Quasilinear parabolic problems with nonlinear boundary conditions

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and Z T ∇ = H 1 2 (1+α) p Y T 1 (1+α)(1− 2p D = B ) pp Y T 1 1 (1+α)( − 2 2p N = B ) pp 2− 2 p(1+α) Y0 = Bpp (Ω), Y1 = B (J; Lp(Ω)) ∩ Lp(J; H 1 p(Ω)), 2− 1 p (J; Lp(ΓD)) ∩ Lp(J; Bpp (ΓD)), (6.3) 1− 1 p (J; Lp(ΓN)) ∩ Lp(J; Bpp (ΓN)), (6.4) 2 2 2− − 1+α p(1+α) pp α > 1/p being assumed in the definition of Y1. For Λ ∈ {ZT , ZT ∇ , Y T D , Y T N , XT 1 ), we as usual denote by 0Λ the subspace of all functions h ∈ Λ with h|t=0 = 0 and ∂th|t=0 = 0, in case that these traces exist. If u ∈ ZT , then this corresponds, as we know from Theorem 4.3.1, to the regularity classes (Ω), ∇u ∈ (Z T ∇) n , ∇ 2 u ∈ (X T ) n×n , γDu ∈ Y T D , γN∇u ∈ (Y T N ) n , u|t=0 ∈ Y0, ∂tu|t=0 ∈ Y1. Consequently, Y0 is the natural space for u0, and if α > 1/p and u ∈ Z T is a solution of (6.1), then we have to ensure that u1 := ∂tu|t=0, which is given by u1(x) = f(0, x, u0(x), ∇u0(x)), x ∈ Ω, belongs to Y1. Of course, we have to assume that Observe also that as well as u0(x) ∈ U0, ∇u0(x) ∈ U1, x ∈ Ω. (6.5) Z T ↩→ C(J; Y0) ↩→ C(J × Ω) Z T 2 1− p(1+α) ∇ ↩→ C(J; Bpp (Ω)) ↩→ C(J × Ω), provided that 1 − 2/p(1 + α) > n/p, which is equivalent to p > 2 1+α + n (6.6) and which will be assumed throughout this section. Notice further that we have to take into account the three compatibility conditions b D (0, x, u0(x)) = 0, x ∈ ΓD, b N (0, x, u0(x), ∇u0(x)) = 0, x ∈ ΓN, b D t (0, x, u0(x)) + b D ξ (0, x, u0(x))u1(x) = 0, x ∈ ΓD, if α > 3 2p−1 . Here and in the subsequent lines we assume that b D and b N are as smooth as needed to make the formulas meaningful. Precise regularity statements will be given later on. We now set out to reformulate (6.1) as a fixed point problem in an appropriate subset of Z T . To this end, we first put A0(x) = −a(0, x, u0(x), ∇u0(x)), x ∈ Ω. Further we fix a function φ ∈ Z T0 which satisfies φ|t=0 = u0 and ∂tφ|t=0 = u1, the latter being demanded in case that α > 1/p. In view of (6.5) and (6.6), we see that, for T 96
sufficiently small, say T ≤ T1 ≤ T0, we have φ(t, x) ∈ U0 and ∇φ(t, x) ∈ U1 for all t ∈ J and x ∈ Ω. So for such T we may define operators B◦ K (φ) and Rφ K , K = D, N, by means of and B ◦ D(φ)u(t, x) = b D ξ (t, x, φ(t, x))u(t, x), t ∈ J, x ∈ ΓD, B ◦ N(φ)u(t, x) = b N ξ (t, x, φ(t, x), ∇φ(t, x))u(t, x) + b N η (t, x, φ(t, x), ∇φ(t, x)) · ∇u(t, x), t ∈ J, x ∈ ΓN, R φ K (u) = BK(u) − BK(φ) − B ◦ K(φ)(u − φ), K = D, N. Obviously, (6.1) restricted to J is equivalent to ⎧ ⎪⎨ ⎪⎩ ∂tu + dk ∗ A0 : ∇2u = F (u) + dk ∗ G(u) +dk ∗ ((A0 − A(u)) : ∇2u) (J × Ω) B◦ D (φ)u = −BD(φ) + B◦ D (φ)φ − Rφ D (u) B (J × ΓD) ◦ N (φ)u = −BN(φ) + B◦ N (φ)φ − Rφ N (u) (J × ΓN) (Ω). u|t=0 = u0 In other words, u ∈ Z T solves a problem of the form ⎧ ⎪⎨ ⎪⎩ ∂tv + dk ∗ A0 : ∇ 2 v = h (J × Ω) B ◦ D (φ)v = ψD (J × ΓD) B◦ N (φ)v = ψN (J × ΓN) (Ω). v|t=0 = v0 (6.7) (6.8) with certain functions on the right-hand side. Given data h, ψD , ψN and v0, (6.8) is a linear problem. For our construction, it is essential to understand this problem, in particular, one needs a precise characterization of unique solvability of (6.8) in ZT in terms of regularity and compatibility conditions for the data. Let us assume for the moment that we have such a result at our disposal - possibly on a yet smaller time-interval - and that the right-hand sides of the following both problems (6.9) and (6.11), which are of the form (6.8), fulfill the conditions needed to get a unique solution in ZT in either case. Then it makes sense to define the reference function w ∈ ZT as solution of the linear problem ⎧ ⎪⎨ ∂tw + dk ∗ A0 : ∇ ⎪⎩ 2w = F (φ) + dk ∗ G(φ) (J × Ω) B◦ D (φ)w = −BD(φ) + B◦ D (φ)φ (J × ΓD) B◦ N (φ)w = −BN(φ) + B◦ (6.9) N (φ)φ (J × ΓN) (Ω). Given ρ > 0, let w|t=0 = u0 Σ(ρ, T, φ) = {v ∈ Z T : v|t=0 = u0, ∂tv|t=0 = u1 (if α > 1/p), |v − w| Z T ≤ ρ}, which is a closed subset of Z T . Since Z T ↩→ C(J; C 1 (Ω)), we may further put µw(T ) = max{|w(t, x) − u0(x)| + |∇w(t, x) − ∇u0(x)| : t ∈ J, x ∈ Ω}. Apparently, µw(T ) → 0 as T → 0, due to w|t=0 = u0. 97
