On a degenerate parabolic equation from double phase convection

The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a(x) and b(x) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a(x) + b(x) > 0, x ∈ and the boundary value condition should be imposed. In this paper, the condition a(x) + b(x) > 0, x ∈ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and ut ∈ L(QT ) is shown. The stability of weak solutions is studied according to the different integrable conditions of a(x) and b(x). To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by a(x)b(x)|x∈∂ = 0 is found for the first time. MSC: Primary 35B65; 49N60; secondary 35K10; 35K55


Introduction
Consider the parabolic equation with a nonlinear convective term which arises from the double phase problems, as well as from the flows of incompressible turbulent fluids etc. [3]. In this paper, Q T = × (0, T), is a smooth bounded domain in R N , p, q > 1, a(x), b(x) ∈ C( ), f i (s, x, t) is a Lipschitz function when |s| is bounded. Though the initial-boundary value problem of the non-Newtonian fluid equation has been studied far and widely [12,13,24], as a generalized case, equation (1.1) has not provoked researchers' attention until recent years. Since the authors of [20] pointed out that the methods used in studying the well-posedness problem of equation (1.2) are invalid for equation (1.1), more and more works related to equation (1.1) have appeared, one can refer to [4-6, 8-10, 35]. First of all, let us give a simple review of [20]. If q ≥ p > 1 and then a(x)|∇u| p + b(x)|∇u| q ≥ c|∇u| p , provided that |∇u| ≥ 1. By its coercivity, we can minimize, with fixed boundary values, the integral F(u) = a(x)|∇u| p + b(x)|∇u| q dx and the local minimizers in the Sobolev class W 1,p loc ( ). It is expected (however it is not always true!) that any local minimizer u is also a weak solution to the corresponding Euler's first variation, i.e., the PDE in a divergence form where a(x, ∇u) = (a i (x, ∇u)), i = 1, 2, . . . , N , is given by a(x, ξ ) = pa(x)|∇u| p-2 + qb(x)|∇u| q-2 ∇u, satisfying that a(x, ξ ) ≤ c 1 + |ξ | q-1 , ξ ∈ R N .
If u ∈ W Of course, there is a similar difficulty in the evolution problems. Emphasizing the fact that an evolution problem is usually formulated by a differential equation and not as a minimization, the authors of [20] adopted a different point of view and posed this philosophical question: does a counterpart of the minimization property exist in evolution problems? i.e., may a solution of an evolution problem be a variational minimizer? By introducing a new kind of weak solution (called as variational solution in [20]), such a problem has been perfectly solved in [20].
We would like to enlarge a bit upon this point. If a(x) and b(x) satisfy to study the well-posedness of the solutions of equation (1.1), besides the initial value condition similar to the usual non-Newtonian fluid equation (1.2), in the sense of the classical trace, the boundary value condition On the other hand, if a(x) and b(x) only satisfy (1.4) or (1.5), equation (1.1) may be degenerate on the boundary ∂ , how to define a suitable boundary value condition instead of (1.8) becomes an important problem. In fact, for a degenerate parabolic equation, the weak solution u(x, t) may be too weak to define the trace on the boundary. For example, the authors of [25] pointed out that if the weak solution u(x, t) only has the property then C ∞ 0 (Q T ) is not dense in the space B = {u : u satisfies (1.9)}, and so one cannot define the trace on ∂ in the classical way. The author of [25] gave a new way to deal with the boundary value condition, and we will introduce the related content in the last section of this paper.
In recent years, the author of this paper has been interested in the stability of weak solutions to the following equation: including the special cases of that p(x) = p is a constant, provided that If the weak solutions of (1.10) only satisfy (1.9), we also cannot define the trace on the boundary in the classical way. To solve this problem, we have avoided to use the boundary value condition (1.8). Instead, we have found that, to study the uniqueness of weak solution of equation (1.10), condition (1.11) can take place of the boundary value condition (1.8) [26,[29][30][31][32]. Actually, for a degenerate parabolic equation, how to deal with the boundary value condition (1.8) has been an important problem for a long time, and there are many papers devoted to this question, one can refer to [14,15,18,21,27,30] etc. for the details.

