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Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 739097, 23 pages doi:10.1155/2009/739097 Research Article On Initial Boundary Value Problems with Equivalued Surface for Nonlinear Parabolic Equations Fengquan Li Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Correspondence should be addressed to Fengquan Li, fqli@dlut.edu.cn Received 6 January 2009; Revised 12 March 2009; Accepted 22 May 2009 Recommended by Sandro Salsa We will use the concept of renormalized solution to initial boundary value problems with equivalued surface for nonlinear parabolic equations, discuss the existence and uniqueness of renormalized solution, and give the relation between renormalized solutions and weak solutions. Copyright q 2009 Fengquan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let Ω ⊂ R N N ≥ 2 be a bounded domain with Lipschitz boundary ∂ΩΓ. T is a fixed positive constant, Q Ω× 0,T. We consider the following nonlinear parabolic boundary value problems with equivalued surface: ∂u ∂t − N  i,j1 ∂ ∂x i  a ij  x, u  ∂u ∂x j   f  x, t  in Q, u  C  t  a function of t to be determined  on Γ ×  0,T  ,  Γ ∂u ∂n L ds  A  t  ∀ a.e.t∈  0,T  , u  x, 0   0inΩ, P where f ∈ L 2 Q and A ∈ L 2 0,T, n n 1 , ,n N  denotes the unit outward normal vector on Γ and ∂u ∂n L  N  i,j1 a ij  x, u  ∂u ∂x j n i . 1.1 2 Boundary Value Problems There are many concrete physical sources for problem P, for example, in the petroleum exploitation, u denotes the oil pressure, and At is the rate of total oil flux per unit length of the well at the time t; in the combustion theory, u denotes the temperature, for any fixed time t, the temperature distribution on the boundary is a constant to be determined, while, the total heat At through the boundary is given cf. 1–7. For linear equations, the existence, uniqueness of solution to the corresponding problem are well understood cf. 1– 3, for the purpose, the Galerkin method was used. For semilinear equations, the existence of global smooth solution was obtained in 7 in which a comparison principle was established. If a ij x, u is locally Lipschitz continuous with respect to the second variable, the existence and uniqueness of bounded weak solution to problem P have been discussed in 8 under the hypotheses of f ∈ L q Q and A ∈ L r 0,T with q>N/2  1, r>N 2. However, if f ∈ L 2 Q and A ∈ L 2 0,T, we cannot get a bounded weak solution. In order to deal with this situation, we will introduce the concept of renormalized solution to problem P and discuss the existence and uniqueness of renormalized solution. The paper is organized as follows. In Section 2, we introduce the concept of renormalized solution and prove the existence of renormalized solution to problem P.In Section 3, uniqueness and a comparison principle of renormalized solution to problem P are established. In Section 4, we discuss the relation between renormalized solutions and weak solutions for problem P. 2. Existence of