Publication: Linear parabolic equations in locally uniform spaces
Loading...
Full text at PDC
Publication Date
2004
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
World Scientific
Citations
Exportar
Abstract
We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp L-p - L-q-type estimates in uniform spaces for heat and Schrodinger semigroups and analyze the regularizing effect and the exponential type of these semigroups. We also deal with general second-order elliptic operators and study the generation of analytic semigoups in uniform spaces.
Description
UCM subjects
Unesco subjects
Keywords
Citation
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Diff. Equation 83 (1990) 85–108.
R. A. Adams, Sobolev Spaces (Academic Press, 1975).
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Teubner, 1993) pp. 9–126.
H. Amann, Linear and Quasilinear Parabolic Problems (Birkhäuser, 1995).
H. Amann, M. Hieber and G. Simonett, Bounded H ∞ -calculus for elliptic operators, Diff. Int. Eqns. 3 (1994) 613–653.
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems. J. Diff. Eqns. 146 (1998) 336–374.
W. Arendt and C. J. K. Batty, Exponential stability of a diffusion equation with absorption, Diff. Int. Eqns. 6 (1993) 1009–1024.
W. Arendt and C. J. K. Batty, Absorption semigroups and Dirichlet boundary conditions, Math. Ann. 295 (1993) 427–448.
J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal. 56 (2004) 515–554.
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh A116 (1990) 221–243.
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (North-Holland, 1991).
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction (Springer, 1976).
J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, to appear in Czech. J. Math. cf.
D. Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr. 217 (2000) 13–41.
G. Duro and E. Zuazua, Large time behavior for convection-diffusion equations in R N with asymptotically constant diffusion, Comm. PDE 24 (1999) 1283–1340.
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators (Cambridge Univ. Press, 1996).
M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688.
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987) 1103–1133.
E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on R N , Diff. Int. Eqns. 9 (1996) 1147–1156.
E. Feireisl, Ph. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Diff. Eqns. 129 (1996) 239–261.
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice Hall, 1964).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second-Order (Springer, 1998).
J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation I. Compactness methods, Physica D95 (1996) 191–228.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (Springer, 1981).
J. J. L. Velazquez, Higher dimensional blow up for semilinear parabolic equations, Comm. Partial Diff. Eqns. 17 (1992) 1567–1686.
M. Hieber, P. Koch Medina and S. Merino, Linear and semilinear parabolic equation on BUC(R N ) , Math. Narch. 179 (1996) 107–118.
T. Kato, The Cauchy problem for quasi–linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58 (1975) 181–205.
A. Lunardi, Analytic Semigroup and Optimal Regularity in Parabolic Problems (Birkhäuser, 1995). MR1329547 (96e:47039)
J. Matos and P. Souplet, Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations, Preprint. cf. MR2083878 (2005e:35026)
S. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on R N , J. Diff. Eqns. 132 (1996) 87–106. MR1418501 (97h:35102)
A. Mielke, The complex Ginzburg–Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity 10 (1997) 199–222. MR1430749 (97m:35242)
A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity 8 (1995) 743–768. MR1355041 (97e:58207)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, 1983). MR0710486 (85g:47061)
B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982) 447–526. MR0670130 (86b:81001a)
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Veb Deutscher, 1978). MR0503903 (80i:46032b)