The definition of weak solution and the main results
In the first place, we give some basic concepts.
Assume that ν(x) is a positive measurable function defined in . Define the weighted Lebesgue space L p (ν, ), 1 < p < ∞, as the space of all real-valued functions u for which Further we suppose that Now, we denote by W 1,p (ν, ) the space of all real-valued functions u such that the derivative in the sense of distributions fulfills u ∈ L p ( ) and ν 1/p |∇u| ∈ L p ( ) with the norm with some g * ≥ 1 p-1 . Then W 1,p (ν, ) is continuously imbedded into W 1,p 1 ( ), where p 1 = pg * g * +1 .
Remark 2.2 By virtue of compact imbedding theorems (see [6]) and Lemma 2.1, we obtain that the imbedding Therefore, if we also suppose that the number g * from Lemma 2.1 satisfies g * > N p . then W 1,p (μ, ) is compactly imbedded into L p ( ).
In the second place, we introduce the definition of weak solution.
and the boundary value condition (1.8) is satisfied in the sense of trace.
In the third place, we give the main results.
If we do not require u t ∈ L 2 (Q T ), instead, u t ∈ L 1 (0, T; W -1,q ( )) (or a more general Banach space), condition (2.10) or condition (2.11) may not be necessary. Also, condition (2.8) is only used in the proof of the L ∞ -norm estimate of u, and we conjecture it can be replaced by the condition In this case, if u 0 (x) ≥ 0, then by the maximal value theorem, one may prove that there is a nonnegative weak solution u(x, t) satisfying Moreover, by considering the minimality for the variational solution, the existence and the regularity of weak solutions was studied in [20] when However, the main aim of this paper is not to study the existence of weak solution to equation (1.1), we do not pay attention to whether conditions (2.10)(2.11) are optimal or not. Also, we do not try to compare Theorem 2.4 with the results of weak solutions given in [20]. We only give a result on the existence of weak solution for the completeness of the paper. We mainly focus on the stability of weak solutions to equation (1.1) when the coefficients a(x) and b(x) may be degenerate on the boundary ∂ .

14)
and with different initial values u 0 (x) and v 0 (x) respectively, then If condition (2.13) is invalid, there are three cases (a) a(x) a(x) By Proposition 3.3, in cases (a) and (b), we still can impose the boundary value condition (2.14) and obtain stability (2.15). If a(x), b(x) satisfy (c), we cannot impose the boundary value condition (2.14) generally. Fortunately, if there are some restrictions between a(x), b(x) and f i (s, x, t), we are still able to prove the following stability of weak solutions without (2.14).

16)
and for η small enough, Here, η is a small constant and

19)
and for η small enough, Here, η is a small constant and There is an essential difference between Theorem 2.6 and Theorem 2.7. In Theorem 2.6, a(x), b(x) ∈ C 1 ( ) satisfy (1.5), and so while in Theorem 2.7 a(x), b(x) ∈ C 1 ( ) satisfy (1.4), and so Naturally, condition (2.16) (or (2.19)) may not be necessary. In the last section of this paper, by giving a generalization of the classical trace of u ∈ BV (Q T ), we will use a reasonable partial boundary value condition instead of condition (2.16) (or (2.19)) to study the stability of weak solutions.

