n n + 2 (see, e.g., [47,48]). Thus, without loss of generality, we may assume that this system forms an orthonormal basis of L 2 ( Ω ) . Now, we fix a positive integer m and define the approximate solution to (2) as follows:
u ( m ) ( z ) : = i = 1 m c i ( m ) ( t ) ϕ i ( x ) ,
where the coefficients c i ( m ) ( t ) are defined via the identity
Ω u t ( m ) ϕ i ( x ) + a ( x , t , u ( m ) ) D ϕ i ( x ) + λ | u ( m ) | p ( x , t ) 2 u ( m ) ϕ i ( x ) d x = 0 ,
for i = 0 , , m and t ( 0 , T ) with the initial condition
c i ( m ) ( 0 ) = Ω u 0 ϕ i d x , i = 1 , , m .
Then, Equation (15) together with the initial condition (16) generates a system of m ordinary differential equations
c i ( m ) ( t ) = F i t , c 1 ( m ) ( t ) , , c m ( m ) ( t ) , c i ( m ) ( 0 ) = Ω u 0 ϕ i d x , i = 1 , , m ,
since ϕ i ( x ) is orthonormal in L 2 ( Ω ) . By ([49], Theorem 1.44, p. 25), we know that there is, for every finite system (17), a solution c i ( m ) ( t ) , i = 1 , , m on the interval ( 0 , T m ) for some T m > 0 . Therefore, we multiply Equation (15) by the coefficients c i ( m ) ( t ) , i = 1 , , m . Then, we need a priori estimates that permit us to extend the solution to the whole domain ( 0 , T ) . Thus, we integrate the equation over ( 0 , τ ) for an arbitrarily τ ( 0 , T m ) . Next, we sum the resulting equation over i = 1 , , m . Therefore, it follows
Ω τ t u ( m ) · u ( m ) + a ( z , u ( m ) ) · u ( m ) + λ | u ( m ) | p ( x , t ) 2 u ( m ) · u ( m ) d z = 0
for a.e. τ ( 0 , T m ) . Furthermore, we use
Ω τ t u ( m ) · u ( m ) d z 1 2 Ω | u ( m ) ( · , τ ) | 2 d x 1 2 Ω | u 0 | 2 d x
for a.e. τ ( 0 , T m ) , since u 0 L 2 ( Ω ) , ϕ i i = 1 L 2 ( Ω ) and u ( m ) ( · , 0 ) L 2 ( Ω ) 2 u 0 L 2 ( Ω ) 2 , cf. [32]. Then, we derive at
1 2 Ω | u ( m ) ( · , τ ) | 2 d x + Ω τ a ( z , u ( m ) ) · u ( m ) + λ | u ( m ) | p ( · ) d z 1 2 u 0 L 2 ( Ω ) 2
for a.e. τ ( 0 , T m ) . Using the coercivity condition (8) on the left-hand side of (19), this yields
1 2 Ω | u ( m ) ( · , τ ) | 2 d x + ν c ( γ 1 , γ 2 ) Ω τ | u ( m ) | p ( · ) + λ | u ( m ) | p ( · ) d z c u 0 L 2 ( Ω ) 2 + | Ω T | ,
where c = c ( γ 1 , γ 2 , ν , L ) . This estimate holds for a.e. τ ( 0 , T m ) . Therefore, we have shown that u ( m ) is uniformly bounded in W p ( · ) ( Ω T m ) and L ( 0 , T m ; L 2 ( Ω ) ) independently of m. Thus, the solution of system (17) can be continued to the maximal interval ( 0 , T ) and we have
sup 0 τ T Ω | u ( m ) ( · , τ ) | 2 d x + Ω T | u ( m ) | p ( · ) + | u ( m ) | p ( · ) d z c u 0 L 2 ( Ω ) 2 + | Ω T | .
Please notice that, if λ 1 , we can estimate the left-hand side of the second last inequality from below by choosing λ = 1 , while if λ = 0 the term depending on λ disappears. If λ ( 0 , 1 ) , we first of all divide the second last equation by λ, then the constant c depends on λ, i.e., c = c ( γ 1 , γ 2 , ν , L , 1 λ ) , and, finally, we estimate the left-hand side of the resulting estimate from below by using 1 λ 1 .