Renormalized Solution to Problem P In order to prove the existence of renormalized solution to problem P, we make the following assumptions. Let a ij : Ω × R → R be Carath ´ eodory functions with 1 ≤ i, j ≤ N. We assume that a ij ·, 0 ∈ L ∞ Ω and for any given M>0 there exist d M ∈ L ∞ Ω and a positive constant λ 0 such that for every s, s 1 , s 2 ∈ R, ξ ξ 1 , ,ξ N  ∈ R N , and a.e. x ∈ Ω,   a ij  x, s 1  − a ij  x, s 2    ≤ d M  x  | s 1 − s 2 | , | s k | ≤ M, k  1, 2, 2.1 N  i,j1 a ij  x, s  ξ i ξ j ≥ λ 0 | ξ | 2 . 2.2 Set V   v ∈ H 1  Ω  | v | Γ  constant  . 2.3 Under hypotheses 2.1-2.2 and f ∈ L 2 Q, A ∈ L 2 0,T, we cannot obtain an L ∞ estimate on the determined function Ct; thus, we cannot prove the existence of bounded weak solutions to problem P, hence a ij ·,uD j u may not belong to L 2 Q.Inorderto overcome this difficulty, we will use the concept of renormalized solution introduced by DiPerna and Lions in 9 for Boltzmann equations see also 10–12. Boundary Value Problems 3 As usual, for k>0, T k denotes the truncation function defined by T k  v   ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ k, if v>k, v, if | v | ≤ k, −k, if v<−k. 2.4 Set W   ξ ∈ C ∞  Q  | ξ  T   0,ξ  t  | Γ  C  t   an arbitrary function of t   . 2.5 Definition 2.1. A renormalized solution to problem P is a measurable function u : Q → R, satisfying u ∈ L 2 0,T; V  ∩ L ∞ 0,T; L 2 Ω and for all h ∈ C 1 c R, ξ ∈ W, −  Q ξ t  u 0 h  r  drdxdt   Q N  i,j1 a ij  x, u  D j uD i  h  u  ξ  dxdt   Q fh  u  ξdxdt   T 0 A  t  h  u  t  | Γ  ξ  t  | Γ dt, 2.6 lim m → ∞  {  x,t  ∈Q:m≤ | u  x,t  | ≤m1 } N  i,j1 a ij  x, u  D j uD i udxdt  0. 2.7 Remark 2.2. Each term in 2.6 and 2.7 is well defined. Indeed, the first term on the left side of 2.6 is welldefined as |  u 0 hrdr|≤h L ∞ |u| and u ∈ L 2 Q. The second term on the left side of 2.6 should be understood as  {  x,t  ∈Q: | u | <k } N  i,j1 a ij  x, T k  u  D j T k  u  D i  h  T k  u  ξ  dxdt, 2.8 for k>0 such that supp h ⊂ −k, k. Since u ∈ L 2 0,T; V , it is the same for huξ and hut| Γ ξt| Γ . The integral in 2.7 should be understood as  {  x,t  ∈Q:m≤ | u  x,t  | ≤m1 } N  i,j1 a ij  x, T m1  u  D j T m1  u  D i T m1  u  dxdt. 2.9 Remark 2.3. Note that if u is a renormalized solution of problem P,wegetB h u  u 0 hrdr ∈ L 2 0,T; V , B h u t ∈ L 2 0,T; V  L 1 Q;thus,B h u ∈ C0,T; L 1 Ω, hence B h u·, 00 makes sense. Remark 2.4. By approximation, 2.6 holds for any h ∈ W 1,∞ R with compact support and all ξ ∈{ξ ∈ L 2 0,T; V  | ξ t ∈ L 2 Q,ξ·,T0}. 4 Boundary Value Problems Now we can state the existence result for prolem P  as follows. Theorem 2.5. Under hypotheses 2.1-2.2 and f ∈ L 2 Q, A ∈ L 2 0,T, problem P  admits a renormalized solution u ∈ L 2 0,T; V  ∩ L ∞ 0,T; L 2 Ω in the sense of Definition 2.1. In order to prove Theorem 2.5, we will consider the following problem: ∂u n ∂t − N  i,j1 ∂ ∂x i  a  n  ij  x, u n  ∂u n ∂x j   f in Q, u n | Γ×0,T  C n  t  a function of t to be determined  on Γ ×  0,T  ,  Γ ∂u n ∂n L ds  A  