Asymