WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY

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1 Electronic Journal of Differential Equations, Vol (2017), No. 13, pp ISSN: URL: or WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY HUASHUI ZHAN, JIE WEN Communicated by Zhaosheng Feng Abstract. In this article we study the electrorheological fluid equation u t = div(ρ α u 2 u), where ρ(x) = dist(x, ) is the distance from the boundary, C 1 (), and p = min x > 1. We show how the degeneracy of ρ α on the boundary affects the well-posedness of the weak solutions. In particular, the local stability of the weak solutions is established without any boundary value condition. 1. Introduction Let R N be a bounded domain with smooth boundary, and is a measurable function. The evolutionary -Laplacian equation u t = div( u 2 u), (x, t) = (0, T ), (1.1) comes from a new interesting type of fluids called electrorheological fluids [1, 10]. We consider an electromagnetic field with vector of magnetic density B = (0, 0, u(x, t)), where x = (x 1, x 2 ) R 2. Let H = (H 1, H 2, H 3 ) be a magnetic field intensity, J = (J 1, J 2, J 3 ) be a current density, E = (E1, E 2, E 3 ) be an electrostatic field intensity and r be a resistivity. Review Maxwell s equations B + rot E = 0, (1.2) J rot H, (1.3) B = λ H, (1.4) E = r J, (1.5) 2010 Mathematics Subject Classification. 35L65, 35K85, 35R35. Key words and phrases. Electrorheological fluid equation; boundary degeneracy; Hölder s inequality; local stability. c 2017 Texas State University. Submitted August 12, Published January 12,

2 2 H. ZHAN, J. WEN EJDE-2017/13 where λ > 0. By (1.4), we have H 3 = u/λ. Therefore, J 1 H 3 x 2 H 2 x 3 J 2 H 1 x 3 H 3 x 1 = 1 u, λ x 2 = 1 u, λ x 1 J 3 H 2 x 1 H 1 x 2 = 0. Now, if we suppose that r = r 0 J q(x), taking into account (1.6) we know that J = J1 2 + J J 3 2 = 1 λ u, (1.6) where r 0 > 0 is a constant, q(x) is a function which depends on the environment. If a(x) = r 0 > 0, (1.7) λq(x)+1 then E = r J = a(x) u q(x) ( u x 2, u x 1, 0), as a generalization of Ohm s law (1.5). Hence the third coordinate of the vector rot E is (rot E) 3 = E 2 E 1 x 1 x 2 = q(x) u ( a(x) u ) q(x) u (a(x) u ) x 1 x 1 x 2 x 2 = div(a(x) u q(x) u). Using (1.2), according to Mashiyev-Buhrii [9], letting = q(x) + 2, we have with the initial value u t div(a(x) u 2 u) = 0, (x, t), (1.8) and the homogeneous boundary value u t=0 = u 0 (x), x, (1.9) u ΓT = 0, (x, t) Γ T = (0, T ), (1.10) which have been researched widely recently, one can refer to [2, 4, 8, 6]. If r 0 = r(x) is a function, then a(x) in (1.7) may be degenerate on the boundary. For example, if r(x) Σp = 0, where Σ p, then the equation is degenerate on Σ p. We will study the problem by taking a special but basic formula of the diffusion function a(x) = ρ α (x), where ρ(x) = dist(x, ), and α > 0. Then equation (1.8) becomes u t = div(ρ α u 2 u), (x, t) (0, T ). (1.11) If p, the above equation becomes u t = div( ρ α u p 2 u), (1.12) which was first studied by Yin-Wang [13]. They had proved the following results:

