The Singular Diffusion Equation with Boundary Degeneracy

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1 The Singular Diffusion Equation with Boundary Degeneracy QINGMEI XIE Jimei University School of Sciences Xiamen, 36 China HUASHUI ZHAN Jimei University School of Sciences Xiamen, 36 China Abstract: For the heat conduction on a bounded domain with boundary degeneracy, though its diffusion coefficient vanishes on the boundary, it is still possible that the heat flux may transfer across the boundary A known result shows that the key role is the ratio of the diffusion coefficient near the boundary If this ratio is large enough, the heat flux transference has not any relation to the boundary condition but is completely controlled by the initial value This phenomena shows there are some essential differences between the heat flux with boundary degeneracy and that without boundary degeneracy However, under the assumption on the uniqueness of the weak solutions, the paper obtains that the weak solution of the singular diffusion equation with boundary degeneracy, has the same regular properties as the solution of a singular diffusion equation without boundary degeneracy Key Words: Boundary degeneracy, Diffusion equation, Uniqueness, Regular property Introduction In this paper, we study the singular diffusion equation with boundary degeneracy t div(dα u p u), () where (x, t) Q T Ω (, T ), Ω is a bounded domain with appropriately smooth boundary, p >, α >, and d d(x) dist(x, Ω) If α, then the equation () becomes the following classical evolutionary p Laplacian equation t div( u p u) () This equation reflects the more practical process of heat conduction than the classical heat conduction equation u t u does For example, when p >, the solution of the equation () may possess the properties of finite speed of propagation, while u t u always has the properties of infinite speed of propagation and it seems clearly contrary to the practice There are a tremendous amount of related works for equation (), one refers to the reference [8]-[3] and the books [], [6], [7] etc and the reference therein The authors also had done some research for this equation in []-[6] However, unlike the equation (), there is few reference related to the equation () except []-[3] The works []-[3] only studied the Corresponding Author well-posedness of the equation (), while the properties of the corresponding weak solutions, such as the Harnack inequality, the large time behavior etc, are still undone For the equation (), the diffusion coefficient depends on the distance function to the boundary Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary However the reference [] shows that the fact might not coincide with what we image In fact, the exponent α, which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary Let us give the definition of weak solution for equation () as follows: Definition If the function u(x, t) satisfies u C(, T ; L (Ω)) L (Q T ), t L (Q T ), d α u p L (Q T ), and for any test function ϕ C (Q T ), the following integral equality holds Q T ( t ϕ + dα u p u ϕ)dxdt (3) then the function u(x, t) is said to be a weak solution of the equation () E-ISSN: Issue, Volume, February

2 According to reference [], if < α < p, we can impose the Dirichlet boundary condition as usual u(x, t) g(x, t), (x, t) Ω (, T ) () Here and in what follows, we always assume that g(x, t) is a function which can be extended to Q T and the extended function is appropriately smooth If α p, then the heat conduction of equation () is entirely free from the limitations of the boundary condition In other words, the problem of heat conduction is entirely controlled by the initial condition u(x, ) u (x), x Ω (5) In details, the reference [] had got the following important propositions Proposition If < α < p, then for any u satisfying u (x) L (Ω), d α u p L (Ω) (6) there exists at least one weak solution of the initial boundary problem ()-()-(5) Moreover, the solution of the problem is unique Proposition 3 If α p, then the equation () admits at most one weak solution with initial value u, no matter what the boundary value is Moreover, for any u as in proposition, there exists at least one weak solution of the equation () with initial value (5) These two propositions show that there are some essential differences between the equation () and the equation () In this paper, we will discuss the regularity of the solution for the equation () If the equation () exists unique a solution, we will show that its weak solution has the same regular properties as that of the solution for the equation () In turn, it shows that, although the diffusion coefficient degenerating on boundary has directly impact on the heat flow cross-border situation, in terms of smoothness of solutions, whether the diffusion coefficient degenerating on the boundary or not does not play an essential role According to [], the distance function is always almost everywhere derivable and d is true in the distribution sense In what follows, for simplicity, we assume that the distance function d(x) dist(x, Ω) is a derivable function for x Ω The gradient bounded properties of the solution Before to prove the theorem, we will introduce the following lemmas from reference [] Lemma There exists a constant γ only depending on p, q, N such that for any v V q,p (Ω T ) and h p(q+n) N, ( ) v(x, t) h dxdt γ v(x, t) p dxdt Ω T Ω T ( ) p ess sup v(x, t) q dx where V q,p (Ω T ) is the closure of C (Ω T ) in space of V q,p (Ω T ), and V q,p (Ω T ) L (, T ; L q (Ω)) L p (, T ; W,p (Ω)) Lemma 5 Let (n,, ) be a sequence of bounded open sets in Ω T, + If for any q, v L q (Ω T ) and there exist some constants α, λ, C, C >, K >, such that the following inequality holds, v α+λkn+ dxdt + Then ( K C C n v dxdt) α+λkn ess sup v ( C holds, where K K for a n N + Ω C v α +λk n dxdt C C K, K nk (n n), nn ) λk n By the above two lemmas, we are able to get the following theorem Theorem 6 Let Ω be a uniformly C domain If u is the unique solution of the equation () in Q T, and p > max{, N N+ }, then x i L loc(q T ), i,,, N (7) N, E-ISSN: Issue, Volume, February

3 Proof: Since u is the unique solution of equation () in Q T, then we can assume that u is the limitation of following regulation equation s solutions t div[(d + n )α ( u + n ) p u] (8) (x, t) Q T Ω (, T ), u(x, ) u,n (x), x Ω, (9) u(x, t) g(x, t), (x, t) Ω (, T ), () where u,n (x) is the smoothly mollified functions of u (x) Let K Q T be an any compact set Similar to the proof Lemmas 3 in the Chapter of reference [], which discusses the equation (), we are able to get u n q dxdt C(q, K, p, N) () K Denote B ρ (x ) {x : x x < ρ}, and for simplicity, denote u n as u Now, we differentiate (8) about x j, and obtain xj t Let αdiv[(d + n )α d xj ( u + n ) p u] +[(d + n )α ( u + n ) p uxi ] xi x j () T n t t n+, ρ n ρ + ρ n, ρ n (ρ n + ρ n+ ) ρ + B n B ρn (x ), B n B ρn (x ), B n (T n, T ), Q n B n (T n+, T ) 3ρ n+, Assume that n is the cut off functions smoothly in, n (, t) C (B n), then n, t T n, n, (x, t) Q n n n+ ρ, nt n+3 t Let x Ω, B ρ (x ) B ρ Ω We choose v u + n, multiply the two sides of () with nv β u xj, and integrate on ( t, t), then we can obtain the following equalities t t t t α t t t t t t t t α + t t nv β u xj xj t dtdx t t (β + ) B v β+ n dtdx ρ t (β + ) nv β+ dtdx t t nt n v β+ dtdx, (3) β + nv β u xj [(d + n )α v p uxi ] xix j dxdt n[(d+ n )α v p uxi ] xj [v β u xj ] xi dxdt n(d + n )α (v p uxi ) xj (v β u xj ) xi dxdt t t nxi n v β u xj [(d + n )α v p uxi ] xj dxdt nd xj (d + n )α v p uxi (v β u xj ) xi dxdt n nxi (d+ n )α (v p uxi ) xj v β u xj dxdt n nxi d xj (d+ n )α v p uxi v β u xj dxdt () (β + ) nv β+ dxdt B ρ t α t t t t n(d + n )α (v p uxi ) xj (v β u xj ) xi dxdt t t α t t + α t + β + nd xj (d + n )α v p uxi (v β u xj ) xi dxdt n nxi (d + n )α (v p uxi ) xj v β u xj dxdt n nxi d xj (d + n )α v p uxi v β u xj dxdt nv β u xj [(d + n )α d xj v p uxi ] xi dxdt t Using the fact that t n nt v β+ dxdt (5) d xj d, x Ω (6) and by Young inequality, we have v p uxi v β u xj εv p+β u + c(ε)v p+β, E-ISSN: Issue, Volume, February

4 and v p uxi v β u xi x j εv p+β u xi x j u xi x j + c(ε)v p+β, nxi v β u xj (v p uxi ) xj ε n v p+β +εv p+β v + εv p+β u xi x j u xi x j (v p uxi ) xj (v β u xj ) xi v p+β u xi x j u xi x j + + p + β v p+β v β(p ) v p+β 6 ( u v) From these formulas and (5), we can obtain (β + ) nv β+ dx t +( c(ε)) t nv p+β u xi x j u xi x j dxdt +( p+β t c(ε)) t nv p+β v dxdt B ρ β(p ) t + t nv p+β 6 ( u v) dxdt B ρ t c(ε) v p+β ( + t n )dxdt + c t t n v β+ dxdt (7) β + where ε is a appropriately small positive constant Case ) When p, denote From (7), we can obtain C sup [ T n<t<t w v p+β (β + ), λ p + β ( Bn λ n w) λ dx + ( Qn λ n w ) dxdt (+ n ρ ) w dxdt+ n t by Lemma and (8), C w + λ + [ (+ n ρ ) N dxdτ w dxdt+ n t w λ dxdt ],(8) ] k w λ dxdt, which implies that v p+β + β+ N dxdτ + [ C ( + n ρ ) v p+β dxdτ + n t v β+ dxdτ ] + N, where k + N By choosing β kn, the above formula can be changed into v p + kn+ dxdτ [ + C ( + n ρ ) v p + kn dxdτ If + n t n t v k n v k n dxdτ ] k (9) dxdτ ( + n ρ ) v p + kn dxdτ, () then by Hölder inequality, we see that v p + kn ρ dxdτ ( ) p +k n p mes, t which implies that sup v ( ρ ) p () B ρ ( t t,t ) Then we can obtain Theorem 6 If () isn t true, we have v p + kn+ dxdτ + C [ By Lemma 5, sup v Cρ (N+) ] k ( + n ρ ) v p + kn dxdτ v p + kn dxdτ, () where n is a positive integer which makes k n > hold Then we can obtain Theorem 6 according to () Therefore, when p >, i L loc(q T ) E-ISSN: Issue, Volume, February