Next, we want to derive a uniform bound for t u ( m ) in W p ( · ) ( Ω T ) . Therefore, we define a subspace of the set of admissible test functions
W m ( Ω T ) : = η : η = i = 1 m d i ϕ i , d i C 1 ( [ 0 , T ] ) W 0 p ( · ) ( Ω T ) .
Then, we choose a test function
φ ( z ) = i = 1 m d i ( t ) ϕ i ( x ) W m ( Ω T ) with d i ( 0 ) = d i ( T ) = 0 .
Note that t φ exists, since the coefficients d i ( t ) lie in C 1 ( [ 0 , T ] ) . Moreover, we know that C 1 ( [ 0 , T ] , W 0 1 , γ 2 ( Ω T ) ) W 0 p ( · ) ( Ω T ) , and, therefore, we have also φ W 0 p ( · ) ( Ω T ) . Thus, we can conclude by the definition of u ( m ) and (15) that
Ω T u ( m ) φ t d z = Ω T u t ( m ) φ d z = Ω T a ( z , u ( m ) ) · φ + λ | u ( m ) | p ( · ) 2 u ( m ) · φ d z .
Then, we derive by utilizing the growth condition (4) and the generalized Hölder’s inequality (9) the following estimate
Ω T u ( m ) φ t d z Ω T | a ( z , u ( m ) ) | + λ | u ( m ) | p ( · ) 1 · | φ | + | φ | d z θ Ω T | a ( z , u ( m ) ) | + | u ( m ) | p ( · ) 1 · | φ | + | φ | d z c ( 1 + | u ( m ) | p ( · ) 1 + | u ( m ) | p ( · ) 1 ) L p ( · ) ( Ω T ) × φ W p ( · ) ( Ω T ) ,
where c = c ( γ 1 , γ 2 , L , θ ) with θ: = max 1 , λ . Using (10) and (20), we have for every φ W m ( Ω T ) W 0 p ( · ) ( Ω T ) and any m that
Ω T u ( m ) φ t d z c φ W p ( · ) ( Ω T )
with a constant c = c ( γ 1 , γ 2 , ν , L , θ , u 0 L 2 , | Ω T | ) , where c is independent of m. This shows that u t ( m ) W p ( · ) ( Ω T ) with
u t ( m ) W p ( · ) ( Ω T ) c ( γ 1 , γ 2 , ν , L , θ , u 0 L 2 , | Ω T | ) .
Therefore, we have a uniform bound of u t ( m ) in W p ( · ) ( Ω T ) and it follows that
u ( m ) W 0 p ( · ) ( Ω T ) L γ 1 ( 0 , T ; W 0 1 , γ 1 ( Ω ) ) , u t ( m ) W p ( · ) ( Ω T ) L γ 2 ( 0 , T ; W 1 , γ 2 ( Ω ) )
are bounded. This implies the following weak convergences for the sequence u ( m ) (up to a subsequence):
u ( m ) * u weakly * in L ( 0 , T ; L 2 ( Ω ) ) , u ( m ) u weakly in L p ( · ) ( Ω T , R n ) , u t ( m ) u t weakly in W p ( · ) ( Ω T ) .
Moreover, by Theorem 2, we can conclude that the sequence u ( m ) (up to a subsequence) converges strongly in L p ^ ( · ) ( Ω T ) with p ^ ( · ) : = max 2 , p ( · ) to some function u W ( Ω T ) . Thus, we get the desired convergences
u ( m ) u strongly in L p ^ ( · ) ( Ω T ) , u ( m ) u a . e . in Ω T
for the sequence u ( m ) (up to a subsequence).
Furthermore, the growth assumption of a ( z , · ) and the estimate (20) imply that the sequence a ( z , u ( m ) ) m N is bounded in L p ( · ) ( Ω T , R n ) . Consequently, after passing to a subsequence once more, we can find a limit map A 0 L p ( · ) ( Ω T , R n ) with
a ( z , u ( m ) ) A 0 as m .