3 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 3 Theorem 1.1. Let p > 1, and u 0 L (), ρ α u 0 p L 1 (). (1.13) If α < p 1, then there exists an unique solution of equation (1.12) with the initial-boundary conditions (1.9)-(1.10). While, if α p 1, there exists an unique solution of (1.12) only with the initial value (1.9). In other words, if α p 1, the stability of the solutions of (1.12) is true without any boundary condition. Inspired by [13], we studied (1.11) in a similar way as the one described in [15], and obtain a similar theorem. Theorem 1.2. Let p > 1, and u 0 L (), ρ α u 0 L 1 (). (1.14) If α < p 1, then there exists an unique solution of (1.11) with the initial-boundary conditions (1.9)-(1.10). While, if α p + 1, then there exists an unique solution of equation (1.11) with the initial value (1.9). We are interested in this problem because we would like to know how the degeneracy of the diffusion function ρ α affects equation (1.11) essentially. To see that, we suppose that u and v are two classical solutions of (1.11) with the initial values u(x, 0) and v(x, 0) respectively. Then (u v)(u v) t dx + ρ α ( u 2 u v 2 v) (u v) dx = ρ α (u v)( u 2 u v 2 v) ndσ = 0, where n is the outer unit normal vector of. So 1 d (u v) 2 dx 0, 2 dt u(x, t) v(x, t) 2 dx u 0 (x) v 0 (x) 2 dx. (1.15) This implies that the classical solutions (if there are) of equation (1.11) are stable without any boundary value condition, only if that α > 0. Certainly, since equation (1.11) is degenerate on the boundary and may be degenerate or singular at points where u = 0, it only has a weak solution generally, so whether the inequality (1.15) is true or not remains to be verified. Obviously, since is a function, there exists a gap if p 1 α < p + 1 in Theorem 1.2. In our paper, roughly speaking, only if α p 1, we can establish the stability of the weak solutions of equation (1.11) without any boundary value condition. The conclusions not only make a supplementary of the results of [13, 15, 14], but also provide a new and more effective way to establish the stability of the solutions (see Theorems 2.6 and 2.7 below). 2. Basic functional spaces and main results Throughout this article we assume that 1 < C 1 (), and denote p + = max, 1 < p = min.

4 4 H. ZHAN, J. WEN EJDE-2017/13 First of all, we introduce some basic functional spaces. The space L () = {u : u is a measurable real-valued function, u(x) dx < }. is equipped with the Luxemburg norm u L () = inf{λ > 0 : u(x) dx 1}. λ The space (L (), L ()) is a separable, uniformly convex Banach space. The space W 1, () = {u L () : u L ()}. is endowed with the norm u W 1, = u L () + u L (), u W 1, (). We use W 1, 0 () to denote the closure of C 0 () in W 1,. Some properties of the function spaces W 1, () are quoted in the following lemma. Lemma 2.1. (i) The spaces (L (), L ()), (W 1, (), W 1, ()) and W 1, 0 () are reflexive Banach spaces. 1 (ii) -Hölder s inequality. Let q 1 (x) and q 2 (x) be real functions with q + 1(x) 1 q 2(x) = 1 and q 1 (x) > 1. Then, the conjugate space of L q1(x) () is L q2(x) (). And for any u L q1(x) () and v L q2(x) (), we have uvdx 2 u L q 1 (x) () v L q 2 (x) (). (iii) u L () = 1 = u L () > 1 = u p L u L () < 1 = u p+ L u dx = 1, (iv) If p 1 (x) p 2 (x), then L p1(x) () L p2(x) (). (v) If p 1 (x) p 2 (x), then W 1,p2(x) () W 1,p1(x) (). u dx u p+ L, u dx u p L. (vi) -Poincarés inequality. If C(), then there is a constant C > 0, such that u L () C u L (), u W 1, 0 (). This implies that u L () and u W 1, () are equivalent norms of W 1, 0. Zhikov [17] showed that W 1, 0 () {v W 1, 0 () v = 0} = W 1, (). Hence, the property of the space is different from the case when p is a constant. This fact gives a general idea used in studying the well-posedness of the solutions to the evolutionary p-laplacian equation which can not be used directly.

5 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 5 with If the exponent is required to satisfy logarithmic Hölder continuity condition p(y) ω( x y ), x, y, x y < 1 2, lim sup s 0 + ω(s) ln( 1 s ) = C <, then W 1, 0 () = W 1, (). In fact Antontsev-Shmarev [2] established the wellposedness of equation (1.8). Now, we introduce some other Banach spaces used to define the weak solution of the equation. For every fixed t [0, T ], we define V t () = {u(x) : u(x) L 2 () W 1,1 0 (), u(x) L 1 ()}, u Vt() = u 2, + u,, and denote by V t () its dual, where u 2, = u L 2 (), u, = u L (). Also we use the Banach space W ( ) = { u : [0, T ] V t () u L 2 ( ), u L 1 ( ), u = 0 on Γ T }, u W (QT ) = u,qt + u 2,QT. The space W ( ) is the dual of W ( ) (the space of linear functionals over W ( )): w W ( ) if and only if n w = w 0 + D i w i, w 0 L 2 ( ), w i L p (x,t) ( ), i=1 φ W ( ), w, φ = The norm in W ( ) is defined by ( w 0 φ + i ) w i D i φ dx dt. v W ( ) = sup{ v, φ : φ W ( ), φ W (QT ) 1}. Definition 2.2. A function u(x, t) is said to be a weak solution of (1.11) with the initial value (1.9), if u L ( ), u t W ( ), ρ α u L 1 ( ), (2.1) and for any function ϕ L (0, T ; W 1, 0 ()) W ( ), it holds u t, ϕ + (ρ α u 2 u ϕ) dx dt = 0. (2.2) The initial value, as usual, is satisfied in the sense of that u(x, t)φ(x)dx = u 0 (x)φ(x)dx, φ(x) C0 (). (2.3) lim t 0 The main result of this article is stated as follows. Theorem 2.3. Let 1 < p, 0 < α. If u 0 (x) L (), ρ α u 0 L 1 (), (2.4) then (1.11) with initial value (1.9) has a weak solution u in the sense of Definition 2.2. If α < p 1, then (1.11) with initial-boundary values (1.9)-(1.10) has a weak solution u. The boundary value condition (1.10) is satisfied in the sense of trace.

6 6 H. ZHAN, J. WEN EJDE-2017/13 Theorem 2.4. Let u and v be two weak solutions of equation (1.11) with initial values u(x, 0) and v(x, 0) respectively. If p 1 > α > 0, ρ α 1 u 1 dx <, ρ α 1 u 1 dx <, (2.5) then u(x, t) v(x, t) dx u 0 (x) v 0 (x) dx. (2.6) The above theorem is a weaker version of [15, Theorem 1.4] when α < p 1. We can use it to prove the following Theorems. Theorem 2.5. Let u and v be two weak solutions of equation (1.11) with the different initial values u(x, 0), v(x, 0) respectively, and the exponent be required to satisfy logarithmic Hölder continuity condition. If α p 1, u and v satisfy (2.5), u t L 2 ( ) and v t L 2 ( ), then then the stability (2.6) is still true. Theorem 2.6. Let p > 1 and 0 < α < p 1. If u and v are two solutions of equation (1.11) with the differential initial values u 0 (x) and v 0 (x) respectively, then there exists a positive constant β max{ p+ α p 1, 2} such that ρ β u(x, t) v(x, t) 2 dx c ρ β u 0 (x) v 0 (x) 2 dx. (2.7) In particular, for any small enough constant δ > 0, there holds u(x, t) v(x, t) 2 dx cδ β u 0 (x) v 0 (x) 2 dx. (2.8) δ Here, δ = {x : dist(x, ) > δ}, by the arbitrary of δ, we have the uniqueness of the solution. The inequality (2.7) shows the local stability of the solutions. Theorem 2.7. Let p > 1, α p 1, b i (s) be a Lipschitz function, and the exponent be required to satisfy logarithmic Hölder continuity condition. If u, v are two solutions of equation (1.11) with the different initial values u 0 (x), v 0 (x) respectively, then the inequality (2.7) is true, which implies the uniqueness of the solution. The proof of the existence (Theorem 2.3) is quite different from that shown in [13, 15, 14]. There to prove the stability of solutions, the authors used two ways to deal with the cases α < p 1 and α p + 1. In this article, we adopt a similar method to prove Theorems 2.4 and 2.5, and then develop it to prove Theorems 2.6 and 2.7. The methods used here seem to be more effective, and can be extended to the degenerate parabolic equation related to the -Laplacian directly. 3. Proof of Theorem 2.3 Following [3], we have the following lemma. Lemma 3.1. Let q 1. If u ε L (0, T ; L 2 ()) W ( ), u εt W ( ) c, and ( u ε q 1 u ε ) p, c, then there is a subsequence of {u ε } which is relatively compactness in L s ( ) with s (1, ).

