Quasilinear parabolic problems with nonlinear boundary conditions

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6 Nonlinear Problems 95 6.1 Quasilinear problems of second order with nonlinear boundary conditions 95 6.2 Nemytskij operators for various function spaces . . . . . . . . . . . . . . . 102 Bibliography 111 2
Chapter 1 Introduction The present thesis is devoted to the study of the Lp-theory of a class of quasilinear parabolic problems with nonlinear boundary conditions. The main objective here is to prove existence and uniqueness of local (in time) strong solutions of these problems. To achieve this we establish optimal regularity estimates of type Lp for an associated linear problem which allow us to reformulate the original problem as a fixed point equation in the desired regularity class, and we show that under appropriate assumptions the contraction mapping principle is applicable, provided the time-interval is sufficiently small. We describe now the class of equations to be studied. Let Ω be a bounded domain in R n with C 2 -smooth boundary Γ which decomposes according to Γ = ΓD ∪ ΓN with dist(ΓD, ΓN) > 0. For the unknown scalar function u : R+ × Ω → R, we consider the subsequent problem: ⎧ ⎪⎨ ⎪⎩ ∂tu + dk ∗ (A(u) : ∇ 2 u) = F (u) + dk ∗ G(u), t ≥ 0, x ∈ Ω BD(u) = 0, t ≥ 0, x ∈ ΓD BN(u) = 0, t ≥ 0, x ∈ ΓN u|t=0 = u0, x ∈ Ω. (1.1) Here, (dk∗w)(t, x) = � t 0 dk(τ)w(t−τ, x), t ≥ 0, x ∈ Ω, ∂tu means the partial derivative of u w.r.t. t, ∇u = ∇xu is the gradient of u w.r.t. the spatial variables, ∇2u denotes its Hessian matrix, that is (∇2u)ij = ∂xi∂xj u, i, j ∈ {1, . . . , n}, and B : C = �n i=1, j=1 BijCij stands for the double scalar product of two matrices B, C ∈ Rn×n . Furthermore, we have the substitution operators A(u)(t, x) = −a(t, x, u(t, x), ∇u(t, x)), t ≥ 0, x ∈ Ω, F (u)(t, x) = f(t, x, u(t, x), ∇u(t, x)), t ≥ 0, x ∈ Ω, G(u)(t, x) = g(t, x, u(t, x), ∇u(t, x)), t ≥ 0, x ∈ Ω, BD(u)(t, x) = b D (t, x, u(t, x)), t ≥ 0, x ∈ ΓD, BN(u)(t, x) = b N (t, x, u(t, x), ∇u(t, x)), t ≥ 0, x ∈ ΓN, where a is R n×n -valued, and f, g, b D , b N are all scalar functions. The scalar-valued kernel k is of bounded variation on each compact interval [0, T ] with k(0) = 0, and belongs to a certain kernel class with parameter α ∈ [0, 1) which contains, roughly speaking, all ’regular’ kernels that behave like t α for t (> 0) near zero. Note that this 3
Page 1 and 2: Gutachter: Quasilinear parabolic pr
Page 3: Contents 1 Introduction 3 2 Prelimi
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
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Let u1 be the restriction of v1 to
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Proof. We begin with the necessity
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Chapter 4 Linear Problems of Second
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The strategy for solving (4.1) is n
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Since ψj ≡ 1 on supp ϕj, we may
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Turning to (c), let g ∈ Ξi+1 and
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endowed with the norm | · | Y T 2
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We remark that the constant C2 stem
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One can then construct functions a
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analogous to (4.17), shows that S i
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Apply now V#, i+1 := I + k ∗ A#(
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Given a function v ∈ H 2 p(R n+1
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v is a solution of (4.40) on Ji+1 :
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Chapter 5 Linear Viscoelasticity In
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where δij denotes Kronecker’s sy
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problem ⎧ ⎪⎨ ⎪⎩ ∂tv −
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To see the converse direction, supp
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and up solves � Aup − ∆xup
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It can be written as where l(z, ξ)
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elongs to H∞ (Σ π 2 +η × Ση
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which allows us to write the first
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Chapter 6 Nonlinear Problems 6.1 Qu
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sufficiently small, say T ≤ T1
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(d) bD ∈ C(J0 × ΓD × U0), ∃C
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which entails (6.14). Corresponding
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substitution operators to be studie
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for all t, τ ∈ J, ξ, η ∈ K,
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to write where h2(t, τ, x) = h21(t
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Lemma 6.2.3 Let 0 < s < s0 < 1, ρ
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Bibliography [1] Albrecht, D.: Func
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[54] Lunardi, A.: On the heat equat
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kleines T mit Hilfe des Kontraktion
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Personal Details Curriculum Vitae N
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Quasilinear parabolic problems with nonlinear boundary conditions

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