5 and Case ) When p <, we can get v p+β β(p ) N j u xj v p+β v t β(p ) t Then according to (7), nv p+β 6 ( u v) dxdτ nv p+β u dxdτ sup nv β+ (x, τ)dx t <t<t + nv αp v dxdτ C +C Q(ρ, t ) Q(ρ, t ) Q(ρ, t ) v αp+ n dxdτ n nt v β+ dxdτ, where α p p+α Then using Lemma, similar to the discussion of the case (), we can obtain If [ C n t v N( p) + kn+ dxdτ + v N( p) + kn dxdτ ] k + C(+ n ρ ) v (N )( p) + kn dxdτ (3) ( + n ρ ) n t then by Hölder inequality, we have which implies that v (N )( p) + kn dxdτ v N( p) + kn dxdτ, () v N( p) + kn dxdτ ( t + ρ ρ ) N( p)+k n ( p) mes, sup v ( t B ρ ( t ρ,t ) ) p If () isn t true, then from (3), we have [ + v N( p) + kn+ dxdτ ] k γ( + n ρ ) v N( p)+ k n dxdτ Then using Lemma 5, we obtain sup v B ρ ( t,t ) ] C [ρ (N+) v N( p) + kn k n dxdτ Also we can obtain Theorem 6 according to () Therefore, when p <, also i L loc(q T ) The proof of Theorem 6 is complete 3 The continuity of the solution Theorem 7 Supposed that u is a weak solution of equation () in Q T, then for any compact set K Q T, (x, t ), (x, t ) K, u(x, t ) u(x, t ) c( x x + t t ), (5) where c is a constant only dependent on N, p, d(k, Ω) and u L (K) Proof: Obviously we only need to prove that u satisfies (5) in B R (t, T ), for any R > such that B R Ω, t (, T ) Let u ε be the usual mollified function of u, u ε (x, t) J ε u(x, t) T j ε (x y, t τ)u(y, τ)dydτ, where < ε < t < t < T ε Then for any x, x B R, we obtain ) u ε (x, t) u ε (x, t) T j ε (x y, t τ)u(y, τ)dydτ R N T j ε (x y, t τ)u(y, τ)dydτ R N T [j ε (x y, t τ) j ε (x y, t τ)] u(y, τ)dydτ E-ISSN: Issue, Volume, February

6 T T T T According to Theorem 6, we have d[j ε (sx + ( s)x y, t τ)] ds u(y, τ)dsdydτ x [j ε (sx + ( s)x y, t τ)] u(y, τ)dsdydτ(x x ) y [j ε (sx + ( s)x y, t τ)] u(y, τ)dsdydτ(x x ) j ε (sx + ( s)x y, t τ) y u(y, τ)dsdydτ(x x ) (6) u ε (x, t) u ε (x, t) T j ε (sx + ( s)x y, t τ) y u(y, τ) dsdydτ x x c x x, (7) where and in what follows, c is a constant independent of ε ) Let < ε < t < t < t < T, B ( t) (x ), ϕ C (), x B R, t t t Then t ϕ(x)[u ε (x, t ) u ε (x, t )]dx ϕ(x) du ε (x, st + ( s)t ) dsdx ds T ϕ(x) j εt (x y, st + ( s)t τ)u(y, τ)dydτdsdx T t ϕ(x) j ετ (x y, st + ( s)t τ)u(y, τ)dydτdsdx For fixed (x, t) Q t, < ε < t < t < T ε, J ε (x y, t τ) C (Q T ) Let us choose the test function in the definition of generalized solution (6) as ϕ(x) J ε (x y, t τ) Q T ( t ϕ + dα u p u ϕ)dxdt Then, we have T T j ετ (x y, st + ( s)t τ)u(y, τ)dydτ d α y u p y u y j ε (x y, st + ( s)t τ)dydτ Thus, we obtain ϕ(x)[u ε (x, t ) u ε (x, t )]dx T t ϕ(x) d α y u p y u y j ε (x y, st + ( s)t τ)dydτdsdx T t d α y u p y u x ϕj ε (x y, st + ( s)t τ)dydτdsdx T t x ϕ j ε (x y, st + ( s)t τ)d α y u p y udydτdxds t x ϕj ε (d α y u p y u) Let δ(s) C (R) satisfy δ(s), (x, st + ( s)t )dxds (8) R δ(s)ds and δ(s) when s For any h >, we define δ h (s) δ( s h ) h By a approaching process, we know that (8) is also true for any ϕ W, () Choosing ϕ ϕ h (x) in (8), then ( t) x x h h δ h (s)ds ϕ h (x)(u ε (x, t ) u ε (x, t ))dx t δ h (( t) x x h) xi x i x x But J ε (d α y u p y u)(x, st + ( s)t )dxds We notice that, when x, lim ϕ h(x) h δ h (( t) x x h) (9) holds when x x < ( t) h, then δ h c h and mes(\b ( t) h (x )) ch( t) N E-ISSN: Issue, Volume, February