The existence of weak solutions
In this section, we want to prove Theorem 2.4. Let us first consider the following Cauchy-Dirichlet problem: Since the convection function f i (s, x, t) is a C 1 function on R × Q T , i = 1, 2, . . . , N , by the classical existence theory for parabolic equations [17], similar to [8], we know there is a unique weak solution u ε ∈ C 0 ([0, T]; L 2 ( )) ∩ L q (0, T; W 1,q 0 ( )) with ∂ t u ε ∈ L q (0, T; W -1,q ( )). Now, let us show that Then the following properties hold: where l > 1.
Proof We only give the proof provided that condition (2.10) is true. When condition (2.11) is true, this lemma can be verified in a similar way. Let k be a real number and u 0 L ∞ ( ) ≤ k, the function ϕ be defined as (3.4). Define We can see where u εt , ϕ(G k (u ε )) is the dyadic interaction between L p (0, T; W 1,p 0 ( )) and L p (0, T; W -1,p ( )). (3.11) Substituting (3.11) into (3.10), using Lemma 2.1, we can deduce that with (3.12) and (3.13), we have (3.14) Since p ≥ 2, by (3.7), then Plugging (3.15) into (3.14) and taking the supremum for τ ∈ [0, t 1 ], with t 1 ≤ T to be determined later, we have where μ( ) is the Lebesgue measure of . Now, using the embedding inequality [16,24], we can deduce that where γ is a constant independent of t 1 , similar to the proof of Theorem 2.2 in [16], it follows from (3.16) that where r > N+p N is a constant, and so Therefore, thanks to the iteration lemma in [21], from (3.18), we eventually obtain that where D > 0 is a constant depending only on p, N , t 1 , r, . This proves that, for fixed λ validating Lemma 2.1, exists almost everywhere and is bounded. If a(x) Accordingly, based on condition (2.10) or condition (2.10), by (3.20), we obtain Multiplying (2.9) by u εt , we have (3.23) For every term in (1.7), firstly, we have At last, it is not difficult to show that for any given ϕ ∈ C 1 0 (Q T ). So u ∈ L p (0, T; W 1,p loc ( )) ∩ L q (0, T; W 1,q loc ( )), and (2.6) is true. In addition, we can choose the test function ϕ(x, t) = χ [t 1 ,t 2 ] φ(x) in which φ(x) ∈ C ∞ 0 ( ) and χ [t 1 ,t 2 ] is the characteristic function of [t 1 , t 2 ] ⊂ (0, T). Then Let t = t 2 and t 1 → 0. Then we have (2.7). Moreover, by the following proposition, u can be defined as the trace on the boundary ∂ , u is a solution of equation (1.1) with the initial-boundary value conditions (1.7)-(1.8). Theorem 2.4 is proved.

Proposition 3.3 If u(x, t) is a weak solution of equation (1.1) with the initial value condition (1.4) and one of the following conditions is true:
Similarly, if (ii) is true, we also have (3.32).

The stability of the initial-boundary value problem
For small η > 0, we introduce the following functions: Obviously, we have |sh η (s)| ≤ 1 and

Proof of Theorems 2.6
Proof of Theorems 2.6 Since a( Secondly, we have Thirdly, we have which goes to zero as η → 0. Similarly, by (2.18), we have which goes to zero as η → 0.
By the generalized Gronwall inequality [28], we can extrapolate Letting τ → 0, we have the stability (2.15). If u(x, t) and v(x, t) are two solutions of (1.1) with the initial values u 0 (x) and v 0 (x) respectively and with the homogeneous boundary value condition Proof Similar to the proof of Theorem 2.6, we have (5.2)-(5.7). Since a(x) and b(x) satisfy condition (a) or condition (b), Proposition 3.3 means that the partial boundary value condition (5.11) is true in the classical sense of the trace. Then by condition (2.17) it yields The remaining process of the proof can be completed as that of Theorem 2.6.

Proof of Theorems 2.7
In this section, we use a similar method as that used in the proof of Theorem 2.6 to prove Theorem 2.7.
Proof of Theorem 2.7 Let u(x, t) and v(x, t) be two weak solutions of equation (1.1) with the initial values u 0 (x) and v 0 (x) respectively. Different from the proof of Theorem 2.6), a(x) and b(x) may satisfy (2.22).
Directly, we have the following three formulas similar to (5.3)-(5.5): which goes to zero as η → 0. Similarly, by (2.14), we can show that which goes to zero as η → 0.
The proof is similar to that of Corollary 5.1, we omit the details here.

