Quasilinear parabolic problems with nonlinear boundary conditions

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Theorem 2.8.1 (Prüss [63, Thm. 8.6]) Suppose X belongs to the class HT , p ∈ (1, ∞), and let a ∈ L1, loc(R+) be of subexponential growth. Assume that a is 1-regular and θ-sectorial, where θ < π. Then there is a unique operator B ∈ S(Lp(R; X)) such that (Bf)˜(ρ) = 1 â(iρ) ˜ f(ρ), ρ ∈ R, ˜ f ∈ C ∞ 0 (R \ {0}; X). (2.28) Moreover, B has the following properties: (i) B commutes with the group of translations; (ii) (µ + B) −1 Lp(R+; X) ⊂ Lp(R+; X) for each µ > 0, i.e. B is causal; (iii) B ∈ BIP(Lp(R; X)), and θB = φB = θa, where (iv) σ(B) = {1/â(iρ) : ρ ∈ R \ {0}}. θa = sup{|arg â(λ)| : Re λ > 0}; (2.29) The next theorem provides information about the domain of the operator B. Proposition 2.8.1 (Prüss [63, Cor. 8.1]) Suppose X belongs to the class HT , p ∈ (1, ∞). Assume a ∈ K 1 (α, θ) with θ < π, and let B be defined by (2.28). Then D(B) = H α p (R; X). Here H α p (R; X) := D(B α/2 0 ), α ∈ R+, where B0 = −(d 2 /dt 2 ) ∈ BIP(Lp(R; X)), cf. [63, p. 226]. Suppose the assumptions of Proposition 2.8.1 hold. Let J = [0, T ] or J = R+. We put H α p (J; X) = {f|J : f ∈ H α p (R; X)} and endow this space with the norm |f| H α p (J;X) = inf{|g| H α p (R;X) : g|J = f}. We further introduce the subspace 0H α p (J; X) by means of 0H α p (J; X) = {f|J : f ∈ H α p (R; X) and supp f ⊆ R+}. Define then the operator B ∈ S(Lp(J; X)) as the restriction of the operator B constructed in Theorem 2.8.1 to Lp(J; X). This makes sense in virtue of causality. In fact, we have D(B) = D(B| Lp(J;X)) = {f ∈ Lp(J; X) ∩ D(B) : Bf ∈ Lp(J; X)} = {f ∈ Lp(J; X) ∩ H α p (R; X) : Bf ∈ Lp(J; X)} = {f|J : f ∈ H α p (R; X), supp f ⊆ R+, Bf ∈ Lp(R; X), and supp Bf ⊆ R+} = {f|J : f ∈ H α p (R; X) and supp f ⊆ R+} = 0H α p (J; X), the equals sign before the last following from the causality of B. Assuming in addition a ∈ L1(R+) in case J = R+, by Young’s inequality, the operator B is invertible and B −1 w = a ∗ w for all w ∈ Lp(J; X). From Theorem 2.8.1 we further see that B ∈ BIP(Lp(J; X)) and θB ≤ θB = θa. We summarize these observations in the subsequent corollary. Corollary 2.8.1 Let X be a Banach space of class HT , p ∈ (1, ∞), and J = [0, T ] be a compact interval or J = R+. Suppose a ∈ K 1 (α, θ) with θ < π, and assume in addition a ∈ L1(R+) in case J = R+. Then the restriction B := B| Lp(J;X) of the operator B constructed in Theorem 2.8.1 to Lp(J; X) is well-defined. The operator B belongs to the class BIP(Lp(J; X)) with power angle θB ≤ θB = θa and is invertible satisfying B −1 w = a ∗ w for all w ∈ Lp(J; X). Moreover D(B) = 0H α p (J; X). 28
Example 2.8.1 For J = [0, T ] and aα(t) = tα−1 /Γ(α), t > 0, α ∈ (0, 1), the operator B in Corollary 2.8.1 takes the form Bu(t) = d dt � t 0 a1−α(t − s)u(s) ds, t > 0, u ∈ 0H α p (J; X), thus coincides with (d/dt) α , the derivation operator of (fractional) order α. The function Bu is called the fractional derivative of u of order α. We point out that Corollary 2.8.1 will prove extremely useful in establishing maximal Lp-regularity results for parabolic Volterra equations. On the one hand it describes precisely the mapping properties of the convolution operators associated to a K-kernel, on the other hand it enables us to apply the Dore-Venni theorem to operator sums in Lp(J; X) which involve an inverse convolution operator B. We next show how Volterra operators can be used to introduce equivalent norms for the vector-valued Bessel-potential spaces Hα p (J; X). Suppose we are in the situation of Corollary 2.8.1, where we restrict ourselves to the case J = [0, T ]. Assume further that α ∈ (0, 2) \ {1/p, 1 + 1/p} and let µ > 0. For f ∈ Hα p (J; X), we put with some abuse of language |f| (a,µ) H α p (J;X) = ⎧ ⎪⎨ |Bf| Lp(J;X) : α ∈ (0, ⎪⎩ 1 p ) |B(f − f(0))| Lp(J;X) + µ|f(0)|X : α ∈ ( 1 1 p , 1 + p ) |B(f − f(0) − t ˙ f(0))| Lp(J;X) + µ|f(0)|X + µ| ˙ f(0)|X : α ∈ (1 + 1 p , 2). (2.30) This is a well-defined expression in view of Sobolev’s embedding theorem. Observe that | · | (a,µ) H α p (J;X) enjoys the properties of a norm for the space Hα p (J; X). To verify that it is equivalent to the usual norm | · | H α p (J;X) we can employ Corollary 2.8.1 and Sobolev embeddings. Indeed, if α > 1 + 1/p and f ∈ H α p (J; X), we may estimate |f| Hα p (J;X) ≤ |f − f(0) − t ˙ f(0)| Hα p (J;X) + |f(0)| Hα p (J;X) + |t ˙ f(0)| Hα p (J;X) = |a ∗ B(f − f(0) − t ˙ f(0))| Hα p (J;X) + |{t ↦→ 1}| Hα p (J)|f(0)|X +|{t ↦→ t}| Hα p (J)| ˙ f(0)|X ≤ c1 (|B(f − f(0) − t ˙ f(0))| Lp(J;X) + µ|f(0)|X + µ| ˙ f(0)|X) = c1 |f| (a,µ) H α p (J;X), with c1 not depending on f. Conversely, by and |f(0)|X + | ˙ f(0)|X ≤ |f| C 1 (J;X) ≤ CSob|f| H α p (J;X) |B(f − f(0) − t ˙ f(0))| Lp(J;X) ≤ c|f − f(0) − t ˙ f(0)| H α p (J;X) ≤ c(|f| H α p (J;X) + |f(0)| H α p (J;X) + |t ˙ f(0)| H α p (J;X)) we also get an inequality of the form ≤ c(|f| H α p (J;X) + ˜c(|f(0)|X + | ˙ f(0)|X)), |f| (a,µ) H α p (J;X) ≤ c2 |f| H α p (J;X), 29 (2.31)
Page 1 and 2: 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: 2.7 Evolutionary integral equations
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 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 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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