Quasilinear parabolic problems with nonlinear boundary conditions

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Letting gη(λ) = âη(λ)/(1 + ωâη(λ)), Reλ > 0, we thus obtain |gη(λ)| ≤ C0|λ| −α for all λ ∈ C+, where C0 > 0 is independent of λ. Here, we also made use of (2.19) for the kernel aη. Observe that λg ′ η(λ) = gη(λ) λâ′ η(λ) âη(λ) · 1 , Re λ > 0. 1 + ωâη(λ) Thanks to (2.24) and 1-regularity of aη, we therefore get an estimate |λg ′ η(λ)| + |gη(λ)| ≤ C , Re λ > 0, (2.25) |λ| α which, together with α > 0 and holomorphy of gη in C+, implies existence of uη ∈ C(0, ∞) ∩ L1, loc(R+) satisfying ûη(λ) = gη(λ), Re λ > 0, by Proposition 2.1.1. (2.2) entails that uη is of subexponential growth. By inversion of the Laplace transform we then obtain uη + ωaη ∗ uη = aη, hence u + ωa ∗ u = a, where u(t) := uη(t)eηt , t > 0. As in the proof of (iv), we see that the construction of u is independent of η > 0, and that u is of subexponential growth. From û = â/(1 + ωâ) and ω > 0 one deduces that u is θa-sectorial. Furthermore, the function h(λ) := 1 + ωâ(λ) defined on C+ satisfies (2.23) with a suitable constant C > 0 for k = 1, . . . , r. In fact, |λ kˆ � � � (k) k k (k) h (λ)| = ω |λ â (λ)| ≤ C|â(λ)| ≤ C| h(λ)| ˆ � 1 � � �ω + 1/â(λ) � ≤ C1| ˆ h(λ)| for all λ ∈ C+ and k = 1, . . . , r, in virtue of θa < π and r-regularity of a. By the considerations in the proof of (iv), that property of h is passed on to the function ¯h(λ) := 1/h(λ), λ ∈ C+, which is well-defined in view of (2.24). Then û = â ¯ h, and as above, with the aid of Leibniz’ formula, we see that u is r-regular. Concerning (K3), the assumption ζ := lim infµ→0 |â(µ)| > 0 implies lim infµ→0 |û(µ)| > 0, since lim inf µ→0 |û(µ)| ≥ lim inf µ→0 (ω + |1/â(µ)|)−1 = (ω + 1/ζ) −1 . Besides, by (2.24), lim sup µ→∞ |â(µ)|µ α < ∞, and (2.19) applied to a1, we have as well as lim sup µ→∞ lim inf µ→∞ |û(µ)|µα ≥ lim inf µ→∞ |û(µ)|µ α ≤ c −1 lim sup |û(µ)|µ µ→∞ α < ∞, (µ + 1) α ≥ lim inf ω + |1/â(µ + 1)| µ→∞ (µ + 1) α > 0. ω + C(µ + 1) α Hence, the kernel ξ := u satisfies ξ + ωa ∗ ξ = a and belongs to K r (α, θa). Uniqueness follows again from the unique inverse of the Laplace transform. Thus (v) is shown. It remains to prove (vi). Suppose a ∈ K r (α, θa) ∩ L1(R+), ω > 0, and ɛ := ω|a| L1(R+) < 1. We proceed as in the previous part. Fix an arbitrary η > 0 and define gη(λ) := âη(λ)/(1 − ωâη(λ)), Reλ > 0. This time the assumption ɛ < 1 ensures that the denominator in the definition of gη is bounded away from zero. Using this and 1-regularity of aη yields (2.25) in a similar fashion as above. With the aid of Proposition 2.1.1, existence of u ∈ L1, loc(R+) with u − ωa ∗ u = a can then be seen. By the same line of arguments as in the previous part one further shows that u is r-regular. Validity of the conditions in (K3) with exponent α can be proved for u similarly as above. By elementary trigonometry, |arg (1 − ωâ(λ))| ≤ arcsin(ɛ) for all λ ∈ C+, i.e. u is (θa + arcsin(ɛ))-sectorial. Last but not least, u ∈ L1(R+) follows from ɛ < 1 and a ∈ L1(R+) by the Paley-Wiener theorem. Hence, ξ := u possesses all properties claimed in (vi), uniqueness being evident as above. � 26
2.7 Evolutionary integral equations Let X be a Banach space, A a closed linear, but in general unbounded operator in X with dense domain D(A), and a ∈ L1, loc(R+) a scalar kernel of subexponential growth which is not identically zero. We consider the Volterra equation u(t) + � t 0 a(t − s)Au(s) ds = f(t), t ≥ 0, (2.26) where f : R+ → X is a given function, strongly measurable and locally integrable, at least. Observe that in case a(t) ≡ 1 and f differentiable, (2.26) is equivalent to the Cauchy problem ˙u(t) + Au(t) = ˙ f(t), t ≥ 0, u(0) = f(0). Following Prüss [66] (see also [63, Def. 3.1]), we call (2.26) parabolic if â(λ) �= 0 for Re λ > 0, −1/â(λ) ∈ ρ(A), and there is a constant M > 0 such that |(I + â(λ)A) −1 | ≤ M for Re λ > 0. If A belongs to the class S(X) with spectral angle φA, and a is θa-sectorial, then (2.26) is parabolic provided that θA + φA < π, cf. [63, Prop. 3.1]. An important property of parabolic Volterra equations consists in that they admit bounded resolvents whenever the kernel a is 1-regular, see [63, Thm 3.1]. By a resolvent for (2.26) we mean a family {S(t)}t≥0 of bounded linear operators in X which satisfy the following conditions: (S1) S(t) is strongly continuous on R+ and S(0) = I; (S2) S(t)D(A) ⊂ D(A) and AS(t)x = S(t)Ax for all x ∈ D(A), t ≥ 0; (S3) S(t)x + A(a ∗ Sx)(t) = x, for all x ∈ X, t ≥ 0. (S3) is called resolvent equation, cf. [63, Def. 1.3, Prop. 1.1]. One can show that (2.26) admits at most one resolvent, and if it exists, then (2.26) has a unique mild solution u represented by the variation of parameters formula u(t) = d dt � t 0 S(t − s)f(s) ds, t ≥ 0, (2.27) at least for such f for which (2.27) is meaningful, see [63, Section 1.1 and 1.2]. 2.8 Volterra operators in Lp This paragraph looks at convolution operators in Lp which are associated to a K-kernel. After stating two fundamental theorems from the monograph Prüss [63] on the inversion of the convolution in Lp(R; X), we will consider restrictions of it to Lp(J; X) and use them to introduce equivalent norms for the vector-valued Bessel-potential spaces H α p (J; X). We will also study operators of the form (I − a∗) in these spaces. Such operators occur in connection with transformations of Volterra equations. For J = [0, T ] and J = R+, we identify in the sequel Lp(J; X) with the subspace {f ∈ Lp(R; X) : supp f ⊆ R+} of Lp(R; X). 27
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: Using (2.19) for aω and bω yields
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 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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