7 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 7 To study (1.11), let us consider the associated regularized problem u εt div(ρ α ε ( u ε 2 + ε) 2 2 u ε ) = 0, (x, t), (3.1) u ε (x, t) = 0, (x, t) (0, T ), (3.2) u ε (x, 0) = u 0ε (x), x. (3.3) where ρ ε = ρ δ ε + ε, ε > 0, δ ε is the usual mollifier, u ε,0 C0 () and ρ α ε u ε,0 L 1 () is uniformly bounded, and u ε,0 converges to u 0 in W 1, 0 (). It is well-known that the above problem has a unique classical solution [5, 11]. Lemma 3.2. There is a subsequence of u ε (we still denote it as u ε ), which converges to a weak solution u of equation (1.11) with the initial value (1.9). Proof. By the maximum principle, there is a constant c dependent on u 0 L () and independent on ε, such that u ε L ( ) c. (3.4) Multiplying (2.1) by u ε and integrating it over, we have 1 u 2 2 εdx + ρ α ε ( u ε 2 + ε) 2 2 u ε 2 dx dt = 1 2 u 2 0dx c. (3.5) For small enough λ > 0, let λ = {x : dist(x, ) > λ}. Since p > 1, by (3.5) we have T ( T ) u ε dx dt c u ε p 1/p dx dt c(λ). (3.6) 0 λ 0 λ Now, for any v W ( ), v W (QT ) = 1, and u εt, v = ρ α ε ( u ε 2 + ε) 2 2 u ε v dx dt, by Young s inequality, we can show that [ ] u εt, v c ρ α ε u ε dx dt + ( v + v ) dx dt c, then u εt W ( ) c. (3.7) Now, let ϕ C 1 0(), 0 ϕ 1 such that ϕ 2λ = 1 and ϕ \λ = 0. Then so we have By (3.6), and so (ϕu ε ) p (ϕu ε ) t, v = ϕu εt, v u εt, v ; (ϕ(x)u) εt W ( ) u εt W ( ) c. (3.8) T dx dt c(λ)(1 + u ε p dx dt) c(λ), (3.9) 0 λ ( ϕu ε ) p, c(λ). (3.10) By Lemma 3.1, ϕu ε is relatively compactness in L s ( ) with s (1, ). Then ϕu ε ϕu a.e. in. In particular, by the arbitraries of λ, it follows that u ε u a.e. in.

8 8 H. ZHAN, J. WEN EJDE-2017/13 Hence, by (3.4), (3.5), (3.7), there exists a function u and the n-dimensional vector function ζ = (ζ 1,, ζ n ) satisfying and u L ( ), u t W ( ), u ε u, in L ( ), u ε u ρ α ε u ε 2 u ε ζ ζ L 1 (QT ), in L loc ( ), in L 1 (QT ). To prove that u satisfies (1.11), we notice that for any function ϕ C0 ( ), we have [u εt ϕ + ρ α ε ( u ε 2 + ε) 2 2 u ε ϕ] dx dt = 0. (3.11) Then ( u ϕ + ς ϕ) dx dt = 0. (3.12) Now, similar to [15, 14], we can prove that ρ α u 2 u ϕ dx dt = ζ ϕ dx dt (3.13) for any function ϕ C 0 ( ). Thus u satisfies (1.11). Similarly, we can prove (1.9) as in [3] in the same manner. The proof is complete. Lemma 3.3. If α < p 1, and let u be the solution of equation (1.11) with the initial value (1.9), then the trace of u on the boundary can be defined in the traditional way. The above lemma was proved in [15, 14]. Note that Theorem 2.3 is the directly consequence of Lemmas 3.2 and 3.3. For small η > 0, let S η (s) = 4. Proof of Theorem 2.4 s Obviously h η (s) C(R), and 0 h η (τ)dτ, h η (s) = 2 η h η (s) 0, sh η (s) 1, S η (s) 1, lim S η(s) = sgn(s), η 0 ( s ) 1 η +. (4.1) lim ss η 0 η(s) = 0. (4.2) Proof. If α < p 1, by Lemma 3.3, the weak solution of (1.11) can be defined by the trace on the boundary in the traditional way. Let u and v be two weak solutions of (1.11) with the initial values u(x, 0) and v(x, 0) respectively. Let β > 0 and φ(x) = ρ β (x). (4.3)