Our next aim is to show that A 0 = a ( z , u ) for almost every z Ω T . First of all, we should mention that each of u ( m ) satisfies the identity (15) with a test function φ W m ( Ω T ) . This follows by the method of construction (see [36]). Then, we fix an arbitrary m N . Thus, we have for every s m the following equation
Ω T u t ( m ) φ + a ( z , u ( m ) ) φ + λ ( | u ( m ) | p ( x , t ) 2 u ( m ) ) φ d z = 0
for all test functions φ W s ( Ω T ) . Passing to the limit m , we can conclude that, for all test functions φ W s ( Ω T ) , we have
Ω T u t φ + A 0 φ + λ ( | u | p ( x , t ) 2 u ) φ d z = 0
with an arbitrary s N , by the convergence from above. Therefore, it follows that the identity (22) holds for every φ W 0 p ( · ) ( Ω T ) . According to monotonicity assumption (5), we know that for every w W s ( Ω T ) and every s m , the following holds
Ω T [ a ( z , u ( m ) ) a ( z , w ) ] ( u ( m ) w ) d z 0 .
Moreover, it follows from (15) the conclusion from above and the choice of an admissible test function φ = u ( m ) w with w W s ( Ω T ) that
Ω T u t ( m ) φ + a ( z , u ( m ) ) φ + λ ( | u ( m ) | p ( x , t ) 2 u ( m ) ) φ d z = 0 .
Adding (23) and (24), we then have
Ω T u t ( m ) φ + a ( z , u ( m ) ) φ + λ ( | u ( m ) | p ( x , t ) 2 u ( m ) ) φ d z + Ω T [ a ( z , u ( m ) ) a ( z , w ) ] φ d z 0
with a test function φ = u ( m ) w . This yields
Ω T u t ( m ) ( u ( m ) w ) + a ( z , w ) ( u ( m ) w ) + λ ( | u ( m ) | p ( x , t ) 2 u ( m ) ) ( u ( m ) w ) d z 0 .
Then, we test Equation (22) with φ = u ( m ) w , subtract the resulting equation from the last estimate and finally pass to the limit m , yielding
Ω T [ A 0 a ( z , w ) ] ( u w ) d z 0
for all w W s ( Ω T ) . Since W s ( Ω T ) W 0 p ( · ) ( Ω T ) is dense, we are allowed to choose w W 0 p ( · ) ( Ω T ) . Hence, we choose w = u ± ε ζ with an arbitrary ζ W 0 p ( · ) ( Ω T ) . This yields
ε Ω T [ A 0 a ( z , ( u ± ε ζ ) ) ] ζ d z 0 .
Then, passing to the limit ε 0 , we can conclude that
Ω T [ A 0 a ( z , u ) ] ζ d z = 0
for all ζ W 0 p ( · ) ( Ω T ) . This shows that
A 0 = a ( z , u ) for almost every z Ω T .
Moreover, we have to show that u ( · , 0 ) = u 0 . First of all, we should mention that we get from (22) and the integration by parts the following equation
Ω T u φ t a ( z , u ) φ λ ( | u | p ( x , t ) 2 u ) φ d z = Ω ( u · φ ) ( · , 0 ) d x
for all φ W 0 p ( · ) ( Ω T ) with φ ( · , T ) = 0 . Moreover, we can conclude from (24)—similar to the previous estimate—that
Ω T u ( m ) φ t a ( z , u ( m ) ) φ λ ( | u ( m ) | p ( x , t ) 2 u ( m ) ) φ d z = Ω ( u ( m ) · φ ) ( · , 0 ) d x
for all φ W 0 p ( · ) ( Ω T ) with φ ( · , T ) = 0 . Passing to the limit m and using the convergences from above, we get
Ω T u φ t a ( z , u ) φ λ ( | u | p ( x , t ) 2 u ) φ d z = Ω u 0 · φ ( · , 0 ) d x ,
where u ( m ) ( · , 0 ) u 0 as m , since
u ( m ) ( · , 0 ) = i = 1 m c i ( m ) ( 0 ) ϕ i ( x ) = i = 1 m Ω u 0 ϕ i ( x ) d x ϕ i ( x ) i = 1 Ω u 0 ϕ i ( x ) d x ϕ i ( x ) = u 0 as m .
Furthermore, φ ( · , 0 ) is arbitrary. Therefore, we can conclude that u ( · , 0 ) = u 0 . This shows that there exists a weak solution to the Dirichlet problem (2).