7 Therefore using Theorem 6 and (9), we can obtain ϕ h (x)[u ε (x, t ) u ε (x, t )]dx Letting h, we obtain [u ε (x, t ) u ε (x, t )]dx c( t) N+ c( t) N+ Then by the mean value theorem, there exist x such that Noticing that u ε (x, t ) u ε (x, t ) c( t) u ε (x, t ) u ε (x, t ) u ε (x, t ) u ε (x, t ) + u ε (x, t ) u ε (x, t ) + u ε (x, t ) u ε (x, t ) c( t), combining this formula with (6), letting ε, we obtain the following formula u(x, t ) u(x, t ) u(x, t ) u(x, t ) + u(x, t ) u(x, t ) c( x x + t t ) The proof is complete Continuity of the gradient of the solution In what follows, let u be the solution of () and let P (x, t ) Q T, < R, µ > If we denote Q µ (P, R) { M ± iµ (x, t) : x x < R, t R µ p < t < t (R) ess sup Q µ(p,r) M µ (R) max (±u xi ), i,,, N ess sup i N Q µ(p,r) u xi, then we can get the following propositions similarly as Chapter in [], we omit the details here Proposition 8 Assume that M + µ (R) M µ(r) }, and µ satisfies M + µ (R) µ M µ(r) Then there exists ε ε (p, N) such that, when mesq u (P, R) it holds that Q µ (P, R)(M + µ (R) u x ) dxdt ess sup u x M µ (R ) Q µ(p, R ) Proposition 9 Assume that and µ satisfies + M + µ (R) M µ(r) M + µ (R) µ M µ(r) ε (M + µ (R), Then for any ε >, there always exist some constants λ, β (, ) which depend on p, N, ε such that, if mesq u (P, R) Q µ (P, R)(M + µ (R) u x ) dxdt > ε (M + µ ), then } mes {(x, t) Q µ (P, R) : u x (x, t) ( β)m µ + (R) Proposition Assume that and µ satisfies M + µ (R) M µ(r) M + µ (R) µ M µ(r) > λmesq µ (P, R) Further assume that there exist some constants λ, β (, ) such that, if } mes {(x, t) Q µ (P, R) : u x (x, t) ( β)m µ + (R) λmesq µ (P, R), then there exist constants δ, γ (, ) which depend on p, N, λ, β such that M + µ (δr) γm + µ (R) E-ISSN: Issue, Volume, February

8 N Theorem Let p > max{, N+ }, u is the generalized solution of equation () in Q T, then u xj (j,,, N) is locally Hölder continuous in Q T Proof: The proof of the theorem includes three steps Firstly, choose the ε on Proposition 8 Secondly, determine λ, β on Proposition 9 according to ε Lastly, determine δ, γ on Proposition according to λ, β In particular, δ, γ is just dependent on N, p Choosing s (, ), which is close to, such that δ ( s) s(p ) > max{, γ} (3) Letting < η < T, and Ω η T Ω (η, T ), Ω η T Q T be a bounded open set Then we obtain that u is bounded in Ω η T according to Theorem 6 We may assume u Ωη T M (3) Denote M M δ ( s) s(p ), then M Mδ ( s) s(p ) Choose R (, ] such that Q M (P, R ) Ω η T For any < R R, denote t R R s R s (M ) p, ˆQ(P, R) {(x, t); x x < R, t t R < t < t }, M ± i (R) ess sup (±u xi ), i,,, N; ˆQ(P,R) M(R) max ess sup u xi, i N ˆQ(P,R) osc ˆQ(P,R) u x i ess sup u xi ess inf u xi ˆQ(P,R) ˆQ(P,R) M + i (R) + M i (R) In what follows, we will prove that there exist constants ρ (, ) and C > which only depend on N, p, such that osc ˆQ(P,R) u x i CM ( R R ) ρ, < R < R (3) By (3), we immediately know that u xi (i,, N) is Hölder continue in Ω η T, and know that the theorem is true Now we prove (3) Define R sup{r [, R ]; exist j N, θ {+, }, such that M θ j (R) M ( R R ) s p } (33) Assume R > without loss of the generality Otherwise, M θ j (R) < M ( R R ) s p, then we obtain j N, θ {+, }, osc ˆQ(P,R) u x i M + i (R) + M i (R) < M ( R R ) ρ, where ρ s p Therefore formula (3) holds naturally Now according to the definition of M, M, we can obtain < R δ s R < R by (33) Therefore there exists R and δ s R < R < R < R, such that M ± j (R ) M ( R R ) s p j,, N (3) At the