Nonlinear Diffusion in Irregular Domains

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1 Nonlinear Diffusion in Irregular Domains Ugur G. Abdulla Max-Planck Institute for Mathematics in the Sciences, Leipzig 0403, Germany We investigate the Dirichlet problem for the parablic equation u t = u m,m > 0, in a nonsmooth domain Ω IR N+,N 2. In a recent paper [U.G.Abdulla, J. Math. Anal. Appl., 260, 2 (200), ] existence and boundary regularity results were established. In this paper we present uniqueness, comparison and stability theorems. Introduction and Statement of Main Results Consider the equation u t = u m (.) N where u = u(x, t),x =(x,...,x N ) IR N,N 2,t IR +, = 2 / x 2 i,m > 0,m. i= In this paper we continue our study of the Dirichlet problem (DP) for the equation (.) in a general domain Ω IR N+. In the recent paper [] (see also [2]) existence and boundary regularity results were established (see Theorem 2. of []). The purpose of this paper is to establish uniqueness, comparison and stability theorems. Let Ω be bounded open subset of IR N+,N 2. Let the boundary Ω of Ω consists of the closure of open domain BΩ lying on t = 0, an open domain DΩ lying on t = T (0, ) and a (not necessarily connected) manifold SΩ lying in the strip 0 <t T. Denote Ω(τ) ={(x, t) Ω:t = τ} and assume that Ω(t) for t (0,T). The set PΩ =BΩ SΩ is called a parabolic boundary of Ω. Furthermore the class of domains with described structure will be denoted by D 0,T. Let Ω D 0,T is given and

2 2 ψ is an arbitrary continuous nonnegative function defined on PΩ. DP consists in finding a solution to equation (.) in Ω DΩ satisfying initial-boundary condition u = ψ on PΩ. (.2) We shall follow the following notion of weak solutions (super- or subsolutions): Definition. We shall say that the function u(x, t) is a solution (respectively super- or subsolution) of DP (.), (.2), if (a) u is nonnegative and continuous in Ω, locally Hölder continuous in Ω DΩ, satisfying (.2) (respectively satisfying (.2) with = replaced by or ), (b) for any t 0,t such that 0 <t 0 <t T and for any domain Ω D t0,t such that Ω Ω DΩ and BΩ, DΩ,SΩ being sufficiently smooth manifolds, the following integral identity holds ufdx = ufdx + (uf t + u m f)dxdt u m f dxdt, (.3) ν DΩ BΩ Ω SΩ (respectively (.3) holds with = replaced by or ), where f C 2, x,t (Ω ) is an arbitrary function (respectively nonnegative function) that equals to zero on SΩ and ν is the outward-directed normal vector to Ω (t) at(x, t) SΩ. We shall use the same notation as in []: z =(x, t) =(x,...,x N,t) IR N+,N 2,x = N N (x, x) IR N, x =(x 2,...,x N ) IR N, x 2 = x i 2, x 2 = x i 2. For a point z =(x, t) IR N+ we denote by B(z; δ) an open ball in IR N+ of radius δ>0 and with center being in z. i= Let Ω D 0,T be a given domain. Assume that for arbitrary point z 0 =(x 0,t 0 ) SΩ i=2 (or (z 0 = (x 0, 0) SΩ) there exists δ > 0 and a continuous function φ such that, after a suitable rotation of x-axes, we have SΩ B(z 0,δ) = {z B(z 0,δ) : x = φ(x, t)} and sign (x φ(x, t)) = for z B(z 0,δ) Ω. Furthermore, we always suppose in this paper that the conditions of the Theorem 2. of [] are satisfied. We are going now to formulate another pointwise restriction at the point z 0 =(x 0,t 0 ) SΩ, 0 <t 0 <T, which plays a crucial role in the proof of uniqueness of