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Gutachter: Quasilinear parabolic pr
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Contents 1 Introduction 3 2 Prelimi
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Chapter 1 Introduction The present
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to assume that m, c are bounded fun
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We give now an overview of the cont
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conditions of order ≤ 1. Sections
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Chapter 2 Preliminaries 2.1 Some no
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Clearly, φA ∈ [0, π) and φA
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Definition 2.2.3 Let X and Y be Ban
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µ ∈ Σφα }. Let N ∈ N, Tj
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We remark that a theorem of the Dor
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Here f(A, ·) ∈ H0(Σ π 2 +η; B
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Further, K ∞ (α, θa) := {a ∈
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Using (2.19) for aω and bω yields
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2.7 Evolutionary integral equations
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Example 2.8.1 For J = [0, T ] and a
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We conclude this section by illustr
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kernel a. The operator B is inverti
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x := f(0) ∈ X exists and we are l
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with two positive constants C1, C2
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with two positive constants C1 and
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derivative theorem to this pair of
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3.2 A general trace theorem Let X b
Page 48 and 49: 3.3 More time regularity for Volter
Page 50 and 51: Theorem 3.4.2 Suppose X is a Banach
Page 52 and 53: Our next objective is to show neces
Page 54 and 55: Let u1 be the restriction of v1 to
Page 56 and 57: Proof. We begin with the necessity
Page 59 and 60: Chapter 4 Linear Problems of Second
Page 61 and 62: The strategy for solving (4.1) is n
Page 63 and 64: Since ψj ≡ 1 on supp ϕj, we may
Page 65 and 66: Turning to (c), let g ∈ Ξi+1 and
Page 67 and 68: endowed with the norm | · | Y T 2
Page 69 and 70: We remark that the constant C2 stem
Page 71 and 72: One can then construct functions a
Page 73 and 74: analogous to (4.17), shows that S i
Page 75 and 76: Apply now V#, i+1 := I + k ∗ A#(
Page 77 and 78: Given a function v ∈ H 2 p(R n+1
Page 79 and 80: v is a solution of (4.40) on Ji+1 :
Page 81 and 82: Chapter 5 Linear Viscoelasticity In
Page 83 and 84: where δij denotes Kronecker’s sy
Page 85 and 86: problem ⎧ ⎪⎨ ⎪⎩ ∂tv −
Page 87 and 88: To see the converse direction, supp
Page 89 and 90: and up solves � Aup − ∆xup
Page 91 and 92: It can be written as where l(z, ξ)
Page 93 and 94: elongs to H∞ (Σ π 2 +η × Ση
Page 95 and 96: which allows us to write the first
Page 97: Chapter 6 Nonlinear Problems 6.1 Qu
Page 101 and 102: (d) bD ∈ C(J0 × ΓD × U0), ∃C
Page 103 and 104: which entails (6.14). Corresponding
Page 105 and 106: substitution operators to be studie
Page 107 and 108: for all t, τ ∈ J, ξ, η ∈ K,
Page 109 and 110: to write where h2(t, τ, x) = h21(t
Page 111 and 112: Lemma 6.2.3 Let 0 < s < s0 < 1, ρ
Page 113 and 114: Bibliography [1] Albrecht, D.: Func
Page 115 and 116: [54] Lunardi, A.: On the heat equat
Page 117 and 118: kleines T mit Hilfe des Kontraktion
Page 119: Personal Details Curriculum Vitae N
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Quasilinear parabolic problems with nonlinear boundary conditions

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