On the Dirichlet Problem for the Nonlinear. Diffusion Equation in Non-smooth Domains

Transcription

1 On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains Ugur G. Abdulla Faculty of Applied Mathematics and Cybernetics, Baku State University Baku , Azerbaijan Abstract. We study the Dirichlet problem for the parabolic equation u t = u m,m>0 in a bounded, non-cylindrical and non-smooth domain Ω IR N+1,N 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or leftupper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 1 2 is critical as in the classical theory of the one-dimensional heat equation u t = u xx. Key Words: Dirichlet problem; non-smooth domains; nonlinear diffusion; degenerate and singular parabolic equations; boundary regularity. Current address: Max-Planck Institute for Mathematics in the Sciences, Inselstr.22, Leipzig, Germany

2 2 1 Introduction Consider the equation u t = u m, (1.1) N where u = u(x, t),x =(x 1,...,x N ) IR N,N 2,t IR +, = 2 / x 2 i,m > 0. We study the Dirichlet problem (DP) for equation (1.1) in a bounded domain Ω IR N+1. It can be stated as follows: given any continuous function on the parabolic boundary PΩ of Ω, to find a continuous extension of this function to the closure of Ω which satisfies (1.1) in Ω\PΩ. The classical DP for the heat equation (m = 1 in (1.1)) is included to our problem. Another direction, this work fits in with, is the modern theory of nonlinear degenerate and singular parabolic equations. If m>1, the equation (1.1) is a well-known porous medium equation, describing the flow of a compressible Newtonian fluid through a porous medium [24], while the singular case (0 <m<1) arises (for example) in plasma physics [8]. A particular motivation for this work arises from the problem about the evolution of interfaces in problems for porous medium equation. Special interest concerns the cases when support of the initial data contains i=1 a corner or cusp singularity at some points. What about the movement of these kinds of singularities along the interface? To solve this problem, it is important, at the first stage, to develop general theory of boundary-value problems in non-cylindrical domains with boundary surfaces which has the same kind of behaviour as the interface. In many cases this may be non-smooth and characteristic (see e.g. [1-3]). We make now precise the meaning of solution to DP. Let Ω be bounded open subset of IR N+1,N 2. Let the boundary Ω of Ω consists of the closure of a domain BΩ lying on t =0, a 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.

3 3 Let Ω D 0,T is given and ψ is an arbitrary continous nonnegative function defined on PΩ. DP consists in finding a solution to equation (1.1) in Ω DΩ satisfying initial-boundary condition u = ψ on PΩ (1.2) Obviously, in general the equation (1.1) degenerates at points (x, t), where u = 0 and we cannot expect the considered problem to have classical solution. If m 1, we shall follow the following notion of weak solution: Definition 1.1 We shall say that the function u(x, t) is a solution of DP (1.1), (1.2), if (a) u is nonnegative and continuous in Ω, satisfying (1.2). (b) for any t 0,t 1 such that 0 <t 0 <t 1 T and for any domain Ω 1 D t0,t 1 such that Ω 1 Ω DΩ and BΩ 1, DΩ 1,SΩ 1 being sufficiently smooth manifolds, the following integral identity holds ufdx = ufdx + (uf t + u m f)dxdt u m f dxdt, (1.3) ν DΩ 1 BΩ 1 Ω 1 SΩ 1 where f C 2,1 x,t (Ω 1) is an arbitrary function that equals to zero on SΩ 1 and ν is the outwarddirected normal vector to Ω 1 (t) at(x, t) SΩ 1.Ifm = 1, however, the solution is understood in the classical sence. After Wiener published his famous work [27], where he accomplished the long line investigations on the DP for Laplace equation in general domains, the DP for the heat equation was continously under the interest of many mathematicians in this century. In [20], a necessary and sufficient condition for the regularity of a boundary point in the Dirichlet problem for the heat equation in an arbitrary spatial dimension has been announced. The analog of the Wiener s condition, namely the necessary and sufficient condition which is a quasigeometric characterization for a boundary point of an arbitrary bounded open subset of IR N+1 to be regular for the heat equation has been established in [12], necessity being established earlier in [19]. A similar criterion for linear parabolic equation with smooth, variable coefficient was established in [15]. Wiener s type sufficient conditions for boundary regularity in the case of

