Quasilinear parabolic problems with nonlinear boundary conditions

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Lemma 6.2.4 Let 0 < s1 < r < 1. Then there exists a constant C > 0 not depending on T such that [fg] Y 1,s 1 1 for all f, g ∈ Y 1,s1 1 Proof. Clearly Y 1,s1 1 ≤ C([f] 1,s Y 1 |g|∞ + |f|∞[g] 1,s 1 Y 1 + T 1 r−s1 [f]X1 [g]C r 1 + T r−s1 [f]C r[g]X1 ) 1 ∩ C r (J; C(Ω)). ↩→ X1. Suppose f, g ∈ Y 1,s1 1 ∩ C r (J; L∞(Ω)). We estimate [fg] 1,s Y 1 = 1 � T � T � |(ftg)(t, x) + (fgt)(t, x) − (ftg)(τ, x) − (fgt)(τ, x)| = ( 0 0 Ω p |t − τ| 1+s1p dx dτ dt) 1 p � T � T � (|ft(t, x) − ft(τ, x)||g(t, x)|) ≤ ( 0 0 Ω p |t − τ| 1+s1p dx dτ dt) 1 p + � T � T � (|ft(τ, x)||g(t, x) − g(τ, x)|) +( 0 0 Ω p |t − τ| 1+s1p dx dτ dt) 1 p + � T � T � (|f(t, x)||gt(t, x) − gt(τ, x)|) +( 0 0 Ω p |t − τ| 1+s1p dx dτ dt) 1 p + � T � T � (|f(t, x) − f(τ, x)||gt(τ, x)|) +( 0 0 Ω p |t − τ| 1+s1p dx dτ dt) 1 p ≤ [f] 1,s Y 1 |g|∞ + [g]C 1 r 1 ( � T � |ft(τ, x)| 0 Ω p � T dt ( ) dx dτ)1 p + 0 |t − τ| 1+(s1−r)p +|f|∞[g] 1,s Y 1 + [f]C 1 r 1 ( � T � |gt(τ, x)| 0 Ω p � T dt ( ) dx dτ)1 p 0 |t − τ| 1+(s1−r)p ≤ [f] 1,s Y 1 |g|∞ + C1T 1 r−s1 [f]X1 [g]C r 1 + |f|∞[g] 1,s Y 1 + C1T 1 r−s1 [f]C r[g]X1 , 1 where C1 = [2/(r − s1)p] 1/p . � A corresponding estimate can be obtained for [fg] 1,s Y 2 , so together with the inequalities 2 from Section 4.2.2, we see that Y k1,s1 1 ∩ Y 1,s2 2 with k1 ∈ {0, 1} is a multiplication algebra provided the above assumptions on p are fulfilled. We conclude this section by justifying the normalization step which we carried through in Section 6.1 for the coefficients on the boundary. Let f, g ∈ Y T N and f(t, x) > 0, t ∈ J, x ∈ ΓN. By compactness of ΓN, we even have f(t, x) ≥ c, t ∈ J, x ∈ ΓN, for some positive constant c. Set K = (c/2, ∞) and consider the function b : K → R defined by b(ξ) = 1/ξ. Clearly f is K-valued, b ∈ C ∞ (K) and b, b ′ are bounded. Therefore 1/f ∈ Y T N . Since Y T N is a multiplication algebra, we deduce that g/f ∈ Y T N , too. By an analogous argument, this property can also be proved for the space Y T D . 110
Bibliography [1] Albrecht, D.: Functional calculi of commuting unbounded operators. Ph.D. thesis, Monash University, Melbourne, Australia, 1994. [2] Albrecht, D., Franks, E., McIntosh, A.: Holomorphic functional calculi and sums of commuting operators. Bull. Austral. Math. Soc. 58 (1998), pp. 291-305. [3] Antman, Stuart S.: Nonlinear problems of elasticity. Applied Mathematical Sciences 107. Springer, New York, 1995. [4] Appell, J., Zabrejko, P.P.: Nonlinear superposition operators. Cambridge University Press, Cambridge, 1990. [5] Amann, H.: Linear and Quasilinear Parabolic Problems, Vol. I. Abstract Linear Theory. Monographs in Mathematics 89, Birkhäuser, Basel, 1995. [6] Amann, H.: Operator-valued Fourier Multiplier, Vector-valued Besov spaces, and Applications. Math. Nachr. 186 (1997), pp. 5-56. [7] Arendt, W., Batty, Ch., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96, Birkhäuser, Basel, 2001. [8] Bajlekova, E.G.: Fractional evolution equations in Banach spaces. Thesis, Eindhoven University Press, Eindhoven, Univ. of Tech., 2001. [9] Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundl. Math. Wiss. 223, Springer Verlag, Berlin, 1976. [10] Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. In G. Letta and M. Pratelli, editors, Probability and Analysis, Lect. Notes Math. 1206, pp. 61-108, Springer, Berlin, 1986. [11] Chow, T.S.: Mesoscopic physics of complex materials. Graduate Texts in Contemporary Physics. New York, NY: Springer, 2000. [12] Christensen, R.M.: Theory of Viscoelasticity. An Introduction. Academic Press, New York, San Fransisco, London, 1982. [13] Clément, Ph.: Maximal Lp-regularity and R-sectorial operators. RIMS Kokyuroku 1197 (2001), pp. 108-121. [14] Clément, Ph.: On the method of sums of operators. Semi-groupes d’opérateurs et calcul fonctionnel. Ecole d’été, Besançon, France, Juin 1998. Besançon: Université de Franche-Compté et CNRS, Equipe de Mathématiques, Publ. Math. UFR Sci. Tech. Besançon. 16 (1998), pp. 1-30. [15] Clément, Ph., Da Prato, G.: Existence and regularity results for an integral equation with infinite delay in a Banach space. Integral Equations Oper. Theory 11 (1988), pp. 480-500. [16] Clément, Ph., Gripenberg, G., Högnäs, V.: Some remarks on the method of sums. In Gesztesy, Fritz (ed.) et al., Stochastic processes, physics and geometry: New interplays. II. Providence, RI: American Mathematical Society (AMS). CMS Conf. Proc. 29 (2000), pp. 125-134. [17] Clément, Ph., Gripenberg, G., Londen, S.-O.: Regularity properties of solutions of fractional evolution equations. In Lumer, Günter (ed.) et al., Evolution equations and their applications in physical and life sciences. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 215 (2001), pp. 235-246. [18] Clément, Ph., Gripenberg, G., Londen, S.-O.: Schauder estimates for equations with fractional derivatives. Trans. Am. Math. Soc. 352 (2000), pp. 2239-2260. [19] Clément, Ph., Gripenberg, G., Londen, S.-O.: Hölder regularity for a linear fractional evolution equation. In Escher, Joachim (ed.) et al., Topics in nonlinear analysis. Basel: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 35 (1999), pp. 69-82. [20] Clément, Ph., Li, S.: Abstract parabolic quasilinear evolution equations and applications to a groundwater problem. Adv. Math. Sci. Appl. 3 (1994), pp. 17-32. [21] Clément, Ph., Londen, S.-O.: Regularity aspects of fractional evolution equations. Rend. Istit. Mat. Univ. Trieste 31 (2000), pp. 19-30. 111
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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
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3.3 More time regularity for Volter
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Theorem 3.4.2 Suppose X is a Banach
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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
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: Lemma 6.2.3 Let 0 < s < s0 < 1, ρ
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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