Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 74, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STABILITY OF ANISOTROPIC PARABOLIC EQUATIONS WITHOUT BOUNDARY CONDITIONS HUASHUI ZHAN, ZHAOSHENG FENG Abstract. In this article, we consider the equation ut = N∑ i=1 ( ai(x)|uxi | pi(x)−2uxi ) xi , with ai(x), pi(x) ∈ C1(Ω) and pi(x) > 1. Where ai(x) = 0 if x ∈ ∂Ω, and ai(x) > 0 if x ∈ Ω, without any boundary conditions. We propose an an- alytical method for studying the stability of weak solutions. We also study the uniqueness of a weak solution, and establish its stability under certain conditions. 1. Introduction In past decades, the so-called electrorheological fluid equation [1, 15]: ut = div ( a(x)|∇u|p(x)−2∇u ) , (x, t) ∈ QT , (1.1) has received a lot of attention from a rather diverse group of scientists such as physicists and mathematicians [3, 4, 6, 7, 11, 13, 16, 19]. In this work, we consider an anisotropic parabolic equation ut = N∑ i=1 ( ai(x)|uxi |pi(x)−2uxi ) xi , (x, t) ∈ QT , (1.2) with the initial condition u(x, 0) = u0(x), x ∈ Ω, (1.3) but without the boundary condition u(x, t) = 0, (x, t) ∈ ∂Ω× (0, T ), (1.4) where Ω ⊂ RN is a bounded domain with the smooth boundary ∂Ω, QT = Ω × (0, T ), and pi(x) is a C1(Ω) function with pi(x) > 1. Equation (1.2) arises in several scientific fields. For instance, in biology [6, 7] it is suggested as a model to describe the spread of an epidemic disease in heterogeneous environments. In fluid mechanics [2, 5], it is used as the mathematical description for the dynamics of fluids with different conductivities in different directions. For equation (1.1), considerable 2010 Mathematics Subject Classification. 35K15, 35B35, 35K55. Key words and phrases. Parabolic equation; boundary condition; stability; Hölder inequality. c©2020 Texas State University. Submitted December 9, 2019. Published July 15, 2020. 1 2 H. ZHAN, Z. FENG EJDE-2020/74 attention has been devoted to the existence and uniqueness of its solution. One can refer to [8, 9, 10, 12, 14, 17, 18] and the references therein. When a(x) ∈ C1(Ω), and a(x) > 0, x ∈ Ω and a(x) = 0, x ∈ ∂Ω, (1.5) the initial-boundary value problem of equation (1.1) was discussed by means of the parabolic regularized method [19]. In this study, we assume that ai(x) ∈ C1(Ω), and ai(x) > 0, x ∈ Ω and ai(x) = 0, x ∈ ∂Ω, i = 1, 2, . . . , N, (1.6) and denote p0 = min x∈Ω {p1(x), p2(x), . . . , pN−1(x), pN (x)}. Throughout this paper, we assume that p0 > 1. Before stating our main results, let us recall two definitions. Definition 1.1. If u(x, t) satisfies u ∈ L∞(QT ), ∂u ∂t ∈ L2(QT ), uxi ∈ L∞(0, T ;Lpi(x)(ai,Ω)), (1.7) and for ϕ1 ∈ C1 0 (QT ), ϕ2 ∈ L∞(0, T ;W 1,p0 loc (Ω)) and