Uniqueness and existence of solutions in the BVt(Q) space to a doubly nonlinear parabolic problem

Publicacions Matem`atiques, Vol 40 (1996), 527–560. UNIQUENESS AND EXISTENCE OF SOLUTIONS IN THE BVt(Q)SPACE TO A DOUBLY NONLINEAR PARABOLIC PROBLEM J. I. D´ ıaz and J. F. Padial Abstract In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equation b(u)t−div(|∇u−k(b(u))e|p−2(∇u−k(b(u))e))+g(x, u)=f(t, x). This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the BVt(Q) space. We prove some comparison properties (implying uniqueness) when the set of jumping points of the BV solution has N-dimensional null measure and suitable additional conditions as, for instance, b−1locally Lipschitz. The existence of this type of weak solution is based on suitable uniform estimates of the BV norm of an approximated solution. 1. Introduction Let be Ω a bounded open subset of RNwith regular boundary and T>0. We consider the following Cauchy-Dirichlet problem b(u)t−div φ(∇u−k(b(u))e)+g(x, u)=f(t, x)inQ,(1.1) u(t, x)=0onΣ,(1.2) u(0,x)=u0(x)inΩ,(1.3) Both authors are partially supported by the DGICYT (Spain) Project No. PB 93/0443. 528 J. I. D´ ıaz, J. F. Padial where Q:=]0,T[×Ω and Σ :=]0,T[×∂Ω. We also assume some structure conditions such as the ellipticity of the diffusion operator (which is implied by the monotonicity of the power vectorial function (1.4) φ(ξ)=|ξ|p−2ξ∀ξ∈RN with p>1) and the monotonicity of the real continuous function b.In fact, we shall assume that (1.5) bis continuous strictly increasing and b(0) = 0. The continuous functions kand g(x, ·) are assumed satisfying some additional assumptions (see (2.1)-(2.5)). In (1.1) edenotes a given unit vector in RN. Problem (1.1), or some special cases of it, arises in many different physical contexts. For instance, the flow of a gas through a porous medium in a turbulent regime is described by the continuity equation θt+ div v=0 and the nonlinear Darcy’s law |v|q−2v=−K(θ)∇Φ(θ),for some q>2, where θ(t, x) is the volumetric moisture content, vthe velocity, K(θ) the hydraulic conductivity and Φ the total potential (usually given by Φ(θ)= ψ(θ)+zwith ψ(θ) the hydrostatic potential and zthe gravitational potential; see Volker [V69]). If edenotes the unit vector in the vertical direction, we obtain θt−div(|∇ϕ(θ)−K(θ)e|p−2(∇ϕ(θ)−K(θ)e)) = 0 with ϕ(θ)=θ 0 K(s)Φ(s)ds, and p=q q−1 (notice that now 1 <p<2). Functions ϕand kare given by physical experiments. Usually they are nondecreasing functions. For unsaturated media ϕis strictly increasing. In that case, the function u=ϕ(θ) verifies the equation (1.1) with b=ϕ−1,K=kand g=f=0. Another typical example is given by a turbulent gas flowing in a pipeline. If we assume that the gas is perfect and the pipe has uniform cross sectional area, we arrive to the system ρt+(ρv)x=0 ρvt+ρvvx+Px=−λ 2ρ|v|v Uniqueness and existence to a nonlinear equation 529 where ρ,Pand vare the density, pressure, and velocity of the gas which are unknown functions depending on the scalar x(the distance along the pipe) and time t. Using asymptotic methods, it was shown in D´ıaz-Li˜n´an [DLi89] that for large values of time the term ρvt+ρvvxcan be neglected and so the second equation may be replaced by Px=−λ 2ρ|v|v. Then, if we define u=|P|P,usatisfies equation (1.1) with b(u)=u1/2sign(u), k=g=f= 0 and p=3/2. We notice that in this case b−1is locally Lipschitz. Previous results on the mathematical treatment of problem (1.1), (1.2) and (1.3) can be found in the references of the paper D´ıaz-de Thelin [DT94] where the authors pay an special attention to the stabilization question. The main goal of this paper is to improve the uniqueness results of D´ıaz-de Thelin [DT94] where the weak solutions are assumed such that b(u)t∈L1(Q). Based in the works [Vo67], [VoHu69] and [J92], we shall prove in this paper a comparison result in a class of weak solutions such that b(u)t is a bounded Radon measure on Q(i.e. b(u)t∈M b(Q)). A preliminar version of our comparison result was shortly presented in [DPa93]. The present version also contains an enlarged presentation of Chapter 2 of the Ph. D. of the second author ([Pa95]). In [GM92a], [GM92b], and [BeGa95] the authors prove some comparison results in the class of weak solutions such that b(u)∈BV (0,T;L1(Ω)) for some related nonlinear parabolic problems but always assuming p= 2. Recently, using some techniques raised by S. N. Kruzhkov for hyperbolic equations and inspired in Carrillo [C86], Gagneux and Madaune-Tort proved in [GM94] and [GM95] a uniqueness result for case p= 2. Some more general results for the case p= 2 avoiding the assumption b(u)t∈L1(Q) can be found in [P95], [PG96] and [U96] where the authors use that any weak solution satisfies the equation in a “renormalized way”. In Section 2, we introduce the assumptions on the data and the notion of bounded BV solution. Section 3 is devoted to recall several properties on bounded variation functions which will be important for the study of the uniqueness of BV solutions presented later in Section 4. Our main result, a comparison criterium depending on the initial data and the forcing terms, assume a condition on the Hausdorff measure of the set of points where the solutions are not approximately continuous. Finally, the existence of a BV solution is established in Section 5 under some extra information on u0and f. 2. Definition of BV solutions Given a general Banach space B, its dual topological space will be 530 J. I. D´ ıaz, J. F. Padial denoted by B.By·,·B,Bwe denote the duality between Band B.We shall use the Sobolev space W1,p 0(Ω) and its dual W−1,p(Ω) where p>1 and 1 p+1 p= 1. Introducing the space X=Lp(0,T;W−1,p(Ω))+L1(Q) then X=Lp(0,T;W1,p 0(Ω)) ∩L∞(Q) and the duality ·,·X,Xis given by h, vX,X=T 0h−1,vW−1,p(Ω),W 1,p 0(Ω) +Ω h1vdx dt for all h∈X,v∈X, where h=h−1+h1with h1∈L1(Q) and h−1∈Lp(0,T;W−1,p(Ω)). We define the space of bounded variation (with respect to variable t) functions by BVt(Q)=u∈L1(Q):∂u ∂t ∈M b(Q), where Mb(Q) denote the space of bounded Radon measures over Q. In what follows we shall assume a series of conditions on the data:      k:R→R,is continuous, k(b(·)) is H¨older continuous with exponent γ, |k(b(s1)) −k(b(s2))|≤ ˆ C|s1−s2|γ ∀s1,s 2∈R,and γ≥1 pif 1 <p≤2,γ≥1 2if p>2; (2.1)      g:Ω×R→Ris a Caratheodory function such that |g(x, s)|≤ω(|s|)(1 + d(x)) for some d∈L1(Ω) and some increasing continuous function ω; (2.2) g(x, s1)−g(x, s2)≥−C∗(b(s1)−b(s2)) ∀s1,s 2∈R, s1>s 2and for some C∗>0; (2.3) f∈L1(Q)(2.4) and finally (2.5) u0∈L∞(Ω). We start by introducing the notion of weak solution of problem (1.1), (1.2) and (1.3) inspired in [AL83] and [DT94]: Uniqueness and existence to a nonlinear equation 531 Definition 1. We shall say that a function udefined on Qis a weak solution of problem (1.1), (1.2) and (1.3), if u∈Lp(0,T;W1,p 0(Ω)) ∩L∞(Q)(2.6) b(u)t∈Lp(0,T;W−1,p(Ω)) + L1(Q)(2.7) and moreover