A generalization of trace
Let BV ( ) be the BV function space, i.e., | ∂f ∂x i | is a regular measure, and Then C ∞ 0 ( ) is dense in BV ( ), and so the trace of f (x) ∈ BV ( ) on the boundary ∂ is defined as the limit of a sequence f ε (x) as It is well known that a BV function space is the weakest space such that integration by parts is still true. For a degenerate parabolic equation, how to impose a suitable boundary condition has been an important and difficult problem for a long time. For example, if we consider the reaction-diffusion equation if a(u, x, t) is smooth enough, then the weak solution u(x, t) ∈ BV (Q T ) can be proved, and so one can impose the boundary value condition (1.8) in the sense of trace in the classical way [26,30,32]. However, if a(u, x, t) is just a continuous function or just a integral function, then one only can prove that there is a weak solution u(x, t) ∈ L ∞ (Q T ), but u(x, t) may not be a BV function. Equation (7.2) is of hyperbolic-parabolic mixed type. When a ≡ 0, equation (7.1) becomes a first-order hyperbolic equation, if the solution is merely in L ∞ , the author of [23] first extended the trace in a weaker sense by introducing an integral formulation of the boundary condition. [23]'s idea was generalized to deal with well-posedness of weak solutions to the strongly degenerate parabolic equations (7.2) in [1,2,7,11,14,15,18,22]. In this paper, we first consider the evolutionary p-Laplacian equations of the form where D i = ∂ ∂x i , α(x) ∈ C( ), α(x) > 0 in but may be equal to 0 on the boundary ∂ . The author of [25] classified the boundary ∂ into three parts: the nondegenerate boundary 3 , the weakly degenerate boundary 4 , and the strongly degenerate boundary 0 . In detail, the author of [25] denoted that where n = {n i (x)} is the inner normal vector of ∂ . In order to study the well-posedness of weak solutions to equation (7.3), they imposed a partial boundary value condition as where g(x, t) is an appropriately smooth function.
According to Proposition 3.3, it is obvious that on ( 3 ∪ 4 ) × (0, T) the boundary value condition is true in the classical trace sense. So, the trouble lies in that the classical trace of u on the strongly degenerate boundary 0 cannot be defined.
Denote that λ = {x ∈ : d(x) > λ} when λ is a positive infinite variable, and denote by B the closure of the set C ∞ 0 (Q T ) with respect to the norm The author of [25] defined the trace of u ∈ B, u(x, t) = 0 on 2 as ess lim Remark 2.2 in [25] pointed out that the usual trace of u ∈ B, u(x, t) = 0 on 3 ∪ 4 also satisfies (7.5). So, (7.5) is a generalization of the usual trace of u ∈ BV (Q T ) to that of u ∈ B.
Moreover, we can generalize the trace of u ∈ BV (Q T ) to that of u ∈ B by a more general way. Let φ(x) be a weak characteristic function of [33], i.e., φ(x) ∈ C( ) ∩ C 1 ( \ μ ) and In a very recent paper [34], using some idea of [25], we defined the trace of u ∈ B, u(x, t) = 0 on 0 as ess lim and the partial boundary value condition matching up with equation (7.3) is in the sense of (7.5), where Secondly, let us come back to our main equation (1.1). Denote that and If a(x), b(x) satisfy (c), we cannot impose the boundary value condition (7.7) in the sense of the classical trace generally. However, inspired by [25,34], if f i satisfies (5.10), by checking the proof of Corollary 5.1, then we may generalize the trace of u ∈ BV (Q T ) to that of u ∈ B p ∩ B q , u(x, t) = 0 as ess lim Accordingly, if a(x), b(x) satisfy (c), in order to study the uniqueness of weak solution to equation (1.1), we can impose the partial boundary value condition (7.9) in the sense of (7.8), where p = x ∈ ∂ : for any small r > 0, p-1 dy = +∞ and q = x ∈ ∂ : for any small r > 0, Naturally, there are other ways to generalize the trace. For example, similar to [25,34], one also can generalize the trace of u ∈ BV (Q T ) to that of u ∈ B p ∩ B q , u(x, t) = 0 as ess lim In this weak sense of trace, one also can study the stability of weak solution to equation (1.1) with the partial boundary value condition (7.9), provided f i satisfies The details are omitted here.

About the regularity
The following parabolic equation with p, q-growth u t = div |∇u| p-2 ∇u + |∇u| q-2 ∇u , (x, t) ∈ Q T , (8.1) was studied in [8]. Actually, the main equation considered in [8] has a more general sense.
Recalling the main equation considered in this paper u t = div a(x)|∇u| p-2 ∇u + b(x)|∇u| q-2 ∇u +  .7) are true correspondingly. We are ready to discuss this problem thoroughly in the future; in particular, we are concerned with the boundary estimates about the weak solution u(x, t) and the estimate of its gradient |∇u| near the boundary.