9 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 9 Then, we can choose S η (φ(u v)) as the test function, and find S η (φ(u v)) dx + ρ α ( u 2 u v 2 v) φ (u v)s η(φ(u v))dx + ρ α ( u 2 u v 2 v) φ(u v)s η(φ(u v))dx = 0. Thus, we have S η (φ(u v)) lim η 0 and dx = = (4.4) sgn(φ(u v)) dx sgn(u v) dx = d (4.5) dt u v 1, ρ α ( u 2 u v 2 v) (u v)s η(φ(u v))φ(x)dx 0, (4.6) ρ α ( u 2 u v 2 v) φ(u v)s η(φ(u v))dx c ρ α 1 ( u 2 u v 2 v) {x:ρ β u v <η} φ(u v)s η(φ(u v)) dx, which tends to 0 as η 0, because of (2.5) and Now, let η 0 in (4.4). Then lim φ(u η 0 v)s η(φ(u v)) = 0. (4.7) d dt u v 1 c u v 1. (4.8) This implies u(x, t) v(x, t) dx c(t ) u 0 v 0 dx. (4.9) The proof is complete. 5. Proof of Theorem 2.5 Proof. If α p 1, the weak solution of equation (1.11) lacks the regularity on the boundary, we can not define the trace on. Denote let β > 0 and λ = {x : dist(x, ) > λ}, (5.1) φ(x) = [dist((x, \ λ )] β = d β λ. (5.2) Let u and v be two weak solutions of equation (1.11) with the initial values u(x, 0) and v(x, 0) respectively. We can choose S η (φ(u ε v ε )) as the test function,

10 10 H. ZHAN, J. WEN EJDE-2017/13 where u ε and v ε are the mollified function of the solutions u and v respectively. Then S η (φ(u ε v ε )) dx λ + ρ α ( u 2 u v 2 v) φ (u ε v ε )S η(φ(u ε v ε ))dx λ (5.3) + ρ α ( u 2 u v 2 v) φ(u ε v ε )S η(φ(u ε v ε ))dx λ = 0. For any given λ > 0, denoting λ = λ (0, T ), by (iii) of Lemma 2.1 and (2.1) in Definition 2.2, we know that u L ( λ ), v L ( λ ). Thus according to the definition of the mollified functions u ε and v ε, the exponent is required to satisfy the logarithmic Hölder continuity condition, by [4, 17, 8], we have u ε L ( ), v ε L ( ), u ε u, v ε v, a.e. in, (5.4) u ε 1,λ u 1,λ, v ε 1,λ v 1,λ, u ε u, v ε v, in L ( λ ). Since 0 S η(φ(u ε v ε )) 2 η, it follows that (u ε v ε )S η(φ(u ε v ε )) L ( λ ) c(η) (u ε v ε ) L ( λ ) c(η), (5.5) For any ϕ L 1 (λ ), it holds (u ε v ε )S η(φ(u ε v ε ))ϕ dx (u v)s η(φ(u v))ϕ dx λ λ = (u ε v ε )[S η(φ(u ε v ε )) S η(φ(u v))]ϕ dx λ + [ (u ε v ε ) (u v)]s η(φ(u v))ϕ dx λ = I 1 + I 2. (5.6) Since u ε u and v ε v in L ( λ ), it follows that while lim I 2 = 0, (5.7) lim I 1 lim (u ε v ε ) L ( λ ) [S η(φ(u ε v ε )) S η(φ(u v))]ϕ L lim (u v) L ( λ ) [S η(φ(u ε v ε )) S η(φ(u v))]ϕ L = 0, by the controlled convergent theorem. Thus we have 1 ( λ ) 1 ( λ ) (5.8) (u ε v ε )S η(φ(u ε v ε )) (u v)s η(φ(u v)), in L ( λ ). (5.9)

11 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 11 Since on λ, one has ρ α φ( u p 2 u v p 2 v) L 1 (λ ) by the weak convergency of (5.9) we have lim ρ α ( u 2 u v 2 v) φ (u ε v ε )S η(φ(u ε v ε )) dx λ = ρ α ( u 2 u v 2 v) φ (u ε v ε )S η(φ(u v))dx. λ At the same time, it is clear that lim ρ α ( u 2 u v 2 v) φ(u ε v ε )S η(φ(u ε v ε )) dx λ = ρ α ( u 2 u v 2 v) φ(u ε v ε )S η(φ(u v))dx, λ (5.10) (5.11) by the controlled convergent theorem. inequality, we have u dx dt <, Also, since u t, v t L 2 ( ), by Hölder s v dx dt <. (5.12) By the controlled convergent theorem, we have lim S η (φ(u ε v ε )) dx = S η (φ(u v)) dx. (5.13) λ λ Now, we let ε 0, and then let λ 0, at last, let η 0 in (5.3). As the proof of (4.5)-(4.9), we arrive at the desired result. 6. Uniqueness in the case 0 < α < p 1 Proof. Let u and v be two solutions of equation (1.11) with initial values u 0 (x) and v 0 (x) respectively. According to the definition of W ( ), L 2 ( ) W ( ), when ϕ L 2 ( ), we have (u v) t, ϕ = ϕ dx dt. (6.1) From the definition of the weak solution, we have ϕ dx dt = ρ α ( u 2 u v 2 v) ϕ dx dt, (6.2) for any ϕ L (0, T ; W 1, 0 ()) L 2 ( ). By Lemma 3.3, α < p 1, then the trace of u on the boundary can be defined in the traditional way. For any fixed τ, s [0, T ], χ [τ,s] is the characteristic function on [τ, s]. Since β 2, and χ [τ,s] (u v)ρ β L 2 ( ) L (0, T ; W 1, 0 ()) (6.3)

12 12 H. ZHAN, J. WEN EJDE-2017/13 we may choose it as a test function in the above equality. Thus, by denoting = [τ, s], we have β (u v)ρ dx dt = ρ α ( u 2 u v 2 v) [(u v)ρ β ] dx dt Q τs (6.4) = ρ α+β ( u 2 u v 2 v) (u v) dx dt Q τs + ρ α ( u 2 u v 2 v)(u v) ρ β dx dt. The first term on the right hand side of (6.4) satisfies ρ α+β ( u 2 u v 2 v) (u v) dx dt 0. (6.5) The second term on the right hand side of (6.4), by (iii) of Lemma 2.1, satisfies (u v)ρ α ( u 2 u v 2 v) ρ β dx dt Q τs u v ρ α ( u 1 + v 1 ) ρ β dx dt c s τ ρ α 1 ( u 1 + v 1 ) ρ α ρ β (u v) L ()dt c s τ ρ α 1 ( u 1 + v 1 ) ρ α +(β 1) (u v) L ()dt s ( ) 1/p c ρ α ( u + v 1 )dx τ ( 1/p1dt ρ α+(β 1) u v dx) s ( 1/p1dt. c ρ α+(β 1) u v dx) τ L 1 () L 1 () (6.6) Here, we used that ρ = 1 almost everywhere, that p 1 = p + or p, and that p (x) = 1, where p 1 = p + or p. Now, from β p+ α p 1, we have ( ) ρ α+(β 1) u v 1/p1 ( ) 1/p1, dx c ρ β u v dx (6.7) 1+ 2 where 1 = {x : 2}, 2 = {x : 1 < < 2}. Then ( ) ρ β u v 1/p1 ( ) dx c ρ β u v 2 1/p1 dx 1 1 ( ) (6.8) 1/p1. c ρ β u v 2 dx