usatisfies the equality (2.8) b(u)t,vX,X+T 0Ω (b(u)−b(u0))vt=0 for all v∈Lp(0,T;W1,p(Ω)) ∩L∞(Q)∩W1,1(0,T;W1,p(Ω)), with v(T,·) = 0 and (2.9) b(u)t,vX,X+T 0Ω φ(∇u−k(b(u))e)·∇v +T 0Ω g(·,u)v=T 0Ω fv for all v∈Lp(0,T;W1,p(Ω)) ∩L∞(Q). By assuming more regularity on b(u)t, we arrive to the following notion: Definition 2. Let ube a weak solution of problem (1.1), (1.2) and (1.3). We shall say that uis a BV solution of (1.1), (1.2) and (1.3) if in addition (2.10) b(u)∈BVt(Q). Notice that in that case b(u)tis a bounded Radon measure on Qand so, the duality between the spaces Xand Xcan be also represented by the correspondent integral with respect to the measure b(u)tfor all measurable Borel function v∈X, i.e. (2.11) b(u)t,vX,X≡Q vb(u)t. In what follows, we shall adopt the integral representation of this duality if the test function is a measurable Borel function. 532 J. I. D´ ıaz, J. F. Padial 3. Treatment of discontinuous functions. Some technical lemmas. In the development of next sections we shall need some properties of functions whose first generalized derivatives are bounded regular (signed) measures. The following notions and properties can be found in [Vo67], [VoHu85], [F69] and [EvG92]. Here, and in what follows, Ld(E) will denote the d-dimensional Lebesgue measure of a set E⊂Rdand Hd(E) its d-dimensional Hausdorff measure (d≥1). Let E,Fbe two Lebesgue measurable subsets of Rd. A point x0∈Rdis a point of F-density of the set Eif lim r→0 Ld(Br(x0)∩E∩F) Ld(Br(x0)∩E)=1 where Br(x0)={x∈Rd:|x−x0|<r}. If the above limit it is equal to 0, the point x0is a point of F-rarefaction of the set E. Taking Fas Rd, we denote by E∗the set of points of density of Eand E∗the set of points of rarefaction of E. Finally, the set ∂E =E∗\E∗is called the essential boundary of the set E(in many cases, the essential boundary of a set Ecoincides with the boundary of E, however the boundary and the essential boundary of a set do not always coincide —for example the boundary and essential boundary of a disk minus a radius are not the same [VoHu85]). Let now f:Rd→Rbe a Lebesgue measurable function. The real number ,is called an approximate limit with respect to the set E⊂Rdof the function fas x→x0if for all ε>0 the point x0isapointofE-density of the set {x∈Rd:|f(x)−,|<ε}. We use the notation lim x→x0,x∈Ef(x)=,. Function fis approximately continuous at x0if lim x→x0,x∈Rdf(x)=f(x0). A point x0is called a regular point of the function fif there exists an unit vector υsuch that the approximate limits fυ(x0) and f−υ(x0) exist, where we have denoted fυ(x0) := lim x→x0,Πυ(x0)f(x) with Πυ(x0)={x∈Rd:x−x0,υ>0}. We choose υsuch that fυ(x0)≥f−υ(x0). Such a vector υis called a defining vector. Vol’pert proved in [Vo67] that if x0is a regular point for f(x) and if υis the defining vector for which fυ(x0)=f−υ(x0), then the associate approximate limit of fin x0exists and for any ω∈Rd fω(x0) also exists and it is equal to f(x0). A point verifying this, is called a point of approximate continuity. When fυ(x0)=f−υ(x0) the vector υis uniquely determined (except for the sign of fυ(x0)). A point x0verifying this inequality is called a jump point of fin the direction υ. The set of jump points of a function fis denoted by Γf. From Theorem 9.2 of [Vo67] follows that if f∈BV (G), G⊂Rd, then any point of the Gis either a point of approximate continuity or a jump point Uniqueness and existence to a nonlinear equation 533 of fwith exception of a set Hd−1-dimensional measure zero. For this class of functions, the inward and outward traces of the function fexist on ΓfHd−1-almost everywhere (see e.g. [VoHu85]). Moreover these traces coincide with the approximate limits fυand f−υrespectively and the defining vector υis the outward normal at the point x0of Γf. To extend the differentiation formulas and Green’s formula to the class of BV functions it is necessary to define a certain borelian function ¯ f Hd−1-almost everywhere equal to a given function f. This borelian representant is the so called symmetric mean value of f. Let us indicate its connection with the inward and outward traces of f, and consequently with the approximate limits fυand f−υ. We define ¯ fby the limit (when it exists) ¯ f(x0) = lim ε→0ηε∗f(x0)(x0∈G) where the sequence {ηε}εcorresponds to an averaging kernel (see [VoHu85, Ch. 4, Section 5, Section 6, p. 181]). It can be shown that if x0is a regular point of this function, the above limit exists and does not depends on the averaging kernel. Besides at this point the equality ¯ f(x0)=1 2[fυ(x0)+f−υ(x0)] holds, where υis a defining vector. In particular, if αis a real continuous function, we can define the functional superposition by means of ¯α(f(x)) = 1 0 α(fυ(x))s+α(f−υ(x))(1 −s)ds. Remark 1. An important property is that ¯α(f(x0)) = α(f(x0)) for any x0point of approximate continuity of f. Since any summable function fis approximately continuous Ld-almost everywhere, then the above equality holds Ld-almost everywhere. So, if fis Hd−1-approximately continuous function, then ¯α(f)=α(f)Hd−1-almost every where in G. However, d-dimensional measure is too large when we try to apply differentiation formulas to functions with measures as generalized derivatives. The generalizes of the classical formulas of differentiation by using the symmetric mean value to functions with measures as generalized derivatives are shown in [VoHu85, Chapter 5, Section 1]. By applying these notions to the case of G=Qand d=N+1,we can obtain the following lemma which gives an important property of the functions whose generalized derivatives are summable functions. 534 J. I. D´ ıaz, J. F. Padial Lemma 1. Let u∈W1,1(Q). Then, uis HN-almost everywhere approximately continuous on Q, i.e. HN(Γu)=0. Proof: If u∈W1,1(Q) it is clear that u∈BV (Q). Then there exists Λ⊂Q, with HN(Λ) = 0 and such that Q−Λ={(t, x)∈Q: regular points}. So, for any (t, x)inΓ u−Λ, there exists an unique vector υ=(υt,υx) (depending of the point (t, x) and where υt∈Rand υx∈ RN) which is the inward normal and there exist the approximated limits uυ(t, x) and u−υ(t, x) (see [VoHu85]). Let Sbe a Borel subset of Γu−Λ. Since u∈W1,1(Q), one has that LN+1(S) = 0 (see [Vo67], [VoHu69]). From that, and as u∈W1,1(Q), it follows that S ∂u ∂xi =Q χS ∂u ∂xi dx dt=0,(i=1,2,3,... ,N) S ∂u ∂t =Q χS ∂u ∂t dx dt =0. (χSis the characteristic function of the subset S). Applying now Theorem 2, p. 203 of [VoHu85] we get 0=S (uυ−u−υ)υxidHN, 0=S (uυ−u−υ)υtdHN(i=1,2,3,... ,N). The above equality implies that HN(S) = 0. Finally, as Sis arbitrary, we conclude that HN(Γu)=0. The main result of the general theory of BV functions that we shall use later in order to prove our uniqueness theorem is given in the following lemma: Lemma 2. Let u∈BVt(Q). Then, the measure utis HN-absolutely continuous. Remark 2. If we assume that u∈BVt(Q) and v∈L∞(Q)HN- approximately continuous function on Q, then