3 3 the constructed solution. For an arbitrary sufficiently small δ>0, consider a domain Q(δ) ={(x, t) : x x 0 < (δ + t 0 t) 2,t 0 <t<t 0 + δ}, (.4) Assumption M Let for all sufficiently small positive δ we have φ(x 0,t 0 ) φ(x, t) [t t 0 + x x 0 2 ] µ for (x, t) Q(δ). (.5) where µ> 2 if 0 <m<, and µ> m m+ if m>. Furthermore we denote ν = µ assuming without loss of generality that ν ( 2, 0) if 0 <m< and ν ( m+, 0) if m>. Definition 2.3 Let [c, d] (0,T) be a given segment and SΩ [c,d] = SΩ {(x, t) :c t d}. We shall say that assumption M is satisfied uniformly in [c, d], if there exists δ 0 > 0 and µ>0 as in (.5) such that for 0 <δ δ 0, (.5) is satisfied for all z 0 SΩ [c,d] with the same µ. Our main theorems read: Theorem 2.2 (Uniqueness) Let there exists a finite number of points t i,i =,...,k such that t =0<t 2 <...<t k <t k+ = T and for the arbitrary compact subsegment [δ,δ 2 ] (t i,t i+ ),i =,...,k, assumption M is uniformly satisfied in [δ,δ 2 ]. Then the solution of the DP is unique. Theorem 2.3 (Comparison) Let u be a solution of DP and g be a supersolution (respectively subsolution) of DP. Assume that the assumption of Theorem 2.2 is satisfied. Then u (respectively ) g in Ω. Theorem 2.4 (Stability or L -contraction) Let the assumption of Theorem 2.2 be satisfied. Let g and g 2 are solutions of DP with inital boundary data ψ and ψ 2 respectively. If ψ = ψ 2 on SΩ, then for arbitrary t [0,T] we have g g 2 L (Ω(t)) ψ ψ 2 L (BΩ). 2 Geometric Meaning of the Assumption M Assumption M is of geometric nature. To explain its meaning, for simplicity assume that N =2,d(z 0 ) = and rewrite (.5) as follows: x 0 x [t t 0 +(x 2 x 0 2 )2 ] µ for (x 2,t) Q(δ), where x 0 = φ(x0 2,t 0) and x = φ(x 2,t) for (x 2,t) Q(δ). Consider the hyperbolic paraboloid

4 4 x 2 = t + x2 2 (Figure ) in the x x 2 t-space. Let M δ be the piece of it lying in the half-space {t 0}, between the planes {x =0} and {x = δ 2 } (see Figure 2). The projection of M δ to the plane {x =0} is Q 0 (δ), where as Q 0 (δ) we denote Q(δ) with N =2,x 0 2 =0,t 0 = 0. The surface M δ has the following representation: x = φ(x 2,t) x t, (x 2,t) Q 0 (δ). t x = x 2 x = x 2 x x 2 Figure : Hyperbolic paraboloid x 2 = t + x2 2 Obviously, the function φ satisfies (.5) with = instead of in the critical case when µ = 2 (we also replace Q(δ) with Q 0 (δ) in (.5)). Consider the displacements of M δ, while it is moved on the x x 2 -plane and shifted along the t-axis. Let us now consider the critical case of the assumption M with µ = 2. Namely, we take 2 in (.5). Equivalent formulation of this critical assumption may be given as follows: Assume that after the displacement of the above type M δ occupies such a position that its vertex coincides with the point z 0 =(x 0,t 0 ) SΩ, and for δ being positive and sufficiently small it has no common point with Ω. Similar geometric reformulation of the assumption M may be given just modifying subsurface M δ according to the lower restriction imposed on µ. Thus if 0 <m<, then the exterior touching surface is slightly more regular at the vertex point than related subsurface M δ of the hyperbolic paraboloid. Otherwise speaking, it is slightly more regular than C, 2 x,t at the vertex point. When m changes from to +, the regularity of M δ increases continuously, for each m

5 5 t δ x = x 2 δ 2 δ 2 x δ 2 x = x 2 x 2 Figure 2: Piece M δ of the hyperbolic paraboloid from Fig. lying in the half-space {t 0}, between the planes {x =0} and {x = δ 2 }. being slightly more regular than C 2m m+, m m+ x,t at the vertex point. Another limit position of M δ as m + (or µ ) is the upper paraboloid frustrum with vertex at the origin. 3 Proofs of the Main Theorems Proof of Theorem 2.2 Suppose that g and g 2 are two solutions of DP. We shall prove uniqueness by proving that g g 2 in Ω {(x, τ) :t j τ t j+ },j =,...,k (3.) We present the proof of (3.) only for the case j =. The proof for cases j =2,...,k coincides with the proof for the case j =. We prove (3.) with j = by proving that for some limit