ϕ2xi ∈ L∞(0, T ;Lpi(x)(ai,Ω)), it holds ∫∫ QT [∂u ∂t (ϕ1ϕ2) + N∑ i=1 ai(x)|uxi |pi(x)−2uxi(ϕ1ϕ2)xi ] dx dt = 0, (1.8) then we call u(x, t) a weak solution of equation (1.2) with the initial condition (1.3) in the sense of lim t→0 ∫ Ω |u(x, t)− u0(x)| dx = 0. (1.9) Here, Lpi(x)(ai,Ω) is the weighted variable exponent Lebesgue space. One can refer to [11] for the definition of such a space and the corresponding Hölder inequal- ity. Recall that the characteristic function χ of Ω is defined by χ(x) = { 1 if x ∈ Ω, 0 if x ∈ RN \ Ω. Definition 1.2. A nonnegative continuous function χ is said to be a weak charac- teristic function of Ω, if χ(x) { > 0, x ∈ Ω, = 0, x ∈ ∂Ω. (1.10) Apparently, the weak characteristic function is not unique for a bounded domain Ω. For examples, the distance function d(x) = dist(x, ∂Ω) and the diffusion function ai(x) in (1.6) both are the weak characteristic functions. Based on Definition 1.2, we propose a new analytical method, currently called the weak characteristic function method, to study the stability of weak solutions to the nonlinear degenerate parabolic equations independent of the boundary condition. EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 3 Theorem 1.3. Let ai(x) ∈ C1(Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of equation (1.2) with the initial values u0(x) and v0(x) respectively. If for sufficiently large n, there are a weak characteristic function χ(x) of Ω and a constant c such that n (∫ Ω\Ωn ai(x)|χxi(x)|pi(x)dx )1/p+i ≤ c, (1.11) then ∫ Ω |u(x, t)− v(x, t)|dx ≤ c ∫ Ω |u0(x)− v0(x)|dx, (1.12) where p+ i = maxx∈Ω pi(x) and Ωn = {x ∈ Ω : χ(x) > 1/n}. Theorem 1.4. Let ai(x) ∈ C1(Ω) satisfy (1.6), and u(x, t) and v(x, t) be two weak solutions of (1.2) with the initial values u0(x) and v0(x) respectively, If there exists a weak characteristic function χ such that∫ Ω ai(x) ∣∣∣χxi(x) χ(x) ∣∣∣pi(x) dx <∞, (1.13) then the stability (1.12) is true. Theorem 1.5. Let ai(x) ∈ C1(Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of (1.2) with the different initial values u0(x) and v0(x) respectively, but without any boundary condition. If there exist a weak characteristic function χ(x) and a constant c such that ai(x)|χxi(x)|pi(x) χ(x) ≤ c, (1.14) then ∫ Ω χ(x)|u(x, t)− v(x, t)|2dx ≤ c ∫ Ω χ(x)|u0(x)− v0(x)|2dx. (1.15) If we choose χ(x) = min 1≤i≤N {ai(x)}, then (1.14) holds, and∫ Ω min 1≤i≤N {ai(x)}|u(x, t)− v(x, t)|2dx ≤ c ∫ Ω min 1≤i≤N {ai(x)}|u0(x)− v0(x)|2dx . This inequality implies that the uniqueness of weak solution is always true provided that ai(x) satisfies conditions (1.5) and (1.6). Note that by choosing various characteristic functions χ(x), one may obtain different results. For example, choosing χ(x) = N∏ i=1 ai(x), then we obtain χxi(x) = N∑ k=1 ( N∏ j=1,j 6=k aj(x) ) axi = N∏ j=1 aj(x) N∑ k=1 akxi ak and n (∫ Ω\Ωn ai(x)|χxi(x)|pi(x)dx )1/p+i 4 H. ZHAN, Z. FENG EJDE-2020/74 = n (∫ Ω\Ωn ai(x)χpi(x)(x) ∣∣∣ N∑ k=1 akxi ak ∣∣∣pi(x) dx )1/p+i ≤ n 1− p − i p + i (∫ Ω\Ωn ai(x) ∣∣∣ N∑ k=1 akxi ak ∣∣∣pi(x) dx )1/p+i . From Theorem 1.3 we obtain the following result. Corollary 1.6. Let ai(x) ∈ C1(Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of equation (1.2) with the initial values u0(x) and v0(x) respectively. If for the sufficiently large n, it holds n 1− p − i p + i (∫ Ω\Ωn ai(x) ∣∣∣ N∑ k=1 akxi ak ∣∣∣pi(x) dx )1/p+i ≤ c, (1.16) then the stability (1.12) is true. Similarly, since∫ Ω ai(x) ∣∣∣χxi(x) χ(x) ∣∣∣pi(x) dx = ∫ Ω ai(x) ∣∣∣ N∑ k=1 akxi ak ∣∣∣pi(x) dx, by Theorem 1.4, we have the following result. Corollary 1.7. Let ai(x) ∈ C1(Ω) satisfy (1.6), and u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u0(x) and v0(x) respectively, If there exists a characteristic function χ(x) such that∫ Ω ai(x) ∣∣∣ N∑ k=1 akxi ak ∣∣∣pi(x) dx <∞, (1.17) then the stability (1.12) is true. If ai(x) ≡ a(x), then condition (1.14) holds, i.e. equation (1.2) reduces to ut = N∑ i=1 ∂ ∂xi ( a(x)|uxi |p(x)−2uxi ) xi . (1.18) From Theorem 1.5, we have the following result. Corollary 1.8. Let a(x) ∈ C1(Ω) satisfy (1.5) and u(x, t) and v(x, t) be two solu- tions of equation (1.18) with the differential initial values u0(x) and v0(x) respec- tively. Then∫ Ω (a(x))N |u(x, t)− v(x, t)|2dx ≤ c ∫ Ω (a(x))N |u0(x)− v0(x)|2dx. If ai(x) ≡ a(x) and pi(x) ≡ p, then condition (1.16) is equivalent to condition (1.17), which is also equivalent to∫ Ω |axi |p ap−1 dx <∞. (1.19) In this case, equation (1.2) reduces to ut = N∑ i=1 ∂ ∂xi ( a(x)|uxi |pi−2uxi ) . (1.20) EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 5 If (1.19) is true, then the stability (1.12) is true without any boundary condition. As we can see, equation (1.20) is different from the evolutionary p-Laplacian equation: ut = div(a(x)|∇u|p−2∇u). (1.21) It is notable that if we choose appropriate weak characteristic functions, we can obtain nice results on the stability. One can see that the weak characteristic function method can also be generalized to study the stability of weak solutions to a more general degenerate parabolic equation as well as the evolutionary p-Laplacian equations. The remainder of this paper is structured as follows. In Sections 2-4, we prove Theorems 1.3-1.5 respectively, by means of the proposed weak characteristic func- tion method. In Section 5, we extend this method to study the stability of solutions of the evolutionary p-Laplacian equation (1.21). 