by the above lemma and Remark 1 we have that ¯v=v∂u ∂t -almost every where on Q. Proof of Lemma 2: Let Abe a borelian subset of Q. Let AΩbe the projection of Aover the hyperplane {t=0}.IfHN(A) = 0 then LN(AΩ) = 0 (see Vol’pert [Vo67]). For a fixed x∈Ω, we denote by ess VT 0(u(x)) the essential variation of u(x)(t):=u(t, x) as the function of Uniqueness and existence to a nonlinear equation 535 t∈(0,T) given by ≡sup {|u(ti,x)−u(ti−1,x)|}. Since u∈BVt(Q), ess VT 0(u(x)) is defined almost everywhere x∈Ω with respect to the Lebesgue measure and LNsummable in AΩ(see [Vo67] and [EvG92]). Moreover  ∂u ∂t  (]0,T[×AΩ)=ΩA ess VT 0(u(x))dx =0 since LN(AΩ) = 0 and so the statement of the lemma holds. Lemma 3. Assume u∈L∞(Q)and bas in (1.5). If in addition we assume that (3.1) b(u)∈BVt(Q), and (3.2) b−1locally Lipschitz on [−uL∞(Q),uL∞(Q)] then u∈BVt(Q). Proof: To prove this property, we show that utis a bounded Radon measure on Q. Following Vol’pert and Hudajaev [VoHu85, Chapter 4, Section 2] it is enough to prove that there exists a positive constant Ksuch that |ut,ϕ| ≤ KϕL∞(Q)for all ϕ∈C 1 c(Q). In order to do that, we use the fact that ut= lim h→0 u(t+h, x)−u(t, x) hin sense of distributions. From the assumptions (3.1) and (3.2) we obtain the result. Remark 3. Condition (3.2) sometimes is verified in an implicit way. For instance, if (3.3) b=λ1b1+λ2b2,with b1,b 2continuous functions λ1,λ 2≥0,and b−1 1,b 2locally Lipschitz; and (3.4) λ1−L1L2λ2>0 where L1and L2are the Lipschitz constant of b−1 1and b2respectively on the interval [−uL∞(Q),uL∞(Q)], then necessarily b−1is Lipschitz on the interval [−uL∞(Q),uL∞(Q)]. 542 J. I. D´ ıaz, J. F. Padial Lemma 7. Under the assumption of Theorem 1 we have (4.9) lim n→∞ Ω T n(u1(t)−u2(t))[φ(ξ1(t)) −φ(ξ2(t))] ·[∇u1(t)−∇u2(t)] dx ≥0 for a.e. t∈]0,T[, i.e. the diffusion operator is T-acretive in L1(Ω). Proof of Lemma 7: It is a slight improvement of D´ıaz-de Thelin [DT94]. For the sake of completeness we give the detailed proof. For any n∈N, the integral term in (4.9) it can be written as the addition of the integrals I1(n)=Ω T n(u1−u2)[φ(ξ1)−φ(ξ2)] ·[ξ1−ξ2]dx and I2(n)=Ω T n[φ(ξ1)−φ(ξ2)] ·e(k(b(u1)) −k(b(u2))) dx. Here we drop writing the t-dependence. We shall find an estimate on |I2(n)|in terms of I1(n). Due to the assumption (2.1) on k, we need to distinguish the cases 1 <p≤2 and p>2. Case 1<p≤2:Applying Young’s inequality we get (4.10) |I2(n)|≤ ε pΩ T n|φ(ξ1)−φ(ξ2)|p +1 pε Ω T n|k(b(u1)) −k(b(u2))|p for any ε>0. Using the inequality (4.1) to the first term of the right hand side of (4.10), by assumption (2.1) on kand the properties of T n we obtain that |I2(n)|≤εpC pI1(n)+ 2ˆ C εpp2 nγp−1 LNx:0<u 1−u2<2 n. Taking εp=p/C, we get −2ˆ CCp−1 pp2 nγp−1 LNx:0<u 1−u2<2 n≤I1(n)+I2(n). Uniqueness and existence to a nonlinear equation 543 Now, since γp −1>0, we have that lim n→∞ I1(n)+I2(n)≥0 and then (4.9) is proved. Case p>2:By H¨older inequality (4.11) |I2(n)|≤Ω T n|φ(ξ1)−φ(ξ2)|pdx 1 p ·Ω T n|k(b(u1)) −k(b(u2))|pdx 1 p . Using inequality (4.1) where we set β=pand η1=ξ1,η2=ξ2, the first multiplicative factor in (4.11) is bounded by C 1 pΩT nφ(ξ1)−φ(ξ2)[ξ1−ξ2]p 2T n(|ξ1|p+|ξ2|p)1−p 2dx 1 p . Using again the H¨older inequality and the properties of T n, we obtain the estimate Ω T n|φ(ξ1)−φ(ξ2)|pdx 1 p ≤A(n)I 1 2 1(n)n 2−p 2p where A(n)=C 1 pΩ∩{0<u1−u2<2 n} |ξ1|p+|ξ2|pdx 2−p 2p . For the second multiplicative factor in (4.11), we have Ω T n|k(b(u1)) −k(b(u2))|pdx1/p ≤2γn 1−pγ pLNx:0<u 1−u2<2 n1/p from assumption (2.1) on kand (4.5). Combining both inequalities, we arrive to |I2(n)|≤2γA(n)n 2−p 2p+1−pγ pI 1 2 1(n)LNx:0<u 1−u2<2 n 1 p . 