t  ∀ a.e.t∈  0,T  , u n  x, 0   0inΩ, P n  where a n ij x, ua ij x, T n u, i, j  1, 2, ,N. Then problem P n  admits a unique weak solution u n ∈ L 2 0,T; V  ∩ C0,T; L 2 Ω such that u  n ∈ L 2 0,T; V   and satisfies  u  n  t  ,v  V  ,V   Ω N  i,j1 a  n  ij  x, u n  D j u n D i vdx   Ω f  x, t  v  x  dx  A  t  v| Γ , a.e.t∈  0,T  , ∀v ∈ V, 2.10 u n  x, 0   0a.e.x∈ Ω. 2.11 In fact, here we can prove the existence of weak solution for problem P n  via Galerkin method. Let us consider the operator B : L 2  Ω  −→ V, F −→ v, 2.12 where v is the weak solution of the following problem: −Δv  v  F in Ω, v  C  a constant to be determined  on Γ,  Γ ∂v ∂n L ds  0. E By Lax-Milgram Theorem, the above problem exists a unique weak solution v which continuously depending on F. Hence B is a compact self-adjoint operator from L 2 Ω to Boundary Value Problems 5 L 2 Ω. By Riesz-Schauder’s theory, there is a completed orthogonal eigenvalues sequence {w k } of the operator B. Here we may take the special orthogonal system {w k }. Define A : V → V  ,  Aw,v  V  ,V   Ω N  i,j1 a n ij  x, w  D j wD i vdx,  Ft,v  V  ,V   Ω f  x, t  v  x  dx  A  t  v| Γ , a.e.t∈  0,T  , ∀w, v ∈ V. 2.13 Let u m n x, t  m k1 Φ km n w k , then Galerkin equations can be written as   u m n    t  ,w k    Au m n  t  ,w k    F  t  ,w k  , a.e.t∈  0,T  , u m n  x, 0   0a.e.x∈ Ω. 2.14 By using the same arguments as 13, Lemma 30.4, we get a solution u m n ∈ L 2 0,T; V  to the above Galerkin equations such that u m n   ∈ L 2 0,T; V . Moreover, we can easily prove the following estimates:  u m n  L ∞  0,T;L 2  Ω   ≤ C 0 ,  u m n  L 2  0,T;V  ≤ C 0 ,  Au m n  L 2 0,T;V   ≤ C 0 ,    u m n     L 2 0,T;V   ≤ C 0 , 2.15 where C 0 is a positive constant independent of m. The above estimates imply that there exists a subsequence of {u m n } still be denoted by {u m n } such that u m n u n weak ∗ in L ∞  0,T; L 2  Ω   , u m n u n weakly in L 2  0,T; V  ,  u m n   u  n weakly in L 2  0,T; V   , Au m n  Au n weakly in L 2  0,T; V   . 2.16 Thus we can pass to the limit in the above Galerkin equations and obtain the existence of weak solution for problem P n . Since it is easy to prove the uniqueness of weak solution for problem P n , we omit the details. 6 Boundary Value Problems To deal with the time derivative of truncation function, we introduce a time regularization of a function u ∈ L 2 0,T; V .Let u ν  x, t    t −∞ νu  x, s  e ν  s−t  ds, u  x, s   u  x, s  χ  0,T   s  , 2.17 where χ 0,T denotes the characteristic function of a set 0,T and ν>0. This convolution function has been first used in 14see also 10, and it enjoys the following properties: u ν belongs to C0,T; V , u ν x, 00, and u v converges strongly to u in L 2 0,T; V  as ν tends to the infinity. Moreover, we have  u ν  t  ν  u − u ν  , 2.18 and finally if u ∈ L ∞ Q, then u ν ∈ L ∞ Q and  u ν  L ∞ Q ≤  u  L ∞ Q , ∀ν>0. 