2. Proof of Theorem 1.3 Following [13, 19], we denote the variable exponent Sobolev space by W 1,p(x)(Ω). To prove Theorem 1.4, we need the following technical lemma [13, 19]. Lemma 2.1. (i) The spaces ( Lp(x)(Ω), ‖ · ‖Lp(x)(Ω) ) , ( W 1,p(x)(Ω), ‖ · ‖W 1,p(x)(Ω) ) and W 1,p(x) 0 (Ω) are reflexive Banach spaces. (ii) (p(x)-Hölder’s inequality) Let q1(x) and q2(x) be real functions with 1 q1(x) + 1 q2(x) = 1 and q1(x) > 1. Then, the conjugate space of Lq1(x)(Ω) is Lq2(x)(Ω). For any u ∈ Lq1(x)(Ω) and v ∈ Lq2(x)(Ω), it holds∣∣ ∫ Ω uvdx ∣∣ ≤ 2‖u‖Lq1(x)(Ω)‖v‖Lq2(x)(Ω). (2.1) (iii) It holds If ‖u‖Lp(x)(Ω) = 1, then ∫ Ω |u|p(x)dx = 1. If ‖u‖Lp(x)(Ω) > 1, then |u|p − Lp(x)(Ω) ≤ ∫ Ω |u|p(x)dx ≤ |u|p + Lp(x)(Ω) . If ‖u‖Lp(x)(Ω) < 1, then |u|p + Lp(x)(Ω) ≤ ∫ Ω |u|p(x)dx ≤ |u|p − Lp(x)(Ω) . (iv) If p1(x) ≤ p2(x), then Lp1(x)(Ω) ⊃ Lp2(x)(Ω). (v) If p1(x) ≤ p2(x), then W 1,p2(x)(Ω) ↪→W 1,p1(x)(Ω). (vi) (p(x)-Poincaré inequality) If p(x) ∈ C(Ω), then there is a constant C > 0, such that ‖u‖Lp(x)(Ω) ≤ C‖∇u‖Lp(x)(Ω), ∀u ∈W 1,p(x) 0 (Ω). This implies ‖∇u‖Lp(x)(Ω) and ‖u‖W 1,p(x)(Ω) being equivalent to the norms of W 1,p(x) 0 (Ω). For n > 0, let gn(s) = ∫ s 0 hn(τ)dτ, hn(s) = 2n(1− |ns|)+. 6 H. ZHAN, Z. FENG EJDE-2020/74 Obviously, hn(s) ∈ C(R), and hn(s) ≥ 0, |shn(s)| ≤ 1, |gn(s)| ≤ 1, lim η→0 gn(s) = sgn s, lim η→0 sg′n(s) = 0. (2.2) Let u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u0(x) and v0(x) respectively, but without any boundary condition. Let χ(x) be a weak characteristic function of Ω. We define φn(x) = { 1, if x ∈ Ωn, nχ(x), if x ∈ Ω \ Ωn, (2.3) where Ωn = {x ∈ Ω : χ(x) > 1 n}. By a process of limit, we choose ϕ1 = χ[τ,s]φn, ϕ2 = gn(u− v), and take χ[τ,s]φngn(u − v) as the test function. Here, χ[τ,s] is the characteristic function of [τ, s) ⊆ [0, T ). Then we have∫ s τ ∫ Ω φngn(u− v) ∂(u− v) ∂t dx dt+ N∑ i=1 ∫ s τ ∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (uxi − vxi)g′n(u− v)φn(x) dx dt + N∑ i=1 ∫ s τ ∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) × (uxi − vxi)gn(u− v)φnxi dx dt = 0. (2.4) In the third term of the left-hand side of (2.4), we note that∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (uxi − vxi)g′n(u− v)φn(x)dx ≥ 0. (2.5) For the first term of the left hand side of (2.4), in view of ut ∈ L2(QT ), it follows the Lebesgue dominated convergence theorem that lim n→∞ ∫ s τ ∫ Ω φn(x)gn(u− v) ∂(u− v) ∂t dx dt = ∫ Ω |u− v|(x, s)dx− ∫ Ω |u− v|(x, τ)dx. (2.6) Since φnxi = nχxi when x ∈ Ω \ Ωn, by (iii) of Lemma 2.1 we deduce that∣∣∣ ∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) φnxign(u− v)dx ∣∣∣ = ∣∣∣ ∫ Ω\Ωn ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) φnxign(u− v)dx ∣∣∣ ≤ n ∫ Ω\Ωn ai(x) ( |uxi | pi(x)−1 + |vxi |pi(x)−1 ) χxign(u− v)dx ≤ cn (∫ Ω\Ωn ai(x) ( |uxi |pi(x) + |vxi |pi(x) ) dx )1/q+i (∫ Ω\Ωn ai(x)|χxi |pi(x)dx )1/p+i ≤ c [(∫ Ω\Ωn ai(x)|uxi |pi(x)dx )1/q+i + (∫ Ω\Ωn ai(x)|vxi |pi(x)dx )1/q+i ] EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 7 × [ n (∫ Ω\Ωn ai(x)|χxi |pi(x)dx )1/p+i ] ≤ c (∫ Ω\Ωn ai(x)|uxi |pi(x)dx )1/q+i + c (∫ Ω\Ωn ai(x)|vxi |pi(x)dx )1/q+i , where qi(x) = pi(x) pi(x)−1 and q+ i = maxx∈Ω qi(x). Therefore, lim n→∞ ∣∣∣ ∫ s τ ∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) φnxign(u− v) dx dt ∣∣∣ ≤ c lim n→∞ [( ∫ Ω\Ωn ai(x)|uxi |pi(x)dx )1/q+i + (∫ Ω\Ωn ai(x)|vxi |pi(x)dx )1/q+i ] = 0. (2.7) Let η → 0 in (2.4). Then we have∫ Ω |u(x, s)− v(x, s)|dx ≤ ∫ Ω |u(x, τ)− v(x, τ)|dx, (2.8) Because of the arbitrariness of τ , we obtain∫ Ω |u(x, s)− v(x, s)| dx ≤ c ∫ Ω |u0(x)− v0(x)| dx. 3. Proof of Theorem 1.4 Making a minor modification, we can generalize Definition 1.1 to the following version. Definition 3.1. Suppose that u(x, t) satisfies (1.7). If for any function g(s) ∈ C1(R) with g(0) = 0, ϕ1 ∈ C1 0 (Ω) and ϕ2xi ∈ L2(0, T ;Lpi(x)(ai,Ω)) it holds∫∫ QT [∂u ∂t g(ϕ1ϕ2) + N∑ i=1 ai(x)|uxi |pi(x)−2uxigxi(ϕ1ϕ2) ] dx dt = 0, (3.1) and the initial value condition (1.3) is satisfied in the sense of (1.9), then u(x, t) is said to be a weak solution of equation (1.2) with initial condition (1.3). Let u(x, t) and v(x, t) be two weak solutions of (1.2) with the initial values u0(x) and v0(x) respectively, and χ be a weak characteristic function. We choose gn(χ(u− v)) as the test function in Definition 3.1. Then we have∫ Ω gn(χ(u− v)) ∂(u− v) ∂t dx + N∑ i=1 ∫ Ω χ(x)ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (u− v)xig ′ n(χ(u− v))dx + N∑ i=1 ∫ Ω ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxi(u− v)g′n(χ(u− v))dx = 0. (3.2) 8 H. ZHAN, Z. FENG EJDE-2020/74 Let us evaluate each term in the left hand side of (3.2). For the first two terms, we find that lim n→∞ ∫ Ω gn(χ(u− v)) ∂(u− v) ∂t dx = d dt ∫ Ω |u(x, t)− v(x, t)|dx, (3.3)∫ Ω χ(x)ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (u− v)xig ′ n(χ(u− v))dx ≥ 0, and ∣∣∣ ∫ Ω ai(x)(u− v)g′n(χ(u− v)) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxidx ∣∣∣ = ∣∣∣ ∫ {Ω:χ|u−v|<1/n} a − pi(x)−1 pi(x) i ai(x)(u− v)g′n(χ(u− v))a pi(x)−1 pi(x) i × ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxidx ∣∣∣ ≤ (∫ {Ω:χ|u−v|< 1 n} |a 1 pi(x) i (u− v)g′n(χ(u− v))χxi |pi(x)dx )1/pi1 × (∫ {Ω:χ|u−v|<1/n} ai(x) ( |uxi |pi(x) + |vxi |pi(x) ) dx )1/qi1 , (3.4) where pi1 = p+ i or p−i based on (iii) of Lemma 2.1, and similar for qi1. If {x ∈ Ω : |u− v| = 0} has zero measure, since∫ Ω ai(x) ∣∣χxi χ ∣∣pi(x) dx <∞, we derive that∫ {Ω:χ|u−v|<1/n} ∣∣a 1 pi(x) i χxi χ χ(u− v)g′n(χ(u− v)) ∣∣pi(x) dx ≤ c, (3.5) and lim n→∞ (∫ {Ω:χ|u−v|< 1 n} ai(x) ( |uxi |pi(x) + |vxi |pi(x) ) dx )1/qi1 = (∫ {Ω:|u−v|=0} ai(x) ( |uxi |pi(x) + |v|pi(x) ) dx )1/qi1 = 0. (3.6) If {x ∈ Ω : u− v = 0} has a positive measure, then lim n→∞ (∫ {Ω:χ|u−v|< 1 n} ∣∣a 1 pi(x) i χxi χ (u− v)g′n(χ(u− v)) ∣∣pi(x) dx )1/pi1 = (∫ {Ω:|u−v|=0} ai(x) ∣∣χxi χ ∣∣pi(x) lim n→∞ |(u− v)g′n((u− v)χ)|pi(x)dx )1/pi1 = 0. (3.7) In view of (2.2) and condition (1.13), it follows the Lebesgue dominated conver- gence theorem that lim n→∞ ∣∣∣ ∫ Ω ai(x)(u− v)g′n(χ(u− v)) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxidx ∣∣∣ = 0. We now letting η → 0 in (3.2), we have d dt ∫ Ω |u(x, t)− v(x, t)dx ≤ ∫ Ω |u(x, t)− v(x, t)dx. EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 9 By Gronwall’s inequality, we obtain∫ Ω |u(x, t)− v(x, t)| dx ≤ c ∫ Ω |u0(x)− v0(x)| dx, ∀t ∈ [0, T ). 4. Proof of Theorem 1.5 Let u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u0(x) and v0(x) respectively. Then we have∫∫ QT [(∂u ∂t − ∂v ∂t ) ϕ+ N∑ i=1 ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) ϕxi ] dx dt = 0. (4.1) Let ϕ = χ[τ,s](u− v)χ(x), where χ[τ,s] is the characteristic function on [τ, s] and χ(x) is a weak characteristic function of Ω. Denote Qτs = Ω× [τ, s]. Then we have∫∫ Qτs ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) [(u− v)χ]xi dx dt = ∫∫ Qτs ai(x)χ(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (u− v)xi dx dt + ∫∫ Qτs ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (u− v)χxi dx dt. (4.2) Clearly, it has∫∫ Qτs ai(x)χ(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) (u− v)xi dx dt ≥ 0. (4.3) Evaluating the second term on the right hand side of (4.2) yields∣∣∣ ∫∫ Qτs (u− v)ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxi dx dt ∣∣∣ ≤ ∫∫ Qτs |u− v|ai(x) ( |uxi |pi(x)−1 + |vxi |pi(x)−1 ) |χxi | dx dt ≤ c (∫ s τ ∫ Ω ai(x) ( |uxi |pi(x) + |v|pi(x) ) dx dt )1/qi1 × (∫ s τ ∫ Ω ai(x)|χxi |pi(x)|u− v|pi(x) dx dt )1/pi1 ≤ c (∫ s τ ∫ Ω ai(x)|χxi |pi(x)|u− v|pi(x) dx dt )1/pi1 . (4.4) Since ai(x)|χxi | pi(x) χ ≤ c, by (4.4) we have∣∣ ∫∫ Qτs (u− v)ai(x) ( |uxi |pi(x)−2uxi − |vxi |pi(x)−2vxi ) χxi dx dt ∣∣ ≤ c (∫ s τ ∫ Ω χ|u− v|pi(x) dx dt )1/pi1 . (4.5) 10 H. ZHAN, Z. FENG EJDE-2020/74 If pi(x) ≥ 2, then(∫ s τ ∫ Ω χ(x)|u− v|pi(x) dx dt )1/pi1 ≤ c (∫ s τ ∫ Ω χ(x)|u− v|2 dx dt )1/pi1 . If 1 < pi(x) < 2, by the Hölder inequality we have∫ s τ ∫ Ω χ(x)|u− v|pi(x) dx dt ≤ c (∫ s τ ∫ Ω χ(x)|u− v|2 dx dt ) 1 pi2 , where pi2 is maxx∈Ω 2 pi(x) or minx∈Ω 2 pi(x) , depending on ∫ s τ ∫ Ω χ|u−v|pi(x) dx dt ≥ 1 or ∫ s τ ∫ Ω χ|u− v|pi(x) dx dt < 1. Thus, we obtain(∫ s τ ∫ Ω χ(x)|u− v|pi(x) dx dt )1/pi1 ≤ c (∫ s τ ∫ Ω χ(x)|u− v|2 dx dt ) 1 pi1 1 pi2 (4.6) and ∫∫ Qτs (u− v)χ(x) ∂(u− v) ∂t dx dt = ∫ Ω χ(x)[u(x, s)− v(x, s)]2dx− ∫ Ω χ(x)[u(x, τ)− v(x, τ)]2dx. (4.7) In