13 EJDE-2017/13 ELECTRORHEOLOGICAL FLUID EQUATIONS 13 In 2, by Hölder s inequality and (iii) in Lemma 2.1, we obtain ( ) ρ β u v 1/p1 ) dx c ( ρ 2 u v 1/p1 2 L ( 2 2) ( ) q. (6.9) c ρ β u v 2 dx where q < 1. Also we have β (u v)ρ dx dt (6.10) = ρ β [u(x, s) v(x, s)] 2 dx ρ β [u(x, τ) v(x, τ)] 2 dx. From (6.4) (6.10), it follows that ρ β [u(x, s) v(x, s)] 2 dx ρ β [u(x, τ) v(x, τ)] 2 dx s ( qdt c ρ β u(x, t) v(x, t) dx) 2 (6.11) τ ( s ) q, c ρ β u(x, t) v(x, t) 2 dx dt τ where q < 1. Let κ(s) = ρβ [u(x, s) v(x, s)] 2 dx. Then we deduce ( s κ(s) κ(τ) τ c κ(t)dt) q. s τ s τ By the L Hospital Rule, κ κ(s) (τ) c lim s τ ( s τ κ(t)dt) 1 q = c lim κ (s)( s κ(t)dt ) q = 0. (6.12) s τ κ(s) τ Thus, because τ is arbitrary, we have ρ β u(x, τ) v(x, τ) 2 dx ρ β u 0 v 0 2 dx. (6.13) The proof is complete. 7. Uniqueness in the case α p 1 When α p 1, let u be a weak solution of equation (1.11) with the initial value (1.9). Generally, we can not define the trace of u on the boundary. Proof. Let the constant β max{ p+ α p 1, 2}. Denote λ, λ = λ (0, T ) as (5.1)-(5.4), and let ξ λ = d β λ. Let u and v be two solutions of equation (1.11) with the initial values u 0 (x) and v 0 (x) respectively. We choose χ [τ,s] (u ε v ε )ξ λ as a test function, where u ε and v ε are the mollified function of the solutions u and v respectively. Then (u v) t, χ [τ,s] (u ε v ε )ξ λ = (u ε v ε )ξ λ dx dt = ρ α ( u p 2 u v p 2 v) [(u ε v ε )ξ λ ] dx dt. (7.1)

14 14 H. ZHAN, J. WEN EJDE-2017/13 Now, by the weak convergence of (5.4) and ρ α ( u 2 u v 2 v) L 1 (λ ) we obtain lim ρ α ξ λ ( u 2 u v 2 v) (u ε v ε ) dx dt Q τs = ρ α ξ λ ( u 2 u v 2 v) (u v) dx dt. By (5.4)-(5.5) and the Lebesgue controlled convergence theorem, we have lim ρ α ( u 2 u v 2 v)(u ε v ε ) ξ λ dx dt Q τs = ρ α ( u 2 u v 2 v)(u v) ξ λ dx dt. So lim ρ α ( u 2 u v 2 v) [(u ε v ε )ξ λ ] dx dt Q τs = ρ α ξ λ ( u 2 u v 2 v) (u v) dx dt Q τs + ρ α ( u 2 u v 2 v)(u v) ξ λ dx dt. At the same time, by Hölder s inequality, similar to (6.6)-(6.8), we have and (7.2) (7.3) (7.4) ρ α ξ λ ( u 2 u v 2 v) (u v) dx dt 0, (7.5) (u v)ρ α ( u 2 u v 2 v) ξ λ dx dt s ( 1/p1dt c ρ α d (β 1) λ u v dx) τ λ s ( 1/p1dt c ρ α+(β 1) u v dx) τ ( s q, c u v 2 dx dt) τ (7.6) where q < 1. From (5.12), since (u ε v ε )ξ λ L ( ), we can use the Lebesgue controlled convergence theorem to deduce that lim (u ε v ε )ξ λ dx dt = (u v)ξ λ dx dt. Now, after letting ε 0 and λ 0 in (7.1), by a similar argument for (6.10)-(6.13), we arrive at the desired result. Acknowledgments. This research is supported by the NSF of China and NSF of Fujian Province 2015J01592.

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