4 4 general quasilinear uniformly parabolic equations were proved in [14,28]. Another sufficient condition, so called exterior tusk condition which is an analog of the exterior cone condition for elliptic equations, has been established in [11] for the linear heat equation and later in [22] for the linear uniformly parabolic equations. However, it should be mentioned that Wiener s criterion does not explicitly clear the natural analytic question, which we impose in this paper for more general nonlinear equation (1.1). Namely, what about the relation between the solvability of the DP or regulartiy of the boundary points and local modulus of continuity of the boundary manifolds. The importance of this question arises in view of applications which we mentioned earlier. Almost complete answer to this question was given by Petrowsky [21] in the case of one-dimensional linear heat equation u t = u xx. Results concerning one-dimensional reaction-diffusion equation u t = a(u m ) xx + bu β,a > 0,m > 0,b IR, β > 0 have been presented in recent papers of the author [4, 5]. Primarily applying the results of [4], a full description of the evolution of interfaces and of the local solution near the interface for all relevant values of parameters is presented in another recent paper [3]. DP for the porous medium equation in cylindrical domain with smooth boundary have been investigated in [7,16]. At the moment there is a complete well established theory of the boundary value problems in cylindrical domains for general second order nonlinear degenerate parabolic equations (which includes as a particular case (1.1) and (1.4) below) due to [6,7,9,10,16,26,29 etc.] (see the review article [17]). It seems that this paper is the first one which addresses the DP for the high dimensional nonlinear degenerate or singular parabolic equations in non-cylindrical domains with non-smooth boundaries. The approach used in this paper may be well expressed by the citation from the classical work [27] on the DP for Laplace equation. As it was pointed out by Lebesgue and independently by Wiener the Dirichlet problem divides itself into two parts, the first of which is the determination of a harmonic function corresponding to certain boundary conditions, while the second is the investigation of the behaviour of this function in the neighbourhood of the boundary. By using an approximation of both Ω and ψ, we also construct a limit solution

5 5 as a limit of a sequence of classical solutions in regular domains. We then prove a boundary regularity by using barriers and a limiting process. The main result of this paper on the existence and boundary continuity of the solution to DP is formulated in Theorem 2.1 (see also Corollary 2.1) of the Section 2. We introduce in this paper a notion of parabolic modulus of left-lower (or left-upper) semicontinuity of the lateral boundary manifold at the given point (Definition 2.1, Section 2). Our main assumption (Assumption A and (2.1), Section 2) consists in upper (or lower) Hölder condition on the parabolic modulus of left-lower (or left-upper) semincontinuity at each point of the lateral boundary manifold. Moreover, as in the classical theory of one-dimensional heat equation, the critical Hölder exponent is equal to 1 2. This assumption relates to the parabolic nature of the equation (1.1) and does not depend on m. At this point, it should be mentioned that equation (1.1) has no essential importance for our results, rather than being a suitable model example for three different class of parabolic equations, namely singular (0 <m<1), degenerate (m >1) and uniform (m = 1) parabolic equations. For example, by using our techniques the same results may be proved for the following reaction-diffusion-convection equation u t = a u m + b u γ + cu β, (1.4) where a,m,γ,β > 0,b IR N,c IR (see Remark 3.1, Section 3). We believe that the same result is true for more general second order parabolic equations. However, in this paper we restrict ourselves to equation (1.1), in order to make it less technical the presentation of our barrier method to prove the boundary regularity. It should also be mentioned that since our main result on the boundary regularity of a weak solution to equation (1.1) is of the local nature, similar result is true for an arbitrary bounded domain Ω IR N. It should also be mentioned that in this paper we restrict ourselves only with the existence and boundary regularity problems. We address issues regarding uniqueness of the constructed solution and related comparison theorems in a subsequent paper. The organisation of the paper is as follows: In Section 2 we outline the main result. In Section 3 we prove the main result (Theorem 2.1) from Section 2.

6 6 2 Statement of the Main Result We shall use the usual notation: z = (x, t) = (x 1,...x N,t) IR N+1,N 2,x = (x 1, x) = (x 1,x 2,...,x N ) IR N, x = N N (x 2,...,x N ) IR N 1, x 2 = x i 2, x 2 = x i 2. For a point z =(x, t) IR N+1 we denote i=1 by Q(z; δ) an open ball in IR N+1 of radius δ>0 and with center in z. i=2 Let Ω D 0,T be a given domain. Assume that for arbitrary point z 0 =(x 0,t 0 ) SΩ (or z 0 =(x 0, 0) SΩ) there exists δ>0 and a conitnuous function φ such that, after a suitable rotation of x-axes, we have SΩ Q(z 0,δ)={z Q(z 0,δ):x 1 = φ(x, t)}. Suppose also that sign (x 1 φ(x, t)) = const for z Q(z 0,δ) Ω Furthermore, we denote this constant by d(z 0 ). Obviously, by introducing a new variable x 1 = x 1, if necessary, we could have supposed that d(z 0 ) = 1. However, we describe the conditions for both cases d(z 0 )=±1 seperately, in order to distinct left and right boundary points as in the one-dimensional case. Let z 0 =(x 0,t 0 ) SΩ be a given boundary point. For an arbitrary sufficiently small δ>0, consider a parabolic domain P (δ) ={(x, t) : x x 0 <ε 0 (δ + t t 0 ) 1 2,t 0 δ<t<t 0 }, where ε 0 > 0 is an arbitrary fixed number. Definition 2.1 Let ω (δ) =max(φ(x 0,t 0 ) φ(x, t) :(x, t) P (δ)). ω + (δ) =min(φ(x 0,t 0 ) φ(x, t) :(x, t) P (δ)). The function ω (δ) (respectively ω + (δ)) is called the parabolic modulus of left-lower (respectively left-upper) semicontinuity of the function φ at the point (x 0,t 0 ).