view of (4.2)-(4.7), letting λ→ 0 in (4.1) leads to∫ Ω χ(x)[u(x, s)− v(x, s)]2dx− ∫ Ω χ(x)[u(x, τ)− v(x, τ)]2dx ≤ c (∫ s 0 ∫ Ω χ(x)|u(x, t)− v(x, t)|2 dx dt )q , (4.8) where q < 1. By (4.8), it is easy to see that∫ Ω χ(x)|u(x, s)− v(x, s)|2dx ≤ ∫ Ω χ(x)|u(x, τ)− v(x, τ)|2dx. (4.9) Due to the arbitrariness of τ , we obtain∫ Ω χ(x)|u(x, s)− v(x, s)|2dx ≤ ∫ Ω χ(x)|u0(x)− v0(x)|2dx. 5. Stability of p-Laplacian equation In the preceding two sections, we use the weak characteristic function method to prove Theorems 1.3-1.5. In this section, we consider equation (1.21) with the initial value condition (1.3), but without any boundary condition. We apply the proposed weak characteristic function method to prove the stability of solutions of equation (1.21). Proposition 5.1. Let a(x) ∈ C1(Ω) satisfy (1.5), and u(x, t) and v(x, t) be two weak solutions of equation (1.21) with the initial values u0(x) and v0(x) respectively. When p > 1, for the sufficiently large n, it holds n1− (N−1)p+1 Np (∫ Ω\Ωn |∇a|pdx )1/p ≤ c, (5.1) where c is a constant. Then the stability (1.12) is true. EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 11 Proof. Let χ(x) = [a(x)]N . We can choose φngn(u− v) as the test function, then∫ Ω φn(x)gn(u− v) ∂(u− v) ∂t dx + ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇(u− v)g′n(u− v)φn(x)dx + ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇(u− v)gn(u− v)∇φndx = 0. (5.2) Clearly, we see that∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇(u− v)g′n(u− v)φn(x)dx ≥ 0. (5.3) By a straightforward computations, we derive that∣∣∣ ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇φngn(u− v)dx ∣∣∣ = ∣∣∣ ∫ Ω\Ωn a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇φngn(u− v)dx ∣∣∣ = ∣∣∣ ∫ Ω\Ωn a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ngn(u− v)[a(x)]N−1∇adx ∣∣∣ ≤ c (∫ Ω\Ωn a(x) ( |∇u|p + |∇u|p )) p−1 p n (∫ Ω\Ωn a(x)[aN−1|∇a|]pdx )1/p ≤ c (∫ Ω\Ωn a(x) ( |∇u|p + |∇u|p )) p−1 p n1− (N−1)p+1 Np (∫ Ω\Ωn |∇a|pdx )1/p ≤ c (∫ Ω\Ωn a(x) ( |∇u|p + |∇u|p )) p−1 p n1− (N−1)p+1 Np (∫ Ω\Ωn |∇a|pdx )1/p , (5.4) which approaches 0 as n→∞. Hence, by (5.2)-(5.4), the desired result is obtained. � If a(x) = dα(x), then n1− (N−1)p+1 Np (∫ Ω\Ωn |∇a|pdx )1/p ≤ cn1− (N−1)p+1 Np − 1+p(α−1) Nα . (5.5) Let α→∞. It is easy to see that lim α→∞ ( 1− (N − 1)p+ 1 Np − 1 + p(α− 1) Nα ) = p− 1− p2 Np < 0. So we can choose an α such that lim n→∞ n1− (N−1)p+1 Np − 1+p(α−1) Nα = 0. (5.6) Proposition 5.2. Let a(x) ∈ C1(Ω) satisfy (1.5), and u(x, t) and v(x, t) be two weak solutions of the equation ut = div(dα|∇u|p−2∇u) (5.7) with the initial values u0(x) and v0(x) respectively. If p > 1, for the sufficiently large α, then the stability (1.12) is