Quasilinear parabolic problems with nonlinear boundary conditions
Quasilinear parabolic problems with nonlinear boundary conditions Quasilinear parabolic problems with nonlinear boundary conditions
[22] Clément, Ph., Nohel, J.A.: Abstract linear and nonlinear Volterra equations preserving positivity. SIAM J. Math. Anal. 10 (1979), pp. 365-388. [23] Clément, Ph., Nohel, J.A.: Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J. Math. Anal. 12 (1981), pp. 514-535. [24] Clément, Ph., Prüss, J.: An operator-valued transference principle and maximal regularity on vector-valued Lp-spaces. In: Evolution Equ. and Appl. Physical Life Sciences, G. Lumer, L. Weis (eds.), Lect. Notes in Pure Appl. Math. 215 (2001), Marcel Dekker, New York, NY, pp. 67-87. [25] Clément, Ph., Prüss, J.: Completely positive measures and Feller semigroups. Math. Ann. 287 (1990), pp. 73-105. [26] Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl. 54 (1975), pp. 305-387. [27] Da Prato, G., Ianelli, M., Sinestrari, E.: Regularity of solutions of a class of linear integrodifferential equations in Banach spaces. J. Int. Equations 8 (1985), pp. 27-40. [28] Da Prato, G., Ianelli, M., Sinestrari, E.: Temporal regularity for a class of integrodifferential equations in Banach spaces. Boll. U.M.I. 2 (1983), pp. 171-185. [29] Denk, R., Hieber, M., Prüss, J.: R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, to appear in Memoirs AMS. [30] Doetsch, G.: Introduction to the Theory and Application of the Laplace Transformation. Springer, Berlin, Heidelberg, 1974. [31] Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196 (1987), pp. 189-201. [32] Dore, G., Venni, A.: Some results about complex powers of closed operators. J. Math. Anal. 149 (1990), pp. 124-136. [33] Engler, H.: Global smooth solutions for a class of parabolic integro-differential equations. Trans. Amer. Math. Soc. 348 (1996), pp. 267-290. [34] Engler, H.: Strong solutions of quasilinear integro-differential equations with singular kernels in several space dimensions. Electron. J. Differ. Equ. 1995 (1995), pp. 1-16. [35] Escher, J., Prüss, J., Simonett, G.: Analytic solutions for a Stefan problem with Gibbs-Thomson correction, to appear, 2003. [36] Escher, J., Prüss, J., Simonett, G.: Analytic solutions of the free boundary value problem for the Navier-Stokes equation. to appear, 2003. [37] Gripenberg, G.: Global existence of solutions of Volterra integro-differential equations of parabolic type. J. Differential Equations 102 (1993), pp. 382-390. [38] Gripenberg, G.: Nonlinear Volterra equations of parabolic type due to singular kernels. J. Differential Equations 112 (1994), pp.154-169. [39] Gripenberg, G., Londen, S. O., Staffans, O. J.: Volterra Integral and Functional Equations. Cambridge Univ. Press, Cambridge, 1990. [40] Grisvard, P.: Spaci di trace e applicazioni. Rend. Math. 5 (1972), pp. 657-729. [41] Gurtin, M. E.: An Introduction to Continuum Mechanics. Acad. Press, New York, 1981. [42] Hieber, M., Prüss, J.: Functional calculi for linear operators in vector-valued L p -spaces via the transference principle. Adv. Diff. Equ. 3 (1998), pp. 847-872. [43] Hieber, M., Prüss, J.: Maximal regularity of parabolic problems. Monograph in preparation, 2003. [44] Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, Rhode Island, 1957. [45] Hrusa, W.J., Renardy, M.: On a class of quasilinear partial integrodifferential equations with singular