544 J. I. D´ ıaz, J. F. Padial Since 2γA(n)LNx:0<u 1−u2<2 n1/p is uniformly bounded for any n∈Nand the exponent 2−p 2p+1−pγ pis positive, due to (2.1), we conclude that lim n→∞(I1(n)+I2(n)) ≥0 which ends the proof. End of the proof of Theorem 1: By the previous two lemmas, we obtain the key inequality Ω [b(u1(t, x)) −b(u2(t, x))]+dx ≤Ω [b(u01(x)) −b(u02(x))]+dx −Qt [g(x, u1)−g(x, u2)] sign+(u1−u2)dx ds +Qt [f1(t, x)−f2(t, x)] sign+(u1−u2)dx ds. Using the assumption (2.3) on g, the conclusion of the theorem is immediate if C∗is zero. More in general we set vj(t, x)=e−C∗tb(uj(t, x)) for j=1,2. Then sign+(v1−v2) = sign+(u1−u2) and ∂vj ∂t =−C∗vj+ e−C∗t∂b(uj) ∂t are also bounded regular measures in Q. Choosing Tn(v1− v2) as test function and working as before, we obtain Ω [v1(t, x)−v2(t, x)]+dx ≤Ω [v01(x)−v02(x)]+dx −C∗Qt [v1(s, x)−v2(s, x)]+dx ds −Qt [g(x, u1)−g(x, u2)]e−C∗ssign+(v1(s, x)−v2(s, x)) dx ds +Qt [f1(s, x)−f2(s, x)]e−C∗ssign+(v1(s, x)−v2(s, x)) dx ds. By assumption (2.3), one has that −[g(x, u1)−g(x, u2)]e−C∗ssign+(v1−v2) ≤C∗[b(u1)−b(u2)]e−C∗ssign+(v1−v2)=C∗[v1−v2]+ and thus, the conclusion holds. Remark 8. The assumption (4.3) on the measure of the jump points set is merely needed in the proof of Lemma 6. This assumption could be replaced by any other condition implying the conclusion of Lemma 6. In particular, we have Uniqueness and existence to a nonlinear equation 545 Corollary 6. Let u1and u2be bounded BV solutions of (1.1), (1.2) and (1.3). Assume the hypotheses of Theorem 1 but replacing (4.3) by (4.12) there exist two homeomorphisms on R,Ψ1and Ψ2, such that Ψ1(u1),Ψ2(u2)∈W1,1(Q). Then u1and u2verify the comparison criterium given in Theorem 1. Proof: By Lemma 1 condition (4.3) holds for ui,i=1,2. 5. Existence of bounded BV solutions In order to obtain the existence of bounded BV solutions for problem (1.1), (1.2), (1.3) we shall assume some additional conditions on functions fand u0: (5.1) f∈L∞(Q)∩BVt(Q), and (5.2) u0∈L∞(Ω) ∩W1,p 0and φ(∇u0−k(b(u0))e)∈(BV (Ω))N. We state our existence result in the following way: Theorem 2. Assume (1.5) on band also that (5.3) b−1is a locally Lipschitz continuous function. We assume also (2.1), (2.2), (2.3), (5.1) and (5.2). Then there exists a bounded BV solution uof (1.1), (1.2) and (1.3). Moreover u∈ C([0,T]; L1(Ω)). Proof: We start by considering, a sequence of regular problems having a unique solution by the classical theory of partial differential equations. After that, we shall obtain suitable a priori estimates. Finally passing to the limit we shall find a bounded BV solution. In view of the structural assumptions, we shall distinguish two cases, according psatisfies 1 <p< 2or2≥p. Case 1<p<2: Regularization. We define a sequence of uniformly parabolic problems with coefficients and free term bounded regular functions. Consider the following regularized equation in Q (5.4) ∂bm(u) ∂t −div φr(∇u−ks(bm(u))e)+gn(x, u)=fl(t, x), 546 J. I. D´ ıaz, J. F. Padial where we define the vectorial function φrby φr(ξ)=                |ξ|2+1 rp−2 2 ξ|ξ|<1 r, |ξ|p−2ξ|ξ|≥2 r, φr(ξ)1 r≤|ξ|<2 r, for any r∈N, and such that φr∈C1(RN) verifies (5.5) |φr(ξ)|≤|ξ|p−1∀ξ∈RN and (5.6) α|ξ|2 (1 + |ξ|)2−p≤φr(ξ)·ξ≤|ξ|p for any ξin RNα:= 2p−2 2. For any m∈N, we define bm(η)= 1 mη+¯ bm(η) with ¯ bmthe Yosida approximation of b. We recall that ¯ bmconverges uniformly on compacts sets to b,¯ bmis a Lipschitz nondecreasing function such that |¯ bm|≤|b|and that bmand b−1 mare Lipschitz nondecreasing functions; see [Be81]. We take a sequence of functions {ks}∞ s=1 belonging to C∞(R) such that they verify (2.1) and ksconverges to kuniformly on compacts of R. For any integer n, we consider a function gn∈C∞(Ω ×R) satisfying the assumptions (2.2) and (2.3) uniformly on nand such that gn(x, η) converges to gn(x, η)inL1(Ω) for any fixed ηin R, for a.e. x∈Ω, as n→∞. Let fl∈C∞([0,T]ׯ Ω) such that flL∞(Q)≤CfL∞(Q),flW1,1(0,T ;L1(Ω)) ≤CfBVtfor any l∈N and such that flconverges to fin L1(Q)asl→∞. Finally we regularize the initial condition. We consider u0,q ∈C∞ 0(Ω) such that u0,q ∗ ?u 0in L∞(Ω) as q→∞and such that φr(∇u0,q − ks(bm(u0,q))e)BV (Ω)N≤φr(∇u0−k(b(u0))e)BV (Ω)N. The equation (5.4) is uniformly parabolic. So, by well-know result (see e.g. Ladyzenskaja, Solonnikov and Uralceva [LSU68, Chap. V]) Uniqueness and existence to a nonlinear equation 547 there exists a unique classical solution ˆu=um,r,s,n,l,q of (5.4) satisfying ˆu(t, x)=0 onΣ, bm(ˆu(0,x)) = bm(u0,q(x)) on Ω. In what follows, we denote by ˆuthe function um,r,s,n,l,q. In order to study the convergence of the sequence ˆuwe shall need some uniform estimates in suitable functional spaces. A priori estimates. By the maximum principle (5.7) |ˆu(t, x)|≤M1∀(t, x)∈Q, where M1is a positive constant independent of m,r,s,land q. On the other hand, if we denote by ˆv:= bm(ˆu)tand we differentiate equation (5.4) with respect to t, we obtain that (5.8) ˆvt= div ∂φj r ∂ξi∂ ∂xi [(b−1 m)(bm(ˆu))ˆv]−k s(bm(ˆu))eiˆvj=1,... ,N  −g n(x, ˆu)(b−1 m)(bm(ˆu))ˆv+∂fl ∂t . For any η>0, we define the function Hηapproximating the absolute value function in the following way: we first introduce hη(σ)=     2 η1−|σ| η|σ|<η 0|σ|≥η, and finally we define Hη(σ)=σ 0 hη(τ)dτ and Hη(σ)=σ 0 Hη(τ)dτ. It is clear that hη≥0,lim η→0σhη(σ)=0, |Hη|≤1,lim η→0Hη(σ) = sgn0(σ) and lim η→0 Hη(σ)=|σ|. 548 J. I. D´ ıaz, J. F. Padial Multiplying equation (5.8) by Hη(ˆv), and integrating on Qt=]0,t[×Ω, we get Ω Hη(ˆv(t, x)) dx −Ω Hη(ˆv(0,x)) dx ≤−Qt (b−1 m)(bm(ˆu))ˆvhη(ˆv)∂φj r ∂ξi ∂bm(ˆu) ∂xi ∂ˆv ∂xj dx ds +Qt ˆvhη(ˆv)k s(bm(ˆu)) ∂ ∂ξi φj rei ∂ˆv ∂xj dx ds −Qt g n(x, ˆu)(b−1 m)(bm(ˆu))ˆvHη(ˆv)dx ds +Qt ∂fl ∂t Hη(ˆv)dx ds, since ˆu(s, x)=0on[0,T]×∂Ω, and then ˆv(s, x)=0on[0,T]×∂Ω. Passing to the limit when η→0, we obtain the inequality Ω |ˆv(t, x)|dx ≤Ω |ˆv(0,x)|dx +Qt ∂fl ∂t (s, x) sgn0(ˆv)dx ds −Qt g n(x, ˆu)(b−1 m)(bm(ˆu))|ˆv|dx ds from the properties of hηand the monotonicity of the vectorial function φr. Now, by (2.3), we arrive −Qt g n(x, ˆu)(b−1 m)(bm(ˆu))|ˆv|≤C∗Qt |ˆv|b m(ˆu)(b−1 m)(bm(ˆu)). Since (b−1 m)(bm(s))=1/b m(s) wherever b m(s)= 0, we have that the last integral is equal to Qt−{(s,x):b m(ˆu(s,x))=0} |ˆv|dx ds. Taking this into account, one verifies that Ω |ˆv(t, x)|dx ≤Ω |ˆv(0,x)|dx +Qt ∂fl ∂t (s, x) sgn(ˆv)dx ds +C∗Qt |ˆv|dx ds. Using the equation satisfied by ∂ ∂tbm(ˆu(0,x)) and the uniform bounded- Uniqueness and existence to a nonlinear equation 549 ness of the data, we get Ωˆv(t, x))dx ≤Ω ∂ ∂tbm(ˆu(0,x)) dx +Qt ∂fl ∂t  dx ds +C∗Qt |ˆv|(s, x)dx ds ≤Ω |div φ(∇u0,q −ks(bm(u0,q))e)|dx +Ω |gn(x, u0,q)|dx +Ω |fl(0,x)|dx +Qt ∂fl ∂t  dx ds +C∗Qt |ˆv|(s, x)dx ds ≤C1+C∗Qt |ˆv|(s, x)dx ds. Applying Gronwall’s lemma, we obtain that Ω |ˆv|dx ≤eC∗tfor any