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Mathematics 2017, 5(4), 50; doi:10.3390/math5040050
The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth
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
In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation where and denote the partial derivative of u with respect to the time variable t, while denotes the one with respect to the space variable x. Moreover, the vector-field satisfies certain nonstandard -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
The aim of this paper is to establish the existence theory to nonlinear parabolic equations with nonstandard -growth of the following formwhere and the vector-field satisfy certain -growth and monotonicity conditions. More precisely, we will prove that there exists a unique weak solution to the following Dirichlet problem:
Moreover, we will show that two unique weak solutions u and v of (2) with different initial values satisfy the following stability estimate:for a.e. . 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 , 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 . 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 ). Most of the known results concern the stationary case with -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 -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 is studied.
In the context of parabolic problems with -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 -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  (for the case ), the authors explained where they studied the asymptotic behaviour of the solution u to the homogeneous case of the following evolutionary p-Laplace equationthat should be a solution to the stationary problem
For this equation, the first eigenvalue is the minimum of the Rayleigh quotientcf.  and see also for further details . Similarly, the stationary solution of the appropriate nonstandard -problem should be the solution of the corresponding eigenvalue problem of the -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 of dimension and we write : for the space-time cylinder over of height . Here, or , respectively, denote the partial derivative with respect to the time variable t and denotes the one with respect to the space variable x. Moreover, we denote by the parabolic boundary of and we write for points in . Furthermore, we consider vector-fields a: that are assumed to be Carathéodory functions—i.e., is measurable in the first argument for every and continuous in the second one for a.e. —and satisfy the following nonstandard growth and monotonicity properties, for some growth exponent p: and structure constants and :for all and . Furthermore, the growth exponent function p: satisfies the following conditions: there exist constants and , such thathold for any choice of , where : denotes a modulus of continuity. More precisely, we assume that is a concave, non-decreasing function with
Moreover, the parabolic distance is given by : for . In addition, for the modulus of continuity , we assume the following weak logarithmic continuity condition
Finally, we point out that the monotonicity condition (5) implies, by using the growth condition (4) and Young’s inequality, the coercivity propertyfor all and .
1.2. The Function Spaces
The spaces , and denote the usual Lebesgue and Sobolev spaces, while the nonstandard -Lebesgue space is defined as the set of those measurable functions v: for , which satisfy , i.e.,
The set equipped with the Luxemburg normbecomes a Banach space. This space is separable and reflexive (see [34,35]). For elements of the generalized Hölder’s inequality holds in the following form: if and , where , we have(see also ). Moreover, the norm can be estimated as follows
Notice that we will use also the abbreviation for the exponent . Next, we introduce nonstandard Sobolev spaces for fixed . From assumption (6), we know that satisfiesfor any choice of and for every . Then, we define for every fixed the Banach space asequipped with the norm
In addition, we define as the closure of and we denote by its dual. For every the inclusion holds true.
Furthermore, we consider more general nonstandard Sobolev spaces without fixed t. By we denote the Banach spaceequipped by the norm
If , we write instead of . Here, it is worth mentioning that the notion or respectively, indicate that u agrees with g on the lateral boundary of the cylinder , i.e., .
Our next aim is to introduce the dual space of . Therefore, we denote by the dual of the space . Assume that . Then, there exist functions , , such thatfor all . Furthermore, if , we define the norm
Notice that whenever (11) holds, we can write , where has to be interpreted as a distributional derivate. Bywe mean that there exists , such that(see also ). The previous equality makes sense due to the inclusionswhich allow us to identify w as an element of .
Finally, we are in the situation to give the definition of a weak solution to the parabolic nonstandard growth equation (1):
We identify a function as a weak solution of the parabolic equation (1), if and only if andholds, whenever .
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 a.e. on Ω. Here, we shall always mean
In particular, when , then (13) is obviously equivalent with saying .
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.
Let , be an open, bounded Lipschitz domain and satisfies (6) and (7). Then, suppose that the vector-field a: is a Carathéodory function and satisfies the growth condition (4) and the monotonicity condition (5). Moreover, let . Then, there exists a unique weak solution with of (2) and this solution satisfies the following estimate:with and a constant if or and if . Furthermore, for two weak solutions with and different initial values (i.e., ) of (2) the stability estimate (3), i.e.,holds true for a.e. .
Please note that we can conclude from (3) and Hölder’s inequality thatfor a.e. —where —orfor a.e. , respectively.
Moreover, we want to emphasise that we can also prove the existence of a unique weak solution to (2), if we assume that satisfies the growth condition (4), coercivity condition (8) and the monotonicity condition for all and . 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., satisfying and (please see the approach in ).
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 byfor and . Moreover, we cite the following lemma from (, Lemma 2.1), which is established for the case in  and in the case in .
Suppose that . Then, there exists a positive constant c, depending on , such that for all with , we have
Since , we are able to choose . Then, choosing and we consider . This allows us to conclude from Lemma 1 ((cf. , Lemma 2.2) in the case and (, Lemma 2) in the case ) the following lemma.
There exists a constant c: , such that for any , there holdswhere .
Finally, we need the following Theorem (, Theorem 1.3), since this Theorem implies the strong convergence in -Lebesgue spaces and therefore, it is important for our existence result.
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 , which is an orthonormal basis in . Since is separable, it is a span of a countable set of linearly independent functions . Moreover, we have the dense embedding for any (see, e.g., [47,48]). Thus, without loss of generality, we may assume that this system forms an orthonormal basis of . Now, we fix a positive integer m and define the approximate solution to (2) as follows:where the coefficients are defined via the identityfor and with the initial condition
Then, Equation (15) together with the initial condition (16) generates a system of m ordinary differential equationssince is orthonormal in . By (, Theorem 1.44, p. 25), we know that there is, for every finite system (17), a solution , on the interval for some . Therefore, we multiply Equation (15) by the coefficients , . Then, we need a priori estimates that permit us to extend the solution to the whole domain . Thus, we integrate the equation over for an arbitrarily . Next, we sum the resulting equation over . Therefore, it followsfor a.e. . Furthermore, we usefor a.e. , since , and , cf. . Then, we derive atfor a.e. . Using the coercivity condition (8) on the left-hand side of (19), this yieldswhere . This estimate holds for a.e. . Therefore, we have shown that is uniformly bounded in and independently of m. Thus, the solution of system (17) can be continued to the maximal interval and we have
Please notice that, if , we can estimate the left-hand side of the second last inequality from below by choosing , while if the term depending on λ disappears. If we first of all divide the second last equation by λ, then the constant c depends on λ, i.e., and, finally, we estimate the left-hand side of the resulting estimate from below by using .
Next, we want to derive a uniform bound for in . Therefore, we define a subspace of the set of admissible test functions
Then, we choose a test function
Note that exists, since the coefficients lie in . Moreover, we know that and, therefore, we have also . Thus, we can conclude by the definition of and (15) that
Then, we derive by utilizing the growth condition (4) and the generalized Hölder’s inequality (9) the following estimatewhere with θ: . Using (10) and (20), we have for every and any m thatwith a constant , where c is independent of m. This shows that with
Therefore, we have a uniform bound of in and it follows thatare bounded. This implies the following weak convergences for the sequence (up to a subsequence):
Moreover, by Theorem 2, we can conclude that the sequence (up to a subsequence) converges strongly in with to some function . Thus, we get the desired convergencesfor the sequence (up to a subsequence).
Furthermore, the growth assumption of and the estimate (20) imply that the sequence is bounded in . Consequently, after passing to a subsequence once more, we can find a limit map with
Our next aim is to show that for almost every . First of all, we should mention that each of satisfies the identity (15) with a test function . This follows by the method of construction (see ). Then, we fix an arbitrary . Thus, we have for every the following equationfor all test functions . Passing to the limit , we can conclude that, for all test functions we havewith an arbitrary , by the convergence from above. Therefore, it follows that the identity (22) holds for every . According to monotonicity assumption (5), we know that for every and every the following holds
Moreover, it follows from (15) the conclusion from above and the choice of an admissible test function with that
Then, we test Equation (22) with , subtract the resulting equation from the last estimate and finally pass to the limit yieldingfor all . Since is dense, we are allowed to choose . Hence, we choose with an arbitrary . This yields
Then, passing to the limit , we can conclude thatfor all . This shows that
Moreover, we have to show that . First of all, we should mention that we get from (22) and the integration by parts the following equationfor all with . Moreover, we can conclude from (24)—similar to the previous estimate—thatfor all with . Passing to the limit and using the convergences from above, we getwhere as , since
Furthermore, is arbitrary. Therefore, we can conclude that . 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 with of the Dirichlet problem (2). Thus, we have the following weak formulationsandwith the admissible test function , since is the dual of . Hence, we can conclude by subtracting the second equation from the first one that
Using the monotonicity condition (5) and Lemma 2, we derive at
Therefore, we have that for every , since .
Finally, we prove the stability of the weak solution to the Dirichlet problem (2). To this aim, we consider the unique weak solution with to (2) and the unique weak solution with towhere the initial values of both problems are different, i.e., . The existence is guaranteed by Theorem 1. Moreover, we know that . Therefore, we choose as an admissible test function in both weak formulationsandsince is the dual of . Now, we subtract the second equation from the first one. This yields
Using the monotonicity condition (5) and Lemma 2, we derive atwhich implies the stability estimate (3), i.e.,for a.e. . This shows the conclusion of the Theorem. ☐
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.
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.
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