kernels. J. Differential Equations 64 (1986), pp.195-220. [46] Hrusa, W.J., Renardy, M.: A model equation for viscoelasticity with a strongly singular kernel. SIAM J. Math. Anal 19 (1988), pp. 257-269. [47] Kalton, N., Weis, L.: The H ∞ -calculus and sums of closed operators. Preprint, 2000. [48] Komatsu, H.: Fractional powers of operators. Pacific J. Math. 19 (1966), pp. 285-346. [49] Komatsu, H.: Fractional powers of operators II, Interpolation spaces. Pacific J. Math. 21 (1967), pp. 89-111. [50] Ladyzenskaja, O. A., Solonnikov, V. A., Uralceva, N. N.: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. Transl. Math. Monographs, Providence, R. I., 1968. [51] Lancien, F., Lancien, G., Le Merdy, C.: A joint functional calculus for sectorial operators with commuting resolvents. Proc. London Math. Soc. 77 (1998), pp. 387-414. [52] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995. [53] Lunardi, A.: Laplace transform methods in integrodifferential equations. J. Int. Eqns. 10 (1985), pp. 185-211. 112
[54] Lunardi, A.: On the heat equation with fading memory. SIAM J. Math. Anal. 21 (1990), pp.1213- 1224. [55] Lunardi, A., Sinestrari, E.: Existence in the large and stability for nonlinear Volterra equations. J. Int. Eqns. 10 (1985), pp. 213-239. [56] Lunardi, A., Sinestrari, E.: Fully nonlinear integro-differential equations in general Banach space. Math. Z. 190 (1985), pp. 225-248. [57] McIntosh, A.: Operators which have an H ∞ -calculus. In: Miniconference on Operator Theory and PDE, B. Jefferies, A. McIntosh, W. Ricker (eds.), Proc. Center Math. Anal. (1986), pp. 210-231. [58] Monniaux, S., Prüss, J.: A theorem of the Dore-Venni type for noncommuting operators. Trans. Am. Math. Soc. 349 (1997), pp. 4787-4814. [59] Nohel, J.A.: Nonlinear Volterra equations for heat flow in materials with memory. Integral and Functional Differential Equations, T.L. Herdman, S.M. Rankin III, and H.W. Stech, eds., Lecture Notes in Pure and Applied Mathematics 67 (1981), Marcel Dekker, New York, pp. 3-82. [60] Nunziato, J.W.: On heat conduction in materials with memory. Quart. Appl. Math. 29 (1971), pp. 187-204. [61] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983. [62] Pipkin, A. C.: Lectures on Viscoelastic Theory. Applied Mathematical Science 7, Springer, Berlin, 1972. [63] Prüss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser, Basel, 1993. [64] Prüss, J.: Laplace transforms and regularity of solutions of evolutionary integral equations. Preprint, 1996. [65] Prüss, J.: Maximal Regularity for Abstract Parabolic Problems with Inhomogeneous Boundary Data in Lp-spaces. Preprint, 2002. [66] Prüss, J.: Maximal regularity of linear vector valued parabolic Volterra equations. J. Integral Eqns. Appl. 3 (1991), pp. 63-83. [67] Prüss, J.: Poisson estimates and maximal regularity for evolutionary integral equations in Lpspaces. Rend. Istit. Mat. Univ. Trieste Suppl. 28 (1997), pp. 287-321. [68] Prüss, J.: Quasilinear parabolic Volterra equations in spaces of integrable functions. In B. de Pagter, Ph. Clément, E. Mitidieri, editors, Semigroup Theory and Evolution Equations. Lect. Notes Pure Appl. Math. 135 (1991), Marcel Dekker, New York, pp. 401-420. [69] Prüss, J., Sohr, H.: On operators with bounded imaginary powers in Banach spaces. Math. Z. 203 (1990), pp. 429-452. [70] Prüss, J., Sohr, H.: Imaginary powers of elliptic second order differential operators in L p -spaces. Hiroshima Math. J. 23 (1993), pp. 161-192. [71] Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical Problems in Viscoelasticity. Pitman Monographs Pure Appl. Math. 35, Longman Sci. Tech., Harlow, Essex, 1988. [72] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. De Gruyter, Berlin, 1996. [73] Schmeisser, H.