Next, we prove the uniqueness of the weak solution. Therefore, we assume that there exist two weak solutions u and u * C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W 0 p ( · ) ( Ω T ) with t u , t u * W p ( · ) ( Ω T ) of the Dirichlet problem (2). Thus, we have the following weak formulations
Ω T u · φ t a ( z , u ) · φ d z = λ Ω T | u | p ( x , t ) 2 u · φ d z
and
Ω T u * · φ t a ( z , u * ) · φ d z = λ Ω T | u * | p ( x , t ) 2 u * · φ d z
with the admissible test function φ = u u * W 0 p ( · ) ( Ω T ) , since W 0 p ( · ) ( Ω T ) is the dual of W 0 p ( · ) ( Ω T ) . Hence, we can conclude by subtracting the second equation from the first one that
Ω T ( u u * ) ( u u * ) t ( a ( z , u ) a ( z , u * ) ) ( u u * ) d z = Ω T λ ( | u | p ( x , t ) 2 u | u * | p ( x , t ) 2 u * ) ( u u * ) d z .
Using the monotonicity condition (5) and Lemma 2, we derive at
0 Ω T ( u u * ) · ( u u * ) t d z = 1 2 Ω T t ( u u * ) 2 d z .
Therefore, we have that 0 1 2 u ( t ) u * ( t ) L 2 ( Ω ) 2 0 for every t ( 0 , T ) , since u ( · , 0 ) = u * ( · , 0 ) = u 0 .
Finally, we prove the stability of the weak solution to the Dirichlet problem (2). To this aim, we consider the unique weak solution u C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W 0 p ( · ) ( Ω T ) with t u W p ( · ) ( Ω T ) to (2) and the unique weak solution v C 0 ( [ 0 , T ] ; L 2 ( Ω ) ) W 0 p ( · ) ( Ω T ) with t v W p ( · ) ( Ω T ) to
t v div a ( z , v ) = λ ( | v | p ( x , t ) 2 v ) in Ω T , v = 0 on Ω × ( 0 , T ) , v ( · , 0 ) = v 0 on Ω × 0 ,
where the initial values u 0 , v 0 L 2 ( Ω ) of both problems are different, i.e., u 0 v 0 . The existence is guaranteed by Theorem 1. Moreover, we know that u v W 0 p ( · ) ( Ω T ) . Therefore, we choose φ = u v W 0 p ( · ) ( Ω T ) as an admissible test function in both weak formulations
Ω T u · φ t a ( z , u ) · φ d z = λ Ω T | u | p ( x , t ) 2 u · φ d z
and
Ω T v · φ t a ( z , v ) · φ d z = λ Ω T | v | p ( x , t ) 2 v · φ d z ,
since W 0 p ( · ) ( Ω T ) is the dual of W 0 p ( · ) ( Ω T ) . Now, we subtract the second equation from the first one. This yields
Ω T ( u v ) · ( u v ) t ( a ( z , u ) a ( z , v ) ) · ( u v ) d z = λ Ω T [ | u | p ( x , t ) 2 u | v | p ( x , t ) 2 v ] · ( u v ) d z .
Using the monotonicity condition (5) and Lemma 2, we derive at
0 Ω T ( u v ) t · ( u v ) d z = 1 2 0 T t ( u v ) L 2 ( Ω ) d t ,
which implies the stability estimate (3), i.e.,
Ω | u ( x , t ) v ( x , t ) | 2 d x Ω | u 0 ( x ) v 0 ( x ) | 2 d x
for a.e. t [ 0 , T ) . This shows the conclusion of the Theorem. ☐

4. Conclusions

In this manuscript we proved the existence of a unique weak solution to the Dirichlet problem (2). Moreover, we mentioned that we can also use this approach to show the existence of a unique weak solution to more general problems, please see Remark 3. Furthermore, we studied the stability of the unique weak solution to the Dirichlet problem (2). To this aim, we established the stability estimate (3) for two unique weak solutions to (2) with different initial values.Therefore, it turns out that these weak solutions are controlled by their initial value completely.

Acknowledgments

The author wishes to thank the referees for their careful reading of the original manuscript and their comments that eventually led to an improved presentation.

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Zhan, H. On a parabolic equation related to the p-Laplacian. Bound. Value Probl. 2016, 2016, 78–86. [Google Scholar] [CrossRef]
  2. Zhan, H. The stability of evolutionary p(x)-Laplacian equation. Bound. Value Probl. 2017, 2017, 13–24. [Google Scholar]
  3. Zhan, H. The stability of the solutions of an equation related to the p-Laplacian with degeneracy on the boundary. Bound. Value Probl. 2016, 2016, 178–191. [Google Scholar]
  4. Feireisl, E. Mathematical analysis of fluids in motion: From well-posedness to model reduction. Rev. Mat. Complut. 2013, 26, 299–340. [Google Scholar]
  5. Díaz, J.I.; Hetzer, G.; Tello, L. An energy balance climate model with hysteresis. Nonlinear Anal. 2006, 64, 2053–2074. [Google Scholar]
  6. Díaz, J.I.; Tello, L. On a climate model with a dynamic nonlinear diffusive boundary condition. Discret. Contin. Dyn. Syst. S 2008, 1, 253–262. [Google Scholar]
  7. Acerbi, E.; Mingione, G.; Seregin, G.A. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann. Inst. Henri Poincaré Anal. Non Linéaire 2004, 21, 25–60. [Google Scholar]
  8. Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M. Lebesgue and Sobolev Spaces with Variable Exponents; Springer-Verlag: Berlin Heidelberg, Germany, 2011. [Google Scholar]
  9. Ettwein, F.; Růžička, M. Existence of Strong Solutions for Electrorheological Fluids in Two Dimensions: Steady Dirichlet Problem; Geometric Analysis and Nonlinear Partial Differential Equations; Springer: Berlin, Germany, 2003. [Google Scholar]
  10. Růžička, M. Electrorheological Fluids: Modeling and Mathematical Theory; Springer-Verlag: Heidelberg, Germany, 2000. [Google Scholar]
  11. Růžička, M. Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 2004, 49, 565–609. [Google Scholar]
  12. Acerbi, E.; Mingione, G. Regularity results for electrorheological fluids: The stationary case. C. R. Math. Acad. Sci. Paris 2002, 334, 817–822. [Google Scholar]
  13. Acerbi, E.; Mingione, G. Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 2002, 164, 213–259. [Google Scholar]
  14. Aboulaich, R.; Meskine, D.; Souissi, A. New diffusion models in image processing. Comput. Math. Appl. 2008, 56, 874–882. [Google Scholar]
  15. Chen, Y.; Levine, S.; Rao, M. Variable exponent, linear growth functionals in image restoration. SIAM J. Numer. Anal. 2006, 66, 1383–1406. [Google Scholar]
  16. Harjulehto, P.; Hästö, P.; Latvala, V.; Toivanen, O. Critical variable exponent functionals in image restoration. Appl. Math. Lett. 2013, 26, 56–60. [Google Scholar]
  17. Li, F.; Li, Z.; Pi, L. Variable exponent functionals in image restoration. Appl. Math. Comput. 2010, 216, 870–882. [Google Scholar]
  18. Barrett, J.W.; Liu, W.B. Finite Element Approximation of the Parabolic p-Laplacian. SIAM J. Numer. Anal. 1994, 31, 413–428. [Google Scholar]
  19. Berselli, L.C.; Breit, D.; Diening, L. Convergence analysis for a finite element approximation of a steady model for electrorheological fluids. Numer. Math. 2016, 132, 657–689. [Google Scholar]
  20. Breit, D.; Diening, L.; Schwarzacher, S. Finite Element Approximation of the p(·)-Laplacian. SIAM J. Numer. Anal. 2015, 53, 551–572. [Google Scholar]
  21. Carstensen, C.; Liu, W.; Yan, N. A Posteriori Error Estimates for Finite Element Approximation of Parabolic p-Laplacian. SIAM J. Numer. Anal. 2006, 43, 2294–2319. [Google Scholar]
  22. Eleuteri, M.; Harjulehto, P.; Lukkari, T. Global regularity and stability of solutions to elliptic equations with nonstandard growth. Complex Var. Elliptic Equ. 2011, 56, 599–622. [Google Scholar]
  23. Eleuteri, M.; Harjulehto, P.; Lukkari, T. Global regularity and stability of solutions to obstacle problems with nonstandard growth. Rev. Mat. Complut. 2013, 26, 147–181. [Google Scholar]
  24. Antontsev, S.; Shmarev, S. A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions. Nonlinear Anal. 2005, 60, 515–545. [Google Scholar]