2.19 Taking v  u n t in 2.10, then integrating over 0,τ with τ ∈ 0,T, we have  τ 0  Ω d dt | u n  x, t  | 2 dxdt   τ 0  Ω N  i,j1 a  n  ij  x, u n  D j u n D i u n dxdt   τ 0  Ω fu n dxdt   τ 0 A  t  u n  t  | Γ dt. 2.20 By 2.2, trace theorem, H ¨ older’s inequality, Young’s inequality and Gronwall’s inequality, we get  u n  L ∞ 0,T;L 2  Ω   ≤ C 1 , 2.21  u n  L 2  0,T;V  ≤ C 1 , 2.22 where C 1 is a positive constant depending only on f L 2 Q , A L 2 0,T , λ 0 , but independent of n and u n . By 2.21 and 2.22, there is a subsequence of {u n } still denoted by {u n } such that u n u weak ∗ in L ∞  0,T; L 2  Ω   , u n u weakly in L 2  0,T; V  u n | Γ u| Γ weakly in L 2  0,T  . 2.23 Using the same method as 10, we can obtain u n −→ u a.e. in Q  up to some subsequence  . 2.24 Boundary Value Problems 7 Thus for any given k>0, T k  u n  T k  u  weakly in L 2  0,T; V  , strongly in L 2  Q  , a.e. in Q. 2.25 By 15, Lemma 2 and Lemma 3, we h ave u n −→ u strongly in L q  Q  , ∀1 ≤ q<2  4 N , 2.26 u n −→ u strongly in L r  Γ ×  0,T  , ∀2 ≤ r<2  2 N . 2.27 impling that u n | Γ −→ u| Γ , a.e. in  0,T  . 2.28 For any given k>0, it follows from 2.27-2.28 and Vitali’s t heorem that T k  u n | Γ  −→ T k  u| Γ  strongly in L 2  0,T  . 2.29 Set η ν  u    T k  u  ν . 2.30 Similar to 10, this function has the following properties: η ν u t  νT k u − η ν u,η ν u00, |η ν u|≤k, η ν  u  −→ T k  u  strongly in L 2  0,T; V  , as ν tends to the infinity. 2.31 For any fixed h and k with h>k>0, let w n  T 2k  u n − T h  u n   T k  u n  − η ν  u   . 2.32 Then we have the following lemma. Lemma 2.6. Under the previous assumptions, we have  T 0 <  u n  t ,w n >dt≥ ω  n, ν, h  , 2.33 where lim h → ∞ lim ν →∞ lim n → ∞ ωn, ν, h0. Proof. The proof of Lemma 2.6 is the same as 10, Lemma 2.1, and we omit the details. 8 Boundary Value Problems Lemma 2.7. Under the previous assumptions, for any given k>0, we have T k  u n  −→ T k  u  strongly in L 2  0,T; V  . 2.34 Proof. Taking v  w n t in 2.10, then integrating over 0,T,byLemma 2.6, we have  Q N  i,j1 a  n  ij  x, u n  D j u n D i w n dxdt ≤  Q fw n dxdt   T 0 A  t  w n  t  | Γ dt  ω  n, ν, h  . 2.35 Now note that Dw n  0if|u n | >h 4k; then if we set M  h  4k, splitting the integral ontheleftsideof2.35 on the sets {x, t ∈ Q : |u n x, t| >k} and {x, t ∈ Q : |u n x, t|≤k}, ∀n>M,weget  Q N  i,j1 a  n  ij  x, u n  D j u n D i w n dxdt   Q N  i,j1 a ij  x, T M  u n  D j T M  u n  D i w n dxdt ≥  Q N  i,j1 a ij  x, T k  u n  D j T k  u n  D i  T k  u n  − η ν  u   dxdt −  {| u n | >k } N  i,j1   a ij  x, T M  u n  D j T M  u n      D i η ν  u    dxdt. 2.36 While,  {| u n | >k } N  i,j1   a ij  x, T M  u n  D j T M  u n      D i η ν  u    dxdt ≤  {| u n | >k } N  i,j1   a ij  x, T M  u n  D j T M  u n    | D i T k  u  | dxdt   Q N  i,j1   a ij  x, T M  u n  D j T M  u n      D i η ν  u  − D i T k  u    dxdt. 2.37 For any fixed h>0, 2.1 and 2.22 imply that a ij x, T M u n D j T M u n  is bounded in L 2 Q with respect to n, while |D i T k u|χ {|u n |>k} strongly converges to zero in L 2 Q. Moreover it follows from 2.31 that  {| u n | >k } N  i,j1   a ij  x, T M  u n  D j T M  u n      D i η ν  u    dxdt ≤ ω  n, ν  , 2.38 Boundary Value Problems 9 where lim ν →∞ lim n → ∞ ωn, ν0. Equations 2.38, 2.36,and2.35 imply that  Q N  i,j1 a ij  x, T k  u n  D j T k  u n  D i  T k  u n  − η ν  u   dxdt ≤  Q fw n dxdt   T 0 A  t  w n  t  | Γ dt  ω  n, ν   ω  n, ν, h  . 