Quasilinear parabolic problems with nonlinear boundary conditions

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Then (4.18) has a unique solution in the space Z if and only if the data f and g are subject to the subsequent conditions. (i) f ∈ H α p (J; Lp(R n+1 + )); (ii) g ∈ B 1 α(1− 2p ) pp 2 2− pα pp (J; Lp(R n )) ∩ Lp(J; B (iii) f|t=0 ∈ B (R n+1 + ), if α > 1 p ; (iv) f|t=0, y=0 = g|t=0, if α > 2 2p−1 ; 1 1 2(1− − α pα ) pp 1 2− p pp (R n )); (v) ∂tf|t=0 ∈ B (R n+1 + ), if α > 1 + 1 p ; (vi) ∂tf|t=0, y=0 = ∂tg|t=0, if α > 1 + 3 2p−1 . Proof. By means of a variable transformation of the form ¯x = QT ΛQx, where Q is a rotation matrix and Λ is diagonal with Λii = 1/ √ λi (λ1, . . . , λn+1 denoting the positive eigenvalues of the matrix a), problem (4.18) can be reduced to a problem of the same structure but with a = In+1. The assertion follows then from Theorem 3.5.1 applied to X = Lp(Rn ) and the operator A = −∆x ′, which belongs to BIP(X) and has power angle 0, and from the fact that all function spaces occurring in that theorem are preserved under the above variable transformation. � The corresponding result for (4.19) reads Theorem 4.2.2 Let 1 < p < ∞, n ∈ N, a ∈ Sym{n + 1}, and b ∈ R n . Suppose k ∈ K 1 (α, θ), where θ < π and α ∈ (0, 2) \ � 1 p , 2 p−1 , 1 + 1 p � . Assume that a is positive definite. Then (4.19) possesses a unique solution in the space Z if and only if the functions f and h satisfy the following conditions. (i) f ∈ H α p (J; Lp(R n+1 + )); (ii) h ∈ B 1 1 α( − 2 2p ) pp 2 2− pα pp (J; Lp(R n )) ∩ Lp(J; B 1 1− p pp (R n )); (iii) f|t=0 ∈ B (R n+1 + ), if α > 1 p ; (iv) −∂yf|t=0, y=0 + b · ∇x ′f|t=0, y=0 = h|t=0, if α > 2 p−1 ; (v) ∂tf|t=0 ∈ B 1 1 2(1− − α pα ) pp (R n+1 + ), if α > 1 + 1 p . Proof. Use the variable transformation described in the proof of Theorem 4.2.1 and normalize the coefficient in front of the normal derivative to reduce (4.19) to a problem of the same structure with a = In+1. The assertion is then a consequence of Theorem 3.5.1 applied to X = Lp(R n ), A = −∆x ′, and D = b · ∇x ′ (b ∈ Rn ). Note that D ∈ BIP(R(D)) with power angle θD ≤ π/2 (cp. Prüss [65, Section 3]) and thus θD < π − θ/2 = π − max{θ, θA}/2 showing the second angle condition in Theorem 3.5.1. � 4.2.2 Pointwise multiplication Let 1 < p < ∞, s1, s2 ∈ (0, 1), 0 < T2, 0 ≤ T1 ≤ T2, and Ω be an arbitrary domain in R n . We are interested in pointwise multipliers for the intersection space Y T2 := Y T2 1 ∩ Y T2 2 := B s1 pp([0; T2]; Lp(Ω)) ∩ Lp([0, T2]; B s2 pp(Ω)) 64