-J.: Vector-valued Sobolev and Besov spaces. in: Sem. Analysis 1885/86, Teubner Texte Math. 96 (1986), pp. 4-44. [74] Sinestrari, E.: Continouos interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations. J. Int. Equations 5 (1983), pp. 287-308. [75] Sobolevskii, P.E.: Coerciveness inequalities for abstract parabolic equations. Soviet Math. (Doklady) 5 (1964), pp. 894-897. [76] ˇ Strkalj, ˇ Z.: R-Beschränktheit, Summensätze abgeschlossener Operatoren und operatorwertige Pseudodifferentialoperatoren. Thesis, University of Karlsruhe, 2000. [77] Triebel, H.: Higher Analysis. Johann Ambrosius Barth, Leipzig, 1992. [78] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, 1978. [79] Triebel, H.: Theory of Function Spaces. Geest & Portig K.-G., Leipzig, 1983. [80] Weis, L.: Operator-valued Fourier multiplier theorems and maximal Lp-regularity. Preprint, 1999. [81] Weis, L.: A new approach to maximal Lp-regularity. 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. 195-214. [82] Yagi, A.: Coincidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs. C.R. Acad. Sci. Paris 299 (1984), pp. 173-176. [83] Zimmermann, F.: Mehrdimensionale vektorwertige Fouriermultiplikatorensätze und vektorwertige Funktionenräume. Thesis, Christian-Albrechts-Universität Kiel, 1987. 113
- 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: Bibliography [1] Albrecht, D.: Func
- Page 117 and 118: kleines T mit Hilfe des Kontraktion
- Page 119: Personal Details Curriculum Vitae N
[22] Clément, Ph., Nohel, J.A.: Abstract linear and <strong>nonlinear</strong> Volterra equations preserving positivity.<br />
SIAM J. Math. Anal. 10 (1979), pp. 365-388.<br />
[23] Clément, Ph., Nohel, J.A.: Asymptotic behavior of solutions of <strong>nonlinear</strong> Volterra equations <strong>with</strong><br />
completely positive kernels. SIAM J. Math. Anal. 12 (1981), pp. 514-535.<br />
[24] Clément, Ph., Prüss, J.: An operator-valued transference principle and maximal regularity on<br />
vector-valued Lp-spaces. In: Evolution Equ. and Appl. Physical Life Sciences, G. Lumer, L. Weis<br />
(eds.), Lect. Notes in Pure Appl. Math. 215 (2001), Marcel Dekker, New York, NY, pp. 67-87.<br />
[25] Clément, Ph., Prüss, J.: Completely positive measures and Feller semigroups. Math. Ann. 287<br />
(1990), pp. 73-105.<br />
[26] Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles.<br />
J. Math. Pures Appl. 54 (1975), pp. 305-387.<br />
[27] Da Prato, G., Ianelli, M., Sinestrari, E.: Regularity of solutions of a class of linear integrodifferential<br />
equations in Banach spaces. J. Int. Equations 8 (1985), pp. 27-40.<br />
[28] Da Prato, G., Ianelli, M., Sinestrari, E.: Temporal regularity for a class of integrodifferential<br />
equations in Banach spaces. Boll. U.M.I. 2 (1983), pp. 171-185.<br />
[29] Denk, R., Hieber, M., Prüss, J.: R-Boundedness, Fourier Multipliers and Problems of Elliptic and<br />
Parabolic Type, to appear in Memoirs AMS.<br />
[30] Doetsch, G.: Introduction to the Theory and Application of the Laplace Transformation. Springer,<br />
Berlin, Heidelberg, 1974.<br />
[31] Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196 (1987),<br />