6 6 solution u = lim u n the following inequalities are valid (u(x, t) g i (x, t))ω(x)dx 0,i=, 2 (3.2) Ω(t) for every t (0,t 2 ) and for every ω C 0 (Ω(t)) such that ω. Obviously, from (3.2) it follows that g = u = g 2 in Ω {(x, τ) :t τ<t 2 }, (3.3) which implies (3.) with j = in view of continuity of u, g and g 2 in Ω. Since the proof of (3.2) is similar for each i, we shall henceforth let g = g i. Let t (0,t 2 ) be fixed and let ω C 0 (Ω(t)) be an arbitrary function such that ω. To construct the required limit solution, as in [], we approximate Ω and ψ with a sequence of smooth domains Ω n D 0,T and smooth positive functions ψ n. We make a slight modification to the construction of Ω n and ψ n. Let Ψ be a nonnegative and continuous function in IR N+, which coincides with ψ on PΩ. This continuation is always possible. Let ψ n be a sequence of smooth functions such that max(ψ; n ) ψ n (Ψ m + Cn m ) m,n=, 2,..., (3.4) where C> is a fixed constant. For arbitrary subset G IR N+ and ρ>0, we define O ρ (G) = B(z,ρ). z G Since g and Ψ are continuous functions in Ω and g = ψ on PΩ, for arbitrary n there exists ρ n > 0 such that g m (z) Ψ m (z) n m for z O ρn (SΩ) Ω. (3.5) We then assume that Ω n satisfies the following: Ω n D 0,T, Ω n Ω DΩ,SΩ n O ρn (SΩ) (3.6) We now formulate assumptions on SΩ n near its point z n, which are direct implications of the assumption M at the point z 0 SΩ. Without loss of generality assume that d(z 0 )=. Assume that SΩ n in some neighbourhood of its point z n =(x (n), x0,t 0 ) is represented by the function x = φ n (x, t), where {φ n } is a sequence of sufficiently smooth functions and φ n φ

7 7 as n +, uniformly in Q(δ 0 ), where δ 0 > 0 be a sufficiently small fixed number, which does not depend on n. Obviously, we can assume that φ n satisfies assumption M (namely (.5)) at the point (x 0,t 0 ), uniformly with respect to n and with the same exponent µ. Let {δ n } be some sequence of positive real numbers such that δ n 0asn +. Assume also that the sequence {φ n } is chosen such that, for n being large enough, the following inequality is satisfied. φ n (x 0,t 0 ) φ n (x, t) δ ν n [t t 0 + x x 0 2 ] for (x, t) Q(δ n ). (3.7) Obviously, this is possible in view of uniform convergence of φ n to φ. For example, if φ(x, t) coincides with its lower bound φ(x, t) =φ(x 0,t 0 ) [t t 0 + x x 0 2 ] µ, for (x, t) Q(δ 0 ) (namely (.5) is satisfied with = instead of ), then for all large n such that δ n <δ 0 we first choose φ n as follows: φ n (x, t) = φ(x 0,t 0 ) δ ν n[t t 0 + x x 0 2 ]for(x, t) Q(δ n ), φ(x, t) for (x, t) Q(δ 0 )\Q(δ n ). Obviosly, φn satisfies (3.7) and converges to φ uniformly in Q(δ 0 ). Then we easily construct φ n by smoothing φ n at the boundary points of Q(δ n ) satisfying t t 0 + x x 0 2 = δ n. In general, we can do similar construction by taking instead of φn (x, t) the function φn (x, t) = max( φ n (x, t); φ(x, t)), which satisfies (3.7) and converges to φ(x, t)) as n +, uniformly in Q(δ 0 ). Furthermore we will assume that the sequence δ n is chosen as follows: δ n = n γ m ɛ +2ν with 0 <ɛ<( + ν) [ ν ( m + ) + ], (3.8) γ γ where γ =ifm>, while if 0 <m< then γ is chosen such that Let u n be a classical solution to the following problem: m <γ< +ν νm. (3.9) u t = u m, in Ω n DΩn (3.0) u = ψ n on PΩ n. (3.) This is a nondegenerate parabolic problem and classical theory [3,4] implies the existence of a unique C 2+α solution. From maximum principle and (3.4) it follows that n u n M in Ω n,n=, 2,... (3.2)