same time, there exist i, θ, without loss of the generality, we can choose i, θ +, such that Denote M + (δ s R ) > M ( δ s R ) s p (35) R µ M ( R R ) s p (36) We will prove the following formula, Q µ (P, R )(M + (R ) u x ) dxdt ε (M + (R ) mesq µ (P, R) (37) According to the definition of µ, we clearly have Q µ (P, R ) ˆQ(P, R ), M + µ (R ) M + (R ), therefore we can obtain following formula from (35), (36) and (3) M µ + (R ) M + (δ s R ) > max{γ, }µ (38) Then according to (3) and (36), we obtain M + µ (R ) > µ M µ (R ) (39) Let us prove (37) now If (37) isn t true, then according to Proposition 9 and Proposition, we obtain M + µ (δr ) γm + µ (R ) Noticing that Q µ (P, δr ) ˆQ(P, δ s R ), then we can obtain M + (δ s R ) M + µ (δr ) γm + µ (R ) γµ E-ISSN: Issue, Volume, February

9 But this conflicts (38), therefore formula (37) holds Now, according to (38) and Proposition 8, we have ess sup u x M µ + (R ) Q µ(p, R ) µ () As follows, according to (), we will prove that u xm satisfies (3) Differentiate () about x m, obtain xm t αdiv(d α d xm u p u) +d α ( u p u xi ) xi x m The above formula can be changed into where d α xm t (a ij u p u xmx j ) xi αd α div(d α d xm u p u), () a ij δ ij + (p )u x i u xj u, and δ ij is the Kronecker sign as usual Let x x, τ d α µ p (t t ), v(, τ) u xm (x, t), Q (R) {(, τ); < R, R < τ } Then v satisfies v τ [a u p ij µ p v j ] i αd α div(d α d xm u p u) () Therefore according to (3) and (), we obtain C η a ij ( u µ )p η i η j C η, η, (, τ) Q ( R ) Since Ω is bounded domain with appropriately smooth boundary, d(x) dist(x, Ω) is bounded Then according to () (6) and Theorem 6, we obtain αd α div(d α d xm u p u) is bounded This can explain that the formula () is uniformly parabolic in Q ( R ), then according to reference [5] we obtain osc Q (R) v C( R )osc R Q R ( ) v, < R < R, where C >, β (, ) is just dependent on N, p Backing to the variables (x, t), we obtain osc Qu(p,R)u xm C( R R ) βosc Qu(p, R ) u x m u xm, < R < R, m, N (3) ) If R R, according to the definition of R, we obtain osc Qu(p,R)u xm M + m(r) + M m(r) ) If R R R, then M ( R R ) s p () osc ˆQu(p,R) u x m osc ˆQu(p,R) u x m M ( R ) s p R (5) 3) If < R < R, then according to (3) and (5), we obtain osc Qu(p,R)u xm C( R R ) βm ( R R ) s p Letting ρ min{ s β, p }, we obtain osc Qu(p,R)u xm CM ( R R ) ρ ( R R ) ρ CM ( R R ) ρ Since ˆQ(P, R) Q u (p, R), the formula (3) holds According to ), ), 3), we obtain Theorem immediately The proof is complete Acknowledgements: The research was supported by the NSF of Fujian Province of China and supported by SF of Pan-Jinlong in Jimei University, China References: [] Wu Z, Zhao J, Yun J and Li F, Nonlinear Diffusion Equations, New York, Singapore: World Scientific Publishing, [] Yin J and Wang Ch, Properties of the boundary flux of a singular diffusion process, Chin Ann Math, 5B: (), 75-8 [3] Yin J and Wang Ch, Evolutionary weighted p- Laplacian with boundary degeneracy, J of Diff Equa, 37(7), -5 [] Karcher H, Riemannian comparison construction, Global Diff Geom, MAA Studies in Math, 7(989), 7- [5] Gu Liankun, Partial differential equations of second-order parabolic, Xiamen University Publishing, E-ISSN: Issue, Volume, February

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