endowed with the norm | · | Y T 2 defined by where | · | Y T = | · | Lp([0,T ]×Ω) + [ · ] Y T 1 + [ · ] Y T 2 , T > 0 (4.20) � T � T � [f] Y T 1 = ( 0 0 � T � Ω � [f] Y T 2 = ( 0 Ω Ω |f(t, x) − f(τ, x)| p |t − τ| 1+s1p |f(t, x) − f(t, y)| p |x − y| n+s2p dx dτ dt) 1 p , (4.21) dx dy dt) 1 p . (4.22) We consider first products with bounded factors. Suppose that m, f ∈ Y T2 ∩ L∞([0, T2] × Ω) =: Y T2 ∩ L∞. In what is to follow we estimate |mf| Y T 2 , using among other things terms referring to norms of functions on [0, T1] and [T1, T2], respectively. Note that in case T1 = 0 respectively T1 = T2 these terms have to be regarded as zero. It is readily seen that |mf| Lp([0,T2]×Ω) ≤ |mf| Lp([0,T1]×Ω) + |mf| Lp([T1,T2]×Ω) ≤ |m| L∞([0,T1]×Ω)|f| Lp([0,T1]×Ω) + |m| L∞([T1,T2]×Ω)|f| Lp([0,T2]×Ω). By employing the trivial identity m(t, x)f(t, x) − m(t, y)f(t, y) = m(t, x)(f(t, x) − f(t, y)) + (m(t, x) − m(t, y))f(t, y) and Minkowski’s inequality, we further get [mf] Y T 2 2 ≤ |m| L∞([0,T1]×Ω)[f] Y T 1 2 +|m| L∞([T1,T2]×Ω)[f] Y T 2 2 Proceeding similarly as above, we also obtain [mf] Y T 2 1 where we have set � T2 = ( ≤ ( � T2 0 0 � � T1 T1 0 0 � � Ω ≤ |m| L∞([0,T1]×Ω)[f] Y T 1 1 [m] Y T 1 , T 2 1 + [m] T Y 1 |f| L∞([0,T1]×Ω) 2 + [m] Lp([T1,T2];B s 2 pp(Ω)) |f| L∞([0,T2]×Ω). |m(t, x)f(t, x) − m(τ, x)f(τ, x)| p |t − τ| 1+s1p � Ω · · ·) 1 � � T2 T2 p + 2( T1 0 +2(|m| L∞([T1,T2]×Ω)[f] Y T 2 1 � T2 = ( T1 � T2 0 � Ω dx dτ dt) 1 p · · ·) Ω 1 p (4.23) + [m] T Y 1 |f| L∞([0,T1]×Ω) 1 + [m] T Y 1 , T2 |f| L∞([0,T2]×Ω)), 1 |m(t, x) − m(τ, x)| p |t − τ| 1+s1p So with | · | Y T i∩L∞ = | · | Y T i + | · | L∞([0,Ti]×Ω), i = 1, 2, and |m| Y T 1 , T 2 ∩L∞ = |m| L∞([T1,T2]×Ω) + [m] Y T 1 , T 2 1 we have thus proved 65 dx dτ dt) 1 p . (4.24) + [m] Lp([T1,T2];B s2 , (4.25) pp(Ω))
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Gutachter: Quasilinear parabolic pr
Page 3 and 4:
Contents 1 Introduction 3 2 Prelimi
Page 5 and 6:
Chapter 1 Introduction The present
Page 7 and 8:
to assume that m, c are bounded fun
Page 9 and 10:
We give now an overview of the cont
Page 11 and 12:
conditions of order ≤ 1. Sections
Page 13 and 14:
Chapter 2 Preliminaries 2.1 Some no
Page 15 and 16: Clearly, φA ∈ [0, π) and φA
Page 17 and 18: Definition 2.2.3 Let X and Y be Ban
Page 19 and 20: µ ∈ Σφα }. Let N ∈ N, Tj
Page 21 and 22: We remark that a theorem of the Dor
Page 23 and 24: Here f(A, ·) ∈ H0(Σ π 2 +η; B
Page 25 and 26: Further, K ∞ (α, θa) := {a ∈
Page 27 and 28: Using (2.19) for aω and bω yields
Page 29 and 30: 2.7 Evolutionary integral equations
Page 31 and 32: Example 2.8.1 For J = [0, T ] and a
Page 33: We conclude this section by illustr
Page 36 and 37: kernel a. The operator B is inverti
Page 38 and 39: x := f(0) ∈ X exists and we are l
Page 40 and 41: with two positive constants C1, C2
Page 42 and 43: with two positive constants C1 and
Page 44 and 45: derivative theorem to this pair of
Page 46 and 47: 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: Turning to (c), let g ∈ Ξi+1 and
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 and 98: Chapter 6 Nonlinear Problems 6.1 Qu
Page 99 and 100: sufficiently small, say T ≤ T1
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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