  25. Henriques, E.; Urbano, J.M. Intrinsic scaling for PDEs with an exponential nonlinearity. Indiana Univ. Math. J. 2006, 55, 1701–1722. [Google Scholar]
  26. Erhardt, A. Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth. Adv. Nonlinear Anal. 2014, 3, 15–44. [Google Scholar]
  27. Erhardt, A.H. Existence and Gradient Estimates in Parabolic Obstacle Problems with Nonstandard Growth. Ph.D. Thesis, University Erlangen-Nürnberg, Erlangen, Germany, 2013. [Google Scholar]
  28. Erhardt, A.H. Existence of solutions to parabolic problems with nonstandard growth and irregular obstacle. Adv. Differ. Equ. 2016, 21, 463–504. [Google Scholar]
  29. Erhardt, A.H. Higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth. J. Math. Anal. Appl. 2016, 435, 1772–1803. [Google Scholar]
  30. Erhardt, A.H. On the Calderón-Zygmund Theory for Nonlinear Parabolic Problems with Nonstandard Growth Condition. J. Math. Res. 2015, 7, 10–36. [Google Scholar]
  31. Antontsev, S.; Shmarev, S. Evolution PDEs with Nonstandard Growth Conditions; Atlantis Studies in Differential Equations; Atlantis Press: Paris, France, 2015. [Google Scholar]
  32. Erhardt, A.H. Compact embedding for p(x, t)-Sobolev spaces and existence theory to parabolic equations with p(x, t)-growth. Rev. Mat. Complut. 2017, 30, 35–61. [Google Scholar]
  33. Winkert, P.; Zacher, R. Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth. Nonlinear Anal. 2016, 145, 1–23. [Google Scholar]
  34. Diening, L.; Nägele, P.; Růžička, M. Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var. Elliptic Equ. 2012, 57, 1209–1231. [Google Scholar]
  35. Alkhutov, Y.A.; Zhikov, V.V. Existence theorems for solutions of parabolic equations with a variable order of nonlinearity. Proc. Steklov Inst. Math. 2010, 270, 15–26. [Google Scholar]
  36. Antontsev, S.; Shmarev, S. Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 2009, 53, 355–399. [Google Scholar]
  37. Antontsev, S.; Shmarev, S. Parabolic equations with anisotropic nonstandard growth conditions. Free Bound. Probl. 2007, 60, 33–44. [Google Scholar]
  38. Juutinen, P.; Lindqvist, P. Pointwise decay for the solutions of degenerate and singular parabolic equations. Adv. Differ. Equ. 2009, 14, 663–684. [Google Scholar]
  39. Lindqvist, P. On the equation div(|∇u|p − 2u) + λ|∇u|p − 2u = 0. Proc. Amer. Math. Soc. 1990, 109, 157–164. [Google Scholar]
  40. Li, Y.; Xie, C. Blow-up for p-Laplacian parabolic equations. Electron. J. Diff. Equ. 2003, 2003, 1–12. [Google Scholar]
  41. Franzina, G.; Lindqvist, P. An eigenvalue problem with variable exponents. Nonlinear Anal. 2013, 85, 1–16. [Google Scholar]
  42. Colasuonno, F.; Squassina, M. Stability of eigenvalues for variable exponent problems. Nonlinear Anal. 2015, 123, 56–67. [Google Scholar]
  43. Hamburger, C. Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. 1992, 431, 7–64. [Google Scholar]
  44. Giaquinta, M.; Modica, G. Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1986, 3, 185–208. [Google Scholar] [CrossRef]
  45. Fusco, N.; Hutchinson, J. Partial regularity for minimisers of certain functionals having nonquadratic growth. Ann. Mat. Pura Appl. 1989, 155, 1–24. [Google Scholar]
  46. Hardt, R.; Lin, F.H.; Mou, L. Strong convergence of p-harmonic mappings. In Progress in Partial Differential Equations: The Metz Surveys, 3rd ed.; Pitman Research Notes in Mathematics Series; Longman Scientific & Technical: Harlow, UK, 1994; Volume 314, pp. 58–64. [Google Scholar]
  47. Showalter, R.E. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. In Mathematical Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 1997; Volume 49. [Google Scholar]
  48. Temam, R. Navier–Stokes Equations: Theory and Numerical Analysis; AMS Chelsea Publishing: Providence, RI, USA, 2001. [Google Scholar]
  49. Roubíček, T. Nonlinear Partial Differential Equations with Applications, 2nd ed.; International Series of Numerical Mathematics; Birkhäuser Verlag: Basel, Switzerland, 2013; Volume 153. [Google Scholar]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top