pp. 189-201.<br />
[32] Dore, G., Venni, A.: Some results about complex powers of closed operators. J. Math. Anal. 149<br />
(1990), pp. 124-136.<br />
[33] Engler, H.: Global smooth solutions for a class of <strong>parabolic</strong> integro-differential equations. Trans.<br />
Amer. Math. Soc. 348 (1996), pp. 267-290.<br />
[34] Engler, H.: Strong solutions of quasilinear integro-differential equations <strong>with</strong> singular kernels in<br />
several space dimensions. Electron. J. Differ. Equ. 1995 (1995), pp. 1-16.<br />
[35] Escher, J., Prüss, J., Simonett, G.: Analytic solutions for a Stefan problem <strong>with</strong> Gibbs-Thomson<br />
correction, to appear, 2003.<br />
[36] Escher, J., Prüss, J., Simonett, G.: Analytic solutions of the free <strong>boundary</strong> value problem for the<br />
Navier-Stokes equation. to appear, 2003.<br />
[37] Gripenberg, G.: Global existence of solutions of Volterra integro-differential equations of <strong>parabolic</strong><br />
type. J. Differential Equations 102 (1993), pp. 382-390.<br />
[38] Gripenberg, G.: Nonlinear Volterra equations of <strong>parabolic</strong> type due to singular kernels. J. Differential<br />
Equations 112 (1994), pp.154-169.<br />
[39] Gripenberg, G., Londen, S. O., Staffans, O. J.: Volterra Integral and Functional Equations. Cambridge<br />
Univ. Press, Cambridge, 1990.<br />
[40] Grisvard, P.: Spaci di trace e applicazioni. Rend. Math. 5 (1972), pp. 657-729.<br />
[41] Gurtin, M. E.: An Introduction to Continuum Mechanics. Acad. Press, New York, 1981.<br />
[42] Hieber, M., Prüss, J.: Functional calculi for linear operators in vector-valued L p -spaces via the<br />
transference principle. Adv. Diff. Equ. 3 (1998), pp. 847-872.<br />
[43] Hieber, M., Prüss, J.: Maximal regularity of <strong>parabolic</strong> <strong>problems</strong>. Monograph in preparation, 2003.<br />
[44] Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. Amer. Math. Soc. Colloq. Publ. 31,<br />
Amer. Math. Soc., Providence, Rhode Island, 1957.<br />
[45] Hrusa, W.J., Renardy, M.: On a class of quasilinear partial integrodifferential equations <strong>with</strong><br />
singular kernels. J. Differential Equations 64 (1986), pp.195-220.<br />
[46] Hrusa, W.J., Renardy, M.: A model equation for viscoelasticity <strong>with</strong> a strongly singular kernel.<br />
SIAM J. Math. Anal 19 (1988), pp. 257-269.<br />
[47] Kalton, N., Weis, L.: The H ∞ -calculus and sums of closed operators. Preprint, 2000.<br />
[48] Komatsu, H.: Fractional powers of operators. Pacific J. Math. 19 (1966), pp. 285-346.<br />
[49] Komatsu, H.: Fractional powers of operators II, Interpolation spaces. Pacific J. Math. 21 (1967),<br />
pp. 89-111.<br />
[50] Ladyzenskaja, O. A., Solonnikov, V. A., Uralceva, N. N.: Linear and <strong>Quasilinear</strong> Equations of<br />
Parabolic Type. Amer. Math. Soc. Transl. Math. Monographs, Providence, R. I., 1968.<br />
[51] Lancien, F., Lancien, G., Le Merdy, C.: A joint functional calculus for sectorial operators <strong>with</strong><br />
commuting resolvents. Proc. London Math. Soc. 77 (1998), pp. 387-414.<br />
[52] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel,<br />
1995.<br />
[53] Lunardi, A.: Laplace transform methods in integrodifferential equations. J. Int. Eqns. 10 (1985),<br />
pp. 185-211.<br />
112
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