2.39 By 2.25, 2.31,and2.39,weget  Q N  i,j1 a ij  x, T k  u n  D j T k  u n  D i  T k  u n  − T k  u  dxdt ≤  Q fw n dxdt   T 0 A  t  w n  t  | Γ dt  ω  n, ν   ω  n, ν, h  . 2.40 By 2.24-2.25 and the Lebesgue dominated convergence theorem, we have  Q fw n dxdt   Q fT 2k  u − T h  u   T k  u  − η ν  u   dxdt  ω  n  , 2.41 where lim n → ∞ ωn0. 2.31 and 2.41 imply that  Q fw n dxdt   Q fT 2k  u − T h  u  dxdt  ω  n, ν  ; 2.42 thus, we get  Q fw n dxdt  ω  n, ν, h  . 2.43 Similarly to the proof of 2.43, we also have  T 0 A  t  w n  t  | Γ dt  ω  n, ν, h  . 2.44 Therefore we get  Q N  i,j1 a ij  x, T k  u n   D j T k  u n  − D j T k  u   D i  T k  u n  − T k  u  dxdt ≤ ω  n, ν, h  −  Q N  i,j1 a ij  x, T k  u n  D j T k  u  D i  T k  u n  − T k  u  dxdt. 2.45 10 Boundary Value Problems Let n, ν, then and h tend to the infinity, respectively, we get lim n → ∞  Q N  i,j1 a ij  x, T k  u n  D j  T k  u n  − T k  u  D i  T k  u n  − T k  u  dxdt  0. 2.46 Using 2.2, 2.25,and2.46,weobtain2.34. Proof of Theorem 2.5. For any given ξ ∈ W, h ∈ C 1 c R, suppose that supp h ⊂ −k, k, taking v  hu n tξt in 2.10 and integrating over 0,T, w e have  T 0   u n  t ,h  u n  ξ  dt   Q N  i,j1 a n ij  x, u n  D j u n D i  h  u n  ξ  dxdt   T 0 A  t  h  u n  t  | Γ  ξ  t  | Γ dt   Q fh  u n  ξdxdt. 2.47 By 12, Lemma 1.4, we have  T 0   u n  t ,h  u n  ξ  dt  −  Q ξ t  u n 0 h  r  drdxdt. 2.48 However −  Q ξ t  u n 0 h  r  drdxdt −→ −  Q ξ t  u 0 h  r  drdxdt. 2.49 In fact, Noting 2.26 we get      −  Q ξ t  u n 0 h  r  drdxdt   Q ξ t  u 0 h  r  drdxdt             Q ξ t  u n u h  r  drdxdt      ≤  ξ t  L 2 Q  h  L ∞ R  u n − u  L 2 Q −→ 0, as n −→ ∞. 2.50 [...]... for Reservoir Simulation, NorthHolland, Amsterdam, The Netherlands, 1986 7 W X Shen and S M Zheng, “Nonlocal initial- boundary value problems for nonlinear parabolic equations, ” Journal of Fudan University Natural Science, vol 24, no 1, pp 47–57, 1985 Chinese 8 F Li, “Existence and uniqueness of bounded weak solution for non-linear parabolic boundary value problem with equivalued surface, ” Mathematical... 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Corporation Boundary Value Problems Volume 2009, Article ID 739097, 23 pages doi:10.1155/2009/739097 Research Article On Initial Boundary Value Problems with Equivalued Surface for Nonlinear Parabolic. the concept of renormalized solution to initial boundary value problems with equivalued surface for nonlinear parabolic equations, discuss the existence and uniqueness of renormalized solution,. We consider the following nonlinear parabolic boundary value problems with equivalued surface: ∂u ∂t − N  i,j1 ∂ ∂x i  a ij  x, u  ∂u ∂x j   f  x, t  in Q, u  C  t  a function of