7 7 For suffuciently small δ>0 these functions are well-defined and converge to zero as δ 0. Our main assumption on the behaviour of the function φ near z 0 is as follows: Assumption A. There exists a function F (δ) which is defined for all positive sufficiently small δ; F is positive with F (δ) 0asδ 0 and if d(z 0 ) = 1 (respectively d(z 0 )= 1) then ω (δ) δ 1 2 F (δ) (2.1) (respectively ω + (δ) δ 1 2 F (δ)) We prove in the next section that assumption A is sufficient for the regularity of the boundary point z 0. Namely, the constructed limit solution takes the boundary value ψ(z 0 ) at the point z = z 0 continuously in Ω. It is well-known that in the case of the classical heat equation (m =1 in (1.1)) boundary point z 0 =(x 0, 0) SΩ is always regular (see e.g. [21, p. 172]). Hence, in this case the assumption A imposed on every boundary point z 0 SΩ is sufficient for solvability of the DP (see Corollary 2.1 below). It may easily be proved that the solution in this particular case is a unique classical solution. However, in general to provide the regularity of the boundary point z 0 =(x 0, 0) SΩ we need another assumption. Denote x 1 = φ(x) φ(x, 0). Definition 2.2 Let ω0 (δ) =max(φ(x0 ) φ(x) : x x 0 δ) ω + 0 (δ) =min(φ(x0 ) φ(x) : x x 0 δ) The function ω0 (δ) (respectively ω+ 0 (δ)) is called the modulus of lower (respectively upper) semicontinuity of the function x 1 = φ(x) at the point x = x 0. Assumption B. There exists a function F 1 (δ) which is defined for all positive sufficiently small δ; F 1 is positive with F 1 (δ) 0asδ 0 and if d(z 0 ) = 1 (respectively d(z 0 )= 1) then ω 0 (δ) δf 1(δ) (2.2) (respectively ω + 0 (δ) δf 1(δ)) It may easily be verified that if we redefine φ as x 1 = φ(x) φ(x, t 0 ) then assumption B is a consequence of the assumption A at the boundary point z 0 =(x 0,t 0 ) SΩ. However,

8 8 assumption B has a sense for the boundary points z 0 =(x 0, 0) SΩ on the bottom of the lateral boundary manifold. We prove in the next section that assumption B is sufficient for the regularity of the boundary point z 0 =(x 0, 0) SΩ. Namely, the constructed limit solution takes the boundary value ψ(z 0 ) at the point z = z 0 continuously in Ω. Thus our main theorem reads: Theorem 2.1 DP (1.1), (1.2) is solvable in a domain Ω which satisfies the assumption A at every point z 0 SΩ and assumption B at every point z 0 =(x 0, 0) SΩ. Corollary 2.1 There exists a unique classical solution to DP (1.1), (1.2) with m = 1, in a domain Ω which satisfies the assumption A at every point z 0 SΩ. It should be noted that our main result about the boundary regularity is of local nature and, consequently, an existence of different function F (δ) (or F 1 (δ)) for each boundary point in respective assumption A (or B) is allowed. It may be easily observed that assumptions A and B coincide in the case of cylindrical domain Ω. 3 Proof of the Main Result Step 1. Construction of the limit solution. Consider a sequence of domains Ω n D 0,T,n = 1, 2,... with SΩ n, BΩ n and DΩ n being sufficiently smooth manifolds. Assume that {SΩ n }, { BΩ n } and { DΩ n } approximate SΩ, BΩ and DΩ respectively. Moreover, let SΩ n at some neighbourhood of its every point after suitable rotation of x axes has a representation via the sufficiently smooth function x 1 = φ n (x, t). More precisely, assume that SΩ in some neighbourhood of its point z 0 is represented by the function x 1 = φ(x, t), (x, t) P (µ 2 0 ) with some µ 0 > 0, where φ satisfies assumption A from Section 2. Then we also assume that SΩ n in some neighbourhood of its point z n =(x (n) 1, x0,t) is represented by the function x 1 = φ n (x, t), (x, t) P (µ 2 0 ), where {φ n } is a sequence of sufficiently smooth functions and φ n φ as n +, uniformly in P (µ 2 0 ). We can also asssume that φ n satisfies assumption A from Section 2 uniformly with