true. 12 H. ZHAN, Z. FENG EJDE-2020/74 Next, we give further discussions on the constant α in Proposition 5.2. Proposition 5.3. Let a(x) ∈ C1(Ω) satisfy (1.5), and u(x, t) and v(x, t) be two solutions of equation (5.7) with the initial values u0(x) and v0(x) respectively. When p > 1, we have ∫ Ω |∇a|p ap−1 dx ≤ c, (5.8) where c is a constant. Then the stability (1.12) is true. Proof. Let χ(x) = [a(x)]N . We can choose gn(χ(u − v)) = gn(aN (u − v)) as the test function. Then∫ Ω gn(aN (u− v)) ∂(u− v) ∂t dx + ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · aN∇(u− v)g′n(aN (u− v))φn(x)dx + ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇aN (u− v)g′n(aN (u− v))dx = 0. (5.9) Clearly,∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇(u− v)g′n(aN (u− v))aNdx ≥ 0. (5.10) By a direct calculation, we deduce that∣∣ ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇aN (u− v)g′n(aN (u− v))dx ∣∣ = ∣∣ ∫ {Ω:aN |u−v|<1/n} a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇aN (u− v)g′n(aN (u− v))dx ∣∣ = N ∣∣∣ ∫ {Ω:aN |u−v|<1/n} a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) · ∇a a aN (u− v)g′n(aN (u− v))dx ∣∣∣ ≤ c (∫ {Ω:aN |u−v|< 1 n} a(x) (|∇u|p + |∇u|p) ) p−1 p · (∫ {Ω:aN |u−v|< 1 n} a(x) ∣∣∣∣∇aa ∣∣∣∣p aN |(u− v)g′n(aN (u− v))|pdx )1/p . (5.11) As for (3.5)-(3.7), we can derive that lim n→∞ ∣∣∣ ∫ Ω a(x) ( |∇u|p−2∇u− |∇v|p−2∇v ) ·∇aN (u−v)g′n(aN (u−v))dx ∣∣∣ = 0. (5.12) Consequently, using (5.9)-(5.12), we arrive at the desire result. � If a(x) = dα(x), then |∇a|p ap−1 = αpd(α−1)p dα(p−1) = αpdα−p. (5.13) EJDE-2020/74 ANISOTROPIC PARABOLIC EQUATIONS 13 Therefore, we can obtain the following proposition which is identical to the corre- sponding result of [18]. Proposition 5.4. 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Zhen; Orlicz estimates for general parabolic obstacle problems with p(t, x)- growth in Reifenberg domains, Electron. J. Differential Equations,2020 No. 13 (2020), 1-25. [18] J. Yin, C. Wang; Properties of the boundary flux of a singular diffusion process, Chin. Ann. Math., Ser. B, 25 (2004), 175-182. [19] H. Zhan, J. Wen; Evolutionary p(x)-Laplacian equation free from the limitation of the bound- ary value, Electron. J. Differential Equations, 2016 No. 143 (2016), 1-13. Huashui Zhan School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China Email address: 2012111007@xmut.edu.cn 14 H. ZHAN, Z. FENG EJDE-2020/74 Zhaosheng Feng School of Mathematical and Statistical Sciences, University of Texas Rio Grande Val- ley, Edinburg, TX 78539, USA Email address: zhaosheng.feng@utrgv.edu 1. Introduction 2. Proof of Theorem ?? 3. Proof of Theorem ?? 4. Proof of Theorem ?? 5. Stability of p-Laplacian equation Acknowledgments References