t∈[0,T]. Thus (5.9) Ω ∂bm(ˆu) ∂t  dx ≤M2for any t∈[0,T] with M2=eC∗T. From (5.7) and (5.3) (5.10) Ω ∂ ∂tˆu dx ≤M3∀t∈[0,T]. Now, we shall show that there exists M4>0, such that (5.11) Ω |∇ˆu|pdx ≤M4 uniformly in t∈[0,T], m,r,s,n,land q. Firstly, we shall show that there exists an uniform positive constant Msuch that (5.12) Ω |ˆ ξ|pdx ≤M∀t∈[0,T] where ˆ ξ:= ∇ˆu−ks(bm(ˆu))e. To do that, we multiply (5.4) by ˆuand we integrate on Ω: The Ω ˆubm(ˆu)t+Ω φr(ˆ ξ)·∇ˆudx+Ω gn(x, ˆu)ˆudx=Ω flˆu dx. 550 J. I. D´ ıaz, J. F. Padial In what following it will appear several positive constants denoted by Ci, i=2,3,4,... which are independent on tand the parameters m, r,s,n, land q. They will dependent on the exponent p, the measure of Ω and the above estimates. Some of them are also function of some positive parameters εand δwe shall introduce later. We shall only indicate the εand δdependence. By estimates (5.7) and (5.9), the assumption (2.3) on g, (5.1) on fand the properties of gnand fl, there exists a positive constant C2uniform in m,r,s,n,land q, such that Ω φr(ˆ ξ)·ˆ ξdx+Ω φr(ˆ ξ)ks(bm(ˆu))edx ≤C2 for any t. By Young’s inequality, we have Ω φr(ˆ ξ)·ˆ ξdx≤C2+εp pΩ |φr(ˆ ξ)|pdx +1 εppΩ |ks(bm(ˆu))e|pdx where εis an arbitrary positive real number. The last integral is uniformly bounded in view of (5.7) and the assumptions on the sequences {ks}and {bm}. By the properties of φr, the first integral of the right hand side is bounded by Ω|ˆ ξ|pdx, for any tin [0,T]. Hence, (5.13) Ω φr(ˆ ξ)·ˆ ξdx≤C3(ε)+εp pΩ |ˆ ξ|pdx for some positive constant C3=C3(ε). Besides, from the properties of φr, we have that (5.14) αΩ |ˆ ξ|2 (1 + |ˆ ξ|)2−pdx ≤Ω φr(ˆ ξ)·ˆ ξ dx. On the other hand, applying Young’s inequality to Ω |ˆ ξ|p (1+|ˆ ξ|)2−p(1 + |ˆ ξ|)2−pdx with exponents 2 pand 2 2−pwe get Ω |ˆ ξ|pdx ≤p 2δ2/p Ω |ˆ ξ|2 (1 + |ˆ ξ|)2−pdx +δ2/(2−p)2−p 2Ω (1 + |ˆ ξ|)pdx for any δ>0. Hence (5.15) (1 −C4δ2/(2−p))Ω |ˆ ξ|pdx ≤C5(δ)+ p 2δ2/p Ω |ˆ ξ|2 (1 + |ˆ ξ|)2−pdx. Uniqueness and existence to a nonlinear equation 551 Using the estimates (5.13) and (5.14) into (5.15), we obtain the inequality α(1 −C4δ2/(2−p))Ω |ˆ ξ|pdx ≤C5+p 2δ2/p C3+εp pΩ |ˆ ξ|pdx, which implies α1−C4δ2/(2−p)−C6 εp δ2/p Ω |ˆ ξ|pdx ≤C7(ε, δ) for some positive constant C7depending on ε,δ. To verify the estimate (5.12), it is enough now to choose 0 <δ1 and ε1 such that εpδ2/p and 1 −C4δ2/(2−p)−C6εp δ2/p >0. Now (5.12) implies (5.11) from the uniform boundedness of Ω |ks(bm(ˆu))|pdx. Finally, multiplying the relation (5.4) by v∈Xand integrating on Q, we obtain, using the H¨older’s inequality that Q v∂bm(ˆu) ∂t  ≤Q |φr(ˆ ξ)|pdx dt1/p Q |∇v|pdx dt1/p +vL∞(Q)Q |gn(x, ˆu)|dx dt +Q |fl|pdx dt1/p Q |v|pdx dt1/p . The properties (5.5) and (5.6) of φr, the assumptions (2.2) on gand (5.1) on fand the properties on gnand fllead to estimate Q v∂bm(ˆu) ∂t  ≤M5vLp(0,T ;W1,p 0(Ω))capL∞(Q) for some positive constant M5independent on m,r,s,n,land qwhere we used estimates (5.7) and (5.12). In this way, we obtain the following uniform estimate in Lp(0,T;W−1,p(Ω)) + L1(Q) (5.16)     ∂bm(ˆu) ∂t    Lp(o,T ;W−1,p(Ω))+L1(Q) ≤M5. Passing to the limit. By the estimates (5.7), (5.9), (5.10), (5.11) and (5.16) we can find a bounded BV solution of problem (1.1), (1.2) and (1.3) 558 J. I. D´ ıaz, J. F. Padial with Φ(ξ)=1 p|ξ|p,ξ∈RNand C1a positive constant independent on m, s, n, l, q, E thanks to the previous estimates on ˆuand |∇ˆu|. 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