9 9 respect to n. Namely, the parabolic modulus of left-lower semicointinuity of the function φ n at the point (x 0,t 0 ) satisfies (2.1) uniformly with respect to n. We make a similar assumption also regarding the points z n =(x (n), 0) SΩ n on the bottom of the lateral boundary manifold. For arbitrary µ>0,δ >0 consider a cylinder R(µ, δ) ={(x, t) : x x 0 <µ 1, 0 <t<δ}. Assume that SΩ in some neighbourhood of its point z 0 =(x 0, 0) is represented by the continous function x 1 = φ(x, t), (x, t) R(µ 0,δ 0 ) with some µ 0 > 0,δ 0 > 0, where φ(x) φ(x, 0) satisfies assumption B (see (2.2)) from Section 2. Then we also assume that SΩ n in some neighbourhood of its point A n =(x (n) 1, x0, 0) is represented by the function x 1 = φ n (x, t), (x, t) R(µ 0,δ 0 ), where {φ n } is a sequence of sufficiently smooth functions and φ n φ as n, uniformly in R(µ 0,δ 0 ). We suppose that φ n satisfies (2.2) uniformly with respect to n. Assume also that for arbitrary compact subset Ω (0) of Ω DΩ, there exists a number n 0 which depends on the distance between Ω (0) and PΩ, such that Ω (0) Ω n DΩ n for n n 0. Let Ψ be a nonnegative and continuous function in IR N+1 which coincides with ψ on PΩ. This continuation is always possible. Next we take ψ n =Ψ+n 1,n=1, 2,... and consider a Dirichlet Problem (1.1), (1.2), in Ω n, with ψ replaced by ψ n. This is a nondegenerate parabolic problem and classical theory ([13,18,23]) imply the existence of a unique C 2+α solution. From maximum principle it follows that n 1 u n (x, t) M in Ω n,n=1, 2,... (3.1) where M is an upper bound for Ψ and ψ n,n=1, 2,... in some compact which contains Ω and Ω n,n=1, 2,... Next we take a sequence of compact subsets Ω (k) of Ω DΩ such that Ω= Ω (k), Ω (k) Ω (k+1),k =1, 2,... (3.2) k=1 By our construction, for each fixed k, there exists a number n k such that Ω (k) Ω n DΩn for n n k. It is a well-known result of the modern theory of degenerate parabolic equations (which includes (1.1) as a model example) that the sequence of uniformly bounded solutions u n,n n k to equation (1) is uniformly equicontinuous in a fixed compact Ω (k) (see e.g. [10,

10 10 Theorem 1 & Proposition 1 and Theorem 7.1]). From (3.2), by diagonalization argument and the Arzela-Ascoli theorem, we may find a subsequence n and a limit function ũ C(Ω DΩ) such that u n ũ as n +, pointwise in Ω DΩ and the convergence is uniform on compact subsets of Ω DΩ. Now consider a function u(x, t) such that u(x, t) =ũ(x, t) for (x, t) Ω DΩ,u(x, t) =ψ for (x, t) PΩ. Obviously the function u satisfies the integral identity (1.3). It is also continuous in BΩ, since above mentioned result on the equicontinuity of the sequence u n is true up to some neighbourhood of every point z BΩ [10, Theorem 6.1]. Hence, the constructed function u is a solution of the Dirichlet Problem (1.1), (1.2), if it is continuous in PΩ\BΩ. Step 2. Boundary regularity. Let z 0 =(x 0,t 0 ) SΩ. We shall prove that z 0 is regular, namely that lim u(z) =ψ(z 0 )asz z 0,z Ω DΩ (3.3) Without loss of generality assume that d(z 0 ) = 1. First, assume that t 0 = T. If 0 <ψ(z 0 ) < M, we shall prove that for arbitrary sufficiently small ε > 0 the following two inequalities are valid lim inf u(z) ψ(z 0 ) ε as z z 0,z Ω DΩ (3.4) lim sup u(z) ψ(z 0 )+ε as z z 0,z Ω DΩ (3.5) Since ε>0 is arbitrary, from (3.4) and (3.5), (3.3) follows. If ψ(z 0 ) = 0 (or respectively ψ(z 0 ) = M), however, then it is sufficient to prove (3.5) (respectively (3.4)), since (3.4) (respectively (3.5)) follows directly from the fact that 0 u M in Ω. Let ψ(z 0 ) > 0. Take an arbitrary ε (0,ψ(z 0 )) and prove (3.4). For arbitrary µ>0, consider a function w n (x, t) =f(ξ) M 1 (ξ/h(µ)) α, where ξ = h(µ)+φ n (x 0,T) x 1 µ[t t + ε 2 0 x x0 2 ], M 1 = ψ(z 0 ) ε, h(µ) =M 3 µ 1 F (µ 2 ), M 3 =[(M 2 /M 1 ) 1 α 1] 1,M 2 = ψ(z 0 ) ε/2,