8 8 where M is some constant which do not depend on n and M > max(sup ψ, sup ψ n ). As in PΩ PΩ n [], we then prove that for some subsequence n, u = lim n u n is a solution of DP (.), (.2). Furthermore, without loss of generality we write n instead of n. Take an arbitrary sequence of real numbers {α l } such that 0 <α l+ <α l <t,α l 0asl +. (3.3) Let Ω l n =Ω n {(x, τ) :αl <τ<t}, Ω 0 n =Ω n {(x, τ) :0<τ<t},SΩ l n = SΩ n {(x, τ) :αl < τ<t},sω 0 n = SΩ n {(x, τ) :0<τ<t}. Since un is a classical solution of (3.0), it satisfies Ω n(t) u n fdx = Ω n(α l ) u n fdx+ (u n f τ + u m n f)dxdτ Ω l n SΩ l n u m n f dxdτ, (3.4) ν for arbitrary f C 2, x,t (Ωl n) that equals to zero on SΩ l n, and ν = ν(x, τ) is the outward-directed normal vector to Ω n (τ) at(x, τ) SΩ l n. Since g is the weak solution of the DP (.), (.2), we also have gfdx = gfdx + (gf τ + g m f)dxdτ Ω n(t) Ω n(α l ) Ω l n SΩ l n Substracting (3.5) from (3.4), we derive g m f dxdτ. (3.5) ν Ω n(t) (u n g)fdx = Ω n(α l ) (u n g)fdx SΩ l n (u m n gm ) f ν dxdτ + where C n =ifm> (accordingly γ = ) and C n = B n if 0 <m<, and A n = mγ 0 γ (θun +( θ)g γ ) mγ dθ, B n = γ 0 Ω l n γ (un g γ )[C n f τ +A n f]dxdτ, γ (θun +( θ)g γ ) γ dθ. (3.6) The functions A n and B n are Hölder continuous in Ω n. From (3.2) and Definition. we derive n mγ γ A n A, n γ γ B n B for (x, τ) Ω n, (3.7) where A, B are some positive constants which do not depend on n. To choose the test function f = f(x, τ) in (3.6), consider the following problem: C n f τ + A n f =0inΩ 0 n BΩn (3.8a) f =0onSΩ 0 n and f = ω(x) onω n (t). (3.8b)

9 9 This is the linear non-degenerate backward-parabolic problem. From the classical parabolic theory ([3,4]) it follows that there exists a unique classical solution f n Cx,τ 2+β,+β/2 (Ω 0 n) with some β>0. From the maximum principle it follows that f n in Ω 0 n (3.9) By the condition of theorem assumption M is satisfied uniformly on every compact subsegment of (0,t]. We prove that for every fixed l (see (3.3)) there exists a positive constant C(l), which does not depend on n, such that sup z SΩ l n (+ν)(m+ɛ γ ) f n (z) C(l)n +2ν. (3.20) To prove (3.20), we use the modification of the method proposed in [] for proving the boundary regularity of the solution to Dirichlet problem. We use (3.20), in order to estimate the righthand side of (3.6) with f = f n (x, τ), which is a solution of the problem (3.8). We have (u n g)ω(x)dx = (u n g)fdx (u m n g m ) f ν dxdτ I + I 2 (3.2) Ω n(t) By using (3.4) - (3.6), we have I 2 sup f(z) z SΩ l n SΩ l n From (3.20), (3.2) we derive Ω n(α l ) SΩ l n ( ψ m n Ψm + Ψ m g m )dxdτ (C +)n m sup z SΩ l n f(z). (3.22) ɛ(+ν) γ ν(m+ γ ) I 2 (C +)C (l)n +2ν. (3.23) where ɛ and γ are chosen as in (3.8),(3.9). Applying (3.9), we have I u n g dx. (3.24) Ω n(α l ) To estimate the right-hand side, introduce a function u u l n (x, α l ), x Ω n (α l ) n(x) = ψ n (x, α l ), x Ω(α l )\Ω n (α l ). Obviously u l n(x),x Ω(α l ) is bounded uniformly with respect to n, l. From (3.24), we have I u l n g dx. (3.25) Ω(α l )

10 0 Since lim n + ul n(x) =u(x, α l ) for x Ω(α l ), from Lebesgue s theorem it follows that lim n + Ω(α l ) u l n g dx = Ω(α l ) u(x, α l ) g(x, α l ) dx (3.26) Hence, by using (3.22) - (3.25) in (3.2) and passing to the limit n + we have (u g)ω(x)dx Ω(t) Ω(α l ) u g dx. (3.27) Let u(x, α l ) g(x, α l ), x Ω(α l ) U l (x) = 0, x / Ω(α l ) Obviously, U l is uniformly bounded with respect to l. Hence, from (3.27) we derive that (u g)ω(x)dx Ω(t) BΩ U l (x) dx + C 2 meas(ω(α l )\BΩ), (3.28) where the constant C 2 does not depend on l. From Lebesgue s theorem it follows that lim U l (x) dx =0. l + BΩ Hence, passing to the limit l +, from (3.28), (3.2) follows. As it is explained earlier, from (3.2), (3.) with j = follows. Similarly, we prove (3.) (step by step) for each j =2,...,k. Theorem 2.2 is proved. The proofs of the Theorems 2.3 and 2.4 are similar to the given proof. References. U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in nonsmooth domains, Journal of Mathematical Analysis and Applications, 260, 2 (200), U.G. Abdulla, On the Dirichlet problem for reaction-diffusion equations in non-smooth domains, Nonlinear Analysis, 47, 2 (200), O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence RI, G.M.Lieberman, Second Order Parabolic Differential Equations, World Scientific, 996.