11 11 and α is an arbitrary number such that α > m 1. If m > 1, then we assume also that α (m 1) 1. Then we set V n = {(x, t) :φ n (x, t) <x 1 <φ 1n (x, t), (x, t) P (µ 2 )}, φ 1n (x, t) =φ n (x, t)+(1+m 3 )µ 1 F (µ 2 ) µ[t t + ε 2 0 x x0 2 ], In the next lemma we clear the structure of V n. Lemma 3.1 If µ>0 is chosen such that F (µ 2 ) (1 + M 3 ) 1 then the parabolic boundary of V n consists of two boundary surfaces x 1 = φ n (x, t) and x 1 = φ 1n (x, t) (see Figure 1). Proof. We have φ 1n (x, t) φ n (x, t) =µ[δ + t T ε 2 0 x x0 2 ], δ =(1+M 3 )µ 2 F (µ 2 ) and δ (0,µ 2 ]ifµ is chosen as in Lemma 3.1. Then it easily follows that V n = V n, where V n = {(x, t) :φ n (x, t) <x 1 <φ 1n (x, t), (x, t) P (δ )}. Obviously, the assertion of lemma is true for Vn. Lemma is proved. In Figure 1 the domain V n is described in the particular case when φ n (x, t) 0,N =2,x 0 2 =0. t T x 1 = 0 x = (x, t) 1 1n T * x 1 x Figure 1. The domain V n in a particular case when φ n =0,N =2,x 0 2 =0.

12 12 In general, the structure of the domain V n coincides with that given in Figure 1 if we change the variable x 1 with the new one x 1 = x 1 φ n (x, t). More precisely, V n is a domain in IR N+1, lying in the strip T δ <t<t, its boundary consists of a single point lying on {t = T δ },a domain DV n lying on {t = T } and a connected manifold SV n lying in the strip {T δ <t T }. Boundary manifold SV n consists of two boundary surfaces x 1 = φ n (x, t) and x 1 = φ 1n (x, t). Our purpose is to estimate u n in V n via the barrier function w n = max (w n ;(2n) 1 ). Obviously, w n =(2n) 1 for x 1 θ n (x, t); w n = w n for x 1 <θ n (x, t), where θ n (x, t) =(1 (2M 1 n) 1 α )h(µ)+φ n (x 0,T) µ[t t + ε 2 0 x x0 2 ]. In the next lemma we estimate u n via the barrier function w n on the parabolic boundary of V n. For that the special structure of V n plays an important role. Namely, our barrier function takes the value (2n) 1, which is less than a minimal value of u n, on the part of the parabolic boundary of V n which lies in Ω n. Hence it is enough to compare u n and w n on the part of the lateral boundary of Ω n, which may easily be done in view of boundary condition for u n. Lemma 3.2 If µ>0 is chosen large enough, then u n > w n on SV n, for n n 1, (3.6) where n 1 = n 1 (ɛ) be some number depending on ɛ. Proof. From (2.1) it follows that for µ>0 being large enough θ n (x, t) φ 1n (x, t) < 0 for (x, t) P (µ 2 ), and hence w n =(2n) 1 for x 1 = φ 1n (x, t), (x, t) P (µ 2 ). (3.7) Without loss of generality, assume that n>m1 1. From (2.1) it also follows that

13 13 w n = f(h(µ)+φ n (x 0,T) φ n (x, t) µ[t t + ε 2 0 x x0 2 ]) f((m )h(µ)) = M 2 for x 1 = φ n (x, t), (x, t) P (µ 2 ), and hence w n M 2 for x 1 = φ n (x, t), (x, t) P (µ 2 ) (3.8) We can also easily estimate u n on SV n. To estimate u n x1 =φ n(x,t), first we choose n 1 = n 1 (ε) so large that for n n 1 Ψ x1 =φ n(x,t) > Ψ x1 =φ(x,t) ε 8 for (x, t) P (µ 2 0 ). This is possible in view of uniform convergence of {φ n } to φ in P (µ 2 0 ). Then we choose µ>0 large enough in order that Ψ x1 =φ(x,t) >ψ(z 0 ) ε 8 for (x, t) P (µ 2 ). If µ and n are chosen like this, then we have u n x1 =φ n(x,t) >ψ(z 0 ) ε 4 for (x, t) P (µ 2 ). (3.9) Thus, from (3.1) and (3.7) - (3.9), (3.6) follows. Lemma is proved. Lemma 3.3 If µ>0 is chosen large enough, then at the points of V n with x 1 <θ n (x, t), we have Lw n w nt wn m < 0 (3.10) Proof At the points of V n with x 1 <θ n (x, t), we have Lw n = µh 1 (µ)αm 1 α 1 f α 1 α h 2 (µ)αm(αm 1)M 2 α 1 f αm 2 α (1 + 4µ 2 ε 4 0 x x0 2 )+2µh 1 (µ)αmε 2 0 (N 1)M 1 α 1 f αm 1 α. (3.11) If m>1 then from (3.11) and (3.8) it follows that Lw n h 2 (µ)αm 1 α 1 f α 1 α S, (3.12)

14 14 S = M 3 F (µ 2 ) m(αm 1)M 1 α 1 f α(m 1) 1 α +2M 3 mε 2 0 (N 1)F (µ 2 ) f m 1 M 3 F (µ 2 ) m(αm 1)M 1 α 1 M m 1 1 α 2 +2M 3 mε 2 m 1 0 (N 1)M2 F (µ 2 ). Hence, if µ is chosen large enough, from (3.12), (3.10) follows. If 0 <m 1, then from (3.11) and (3.8) we derive that Lw n h 2 (µ)αm 1 α 1 f αm 1 α S, (3.13) S = M 3 F (µ 2 )f 1 m m(αm 1)M 1 α 1 f 1 α +2M3 mε 2 0 (N 1) F (µ 2 ) M 3 F (µ 2 )M 1 m 2 m(αm 1)M 1 α 1 M 1 α 2 +2M 3 mε 2 0 (N 1)F (µ 2 ). If µ is chosen large enough, from (3.13), (3.10) again follows. Lemma is proved. Thus w n is the maximum of two smooth subsolutions of the equation (1.1) in V n. By the standard maximum principle, from Lemma 3.1, (3.6) and (3.10) we easily derive that u n w n in V n, for n n 1. In the limit as n +, we have u w in V, (3.14) where w = max(w;0), in V w(x, t) =f(h(µ)+φ(x 0,T) x 1 µ[t t + ε 2 0 x x0 2 ]), V = {(x, t) :φ(x, t) <x 1 <φ 1 (x, t), x x 0 <ε 0 [δ + t T ] 1 2,T δ <t<t}, φ 1 (x, t) =φ(x, t)+(1+m 3 )µ 1 F (µ 2 ) µ[t t + ε 2 0 x x0 2 ]. Obviously, we have lim z z 0,z V w = lim w = ψ(z 0 ) ε z z 0,z Ω Hence, from (3.14), (3.4) follows.

15 15 Assume now that 0 ψ(z 0 ) <Mand prove (3.5) for an arbitrary ε>0 such that ψ(z 0 )+ε < M. For arbitrary µ>0 consider a function w n (x, t) =f 1 (ξ) [M 1 α + ξh 1 (µ)(m 1 α 4 M 1 α )] α, where ξ is defined as before and h(µ) =M 6 µ 1 F (µ 2 ),M 4 = ψ(z 0 )+ε, M 5 = ψ(z 0 )+ε/2,m 6 =(M 1 α M 1 α 4 )(M 1 α 4 M 1 α 5 ) 1, and α is an arbitrary number such that 0 <α<min (1; m 1 ). Similarly, consider the domains V n (with M 3 replaced by M 6 in the expression of φ 1n (x, t) and δ ) and V n (see Lemma 3.1). We then construct an upper barrier function as follows: w n = min (w n ; M). Obviously, w n = M for x 1 θ n (x, t); w n = w n for x 1 <θ n (x, t). where θ n (x, t) =h(µ)+φ n (x 0,T) µ[t t + ε 2 0 x x0 2 ]. Next, we prove an analog of the Lemma 3.2. Lemma 3.4 If µ>0 is chosen large enough, then u n w n on SV n, for n n 1, (3.15) where n 1 = n 1 (ɛ) be some number depending on ɛ. Proof. From (2.1) it follows that for µ>0 being large enough θ n (x, t) φ 1n (x, t) 0 for (x, t) P (µ 2 ), and hence w n = M for x 1 = φ 1n (x, t), (x, t) P (µ 2 ). (3.16)

16 16 From (2.1) it also follows that w n = f 1 (h(µ)+φ n (x 0,T) φ n (x, t) µ[t t + ε 2 0 x x0 2 ]) f 1 ((M )h(µ)) = M 5 for x 1 = φ n (x, t), (x, t) P (µ 2 ), and hence w n M 5 for x 1 = φ n (x, t)), (x, t) P (µ 2 ). (3.17) Similarly, as in (3.9), we can establish that if µ>0 is large enough and n n 1 (ε) then u n x1 =φ n(x,t) <ψ(z 0 )+ ε 4 for (x, t) P (µ 2 ). (3.18) Thus, from (3.1) and (3.16) - (3.18), (3.15) follows. Lemma is proved. The next lemma is analog of the Lemma 3.3. Lemma 3.5 If µ>0 is chosen large enough, then at the points of V n with x 1 <θ n (x, t), we have Lw n > 0. (3.19) Proof. By using (3.17), at the points of V n with x 1 <θ n (x, t), we have Lw n = µh 1 (µ)α(m 1 1 α 1 α M α α 4 )f1 + mα(1 αm)h 2 (µ) (M 1 1 α M α 4 )2 f αm 2 α 1 (1 + 4µ 2 ε 4 0 x x0 2 ) 2µh 1 (µ)αmε 2 0 (N 1)(M 1 1 α M α 4 αm 1 α )f 1 h 2 (µ)α(m 1 1 α M α 4 )S, (3.20) S = M 6 M α 1 α 5 F (µ 2 )+m(1 αm)(m 1 1 α M α 4 αm 2 )M α 2M 6 mε 2 0 αm 1 α (N 1)M5 F (µ 2 ). Hence, if µ is chosen large enough, from (3.20), (3.19) follows. Lemma is proved. Thus w n is the minimum of two smooth supersolutions of the equation (1) in V n. By the standard maximum principle, from Lemma 3.1, (3.15) and (3.19) we easily derive that u n w n in V n, for n n 1. In the limit as n, we have u w in V, (3.21)

17 17 where w = min(w; M) inv w(x, t) =f 1 (h(µ)+φ(x 0,T) x 1 µ[t t + ε 2 0 x x0 2 ]) and the domain V being defined as in (3.14). Obviously, we have lim z z 0,z V w = lim w = ψ(z 0 )+ε z z 0,z Ω Hence, from (3.21), (3.5) follows. Thus we have proved (3.3) for z 0 =(x 0,T) SΩ when d(z 0 ) = 1. The proof is similar when d(z 0 )= 1. Suppose now that z 0 =(x 0,t 0 ) SΩ with t 0 <T. Clearly, the same proof given in the case t 0 = T, implies the regularity of z 0 regarding subdomain Ω =Ω {t<t 0 }. Namely, (3.3) is valid for z Ω. Hence, it is enough to prove (3.3) for z Ω +, Ω + =Ω {t>t 0 }. The proof of this latter, however, is equivalent to the proof of regularity of the point z 0 =(x 0, 0) SΩ under the assumption B. That easily follows from the fact that assumption B (with redefined φ(x) φ(x, t 0 )) is a consequence of the assumption A. Thus, to complete the proof, it remains just to prove (3.3) for z 0 =(x 0, 0) SΩ. The proof is similar to that given above. Without loss of generality assume again that d(z 0 ) = 1. Let ψ(z 0 ) > 0. Take an arbitrary ε (0,ψ(z 0 )) and prove (3.4). For arbitrary µ>µ 0 consider a function w n (x, t) =f(ξ) M 1 (ξ/h(µ)) α, where ξ = h(µ)+φ n (x 0, 0) x 1 + µ[t x x 0 2 ], M 1 = ψ(z 0 ) ε, h(µ) =M 3 µ 1 F 1 (µ 1 ), M 2 = ψ(z 0 ) ε/2,m 3 = 4[(M 2 /M 1 ) 1 α 1] 1 and α is an arbitrary number such that α > m 1. If m > 1, then we assume also that α (m 1) 1. Then we set

18 18 V n = {(x, t) :φ n (x, t) <x 1 <φ 1n (x, t), (x, t) R(µ, δ)} φ 1n (x, t) =φ n (x 0, 0) + (1 (2M 1 n) 1 α )h(µ)+µ[t x x 0 2 ], where δ = δ(µ) (0,δ ],δ = min(δ 1,δ 2 ),δ 1 =2µ 2 F 1 (µ 1 ) and δ 2 = δ 2 (µ) (0,δ 0 ] is chosen such that φ n (x, 0) φ n (x, t) µ 1 F 1 (µ 1 ) for (x, t) R(µ 0,δ 2 ) (3.22) and for n n 2 (µ). The existence of δ 2 and n 2 follow from the following proposition. Proposition 3.1 For arbitrary µ>µ 0 there exists δ 2 = δ 2 (µ) (0,δ 0 ] and n 2 = n 2 (µ) such that (3.22) is valid for n n 2. Proof. Since {φ n } converges to φ uniformly in R(µ 0,δ 0 ), for arbitrary µ>µ 0 there exists a number n 2 = n 2 (µ) such that for n n 2, we have φ n (x, 0) φ n (x, t) φ(x, 0) φ(x, t)+ 1 2 µ 1 F 1 (µ 1 )inr(µ 0,δ o ) (3.23) Since φ is uniformly continuous in R(µ 0,δ 0 ), there also exists a number δ 2 = δ 2 (µ) (0,δ 0 ] such that φ(x, 0) φ(x, t) 1 2 µ 1 F 1 (µ 1 )inr(µ 0,δ 2 ) (3.24) From (3.23) and (3.24), (3.22) follows. Proposition is proved. Furthermore we shall always suppose that n max (n 2 ; M 1 1 ). If µ>µ 0 is chosen large enough, from (2.2) and (3.22) it follows that φ 1n (x, t) φ n (x, t) <φ n (x 0, 0) φ n (x, 0) + φ n (x, 0) φ n (x, t)+h(µ)+µδ 1 µ 1 µ 1 [(M 3 +4)F 1 (µ 1 ) 1] < 0 for x x 0 = µ 1, 0 t δ. Thus, the parabolic boundary of V n consists of two boundary surfaces x 1 = φ n (x, t),x 1 = φ 1n (x, t), and of the closure of a domain V 0 n = {(x, 0) : φ n (x, 0) <x 1 <φ 1n (x, 0), x x 0 <µ 1 }. In the next lemma, which is analog of the Lemma 3.2, we estimate u n via the barrier function w n on the parabolic boundary PV n of V n.

19 19 Lemma 3.6 If µ>0 is chosen large enough, then u n >w n on PV n, for n n 4, (3.25) where n 4 = n 4 (ɛ, µ) be some number depending on ɛ and µ. Proof. We have w n =(2n) 1 for x 1 = φ 1n (x, t). (3.26) From (2.2) and (3.22) it also follows that if µ is chosen large enough, then w n = f(h(µ)+φ n (x 0, 0) φ n (x, t)+µt µ x x 0 2 ) f(h(µ)+φ n (x 0, 0) φ n (x, 0) + φ n (x, 0) φ n (x, t)+µδ 1 ) f((4m )h(µ)) = M 2 for x 1 = φ n (x, t), (x, t) R(µ, δ). (3.27) From (3.27) it also follows that ω n = f(h(µ)+φ n (x 0, 0) x 1 µ x x 0 2 ) f(h(µ)+φ n (x 0, 0) φ n (x, 0)) M 2 inv 0 n (3.28) We can also easily estimate u n on PV n. To estimate u n x1 =φ n(x,t), first we choose n 3 = n 3 (ε) so large that for n n 3 Ψ x1 =φ n(x,t) > Ψ x1 =φ(x,t) ε 8 for (x, t) R(µ 0,δ 0 ). This is possible in view of uniform convergence of {φ n } to φ in R(µ 0,δ 0 ). Then we choose µ>0 large enough and δ = δ(µ) > 0 small enough in order that Ψ x1 =φ(x,t) >ψ(z 0 ) ε 8 for (x, t) R(µ, δ) and hence, u n x1 =φ n(x,t) >ψ(z 0 ) ε 4 for (x, t) R(µ, δ) (3.29) Similarly, we can establish that if µ>0 is chosen large enough, there exists a number n 3 (ε) such that for n n 3 we have u n >ψ(z 0 ) ε 4 in V 0 n (3.30) Thus, if we take n 4 = max (n 2 ; n 3 ; M1 1 ), then from (3.1) and (3.26) - (3.30), (3.25) follows. The lemma is proved.

20 20 The next step consists in proving that for µ>0 being large enough Lw n < 0inV n. The proof coincides with that given above in Lemma 3.3. As before, by the standard maximum principle we then easily derive that u n w n in V n, for n n 4. In the limit as n, we have u w in V, (3.31) where w(x, t) =f(h(µ)+φ(x 0, 0) x 1 + µt µ x x 0 2 ), V = {(x, t) :φ(x, t) <x 1 <φ 1 (x, t), (x, t) R(µ, δ)}, φ 1 (x, t) =φ(x 0, 0) + h(µ)+µ[t x x 0 2 ]. Obviously, we have lim w = lim w = ψ(z 0 ) ε z z 0,z V z z 0,z Ω Hence, from (3.31), (3.4) follows. To complete the proof it remains to prove (3.5) when 0 ψ(z 0 ) <M. To do that, we consider a barrier function w n (x, t) =f 1 (ξ) [M 1 α + ξh 1 (µ)(m 1 α 4 M 1 α )] α, where ξ is defined as before and h(µ) =M 6 µ 1 F 1 (µ 1 ),M 4 = ψ(z 0 )+ε, M 5 = ψ(z 0 )+ε/2,m 6 =4(M 1 α M 1 α 4 )(M 1 α 4 M 1 α 5 ) 1 and α is an arbitrary number such that 0 <α<min(1; m 1 ). The rest of the proof of (3.5) is similar to the given proof of (3.4) and to that given above in the case when t 0 = T, therefore we omit it. Thus we have completed the proof of the boundary continuity of the constructed limit solution. The Theorem 2.1 is proved. Corollary 2.1 is immediate (see Section 2).

21 21 Remark 3.1 The proof of the Theorem 2.1 in the case of the more general equation (1.4) almost coincides with that given above for the equation (1.1). The main technical difference consists in choosing an exponent α in respective barrier functions w n. In this more general case it depends on the parameters m, β and γ. It should be also mentioned that if c>0 and β>1in (1.4), we have to prevent blow up, say, imposing a restriction on the length of the time intervall: T (0,T ),T = M 1 β /c(β 1),M = sup ψ>0. Within the Step 1 for the construction of the sequence u n we consider the following regularized equation u t = a u m + b u γ + cu β cθ c n β, (3.32) where θ c =(1, if c < 0; 0, if c 0). We then consider the DP in Ω n for the equation (3.32) with the initial boundary data ψ n =Ψ+n 1,n=1, 2,... As before, the classical theory implies the existence of a unique classical solution u n which satisfies n 1 u n (x, t) ψ 1 (t) inω n where [ 1/(1 β) ψ 1 M 1 β + c(1 θ c )(1 β)t], if β 1, (t) = M exp (c(1 θ c )t), if β =1. The rest of the proof almost completely coincides with that given above for equation (1.1). Slight technical modifications are similar to those made in the one-dimensional case [4]. Remark 3.2 One may show by standard methods that the weak solution to DP is a classical solution in a neighbourhood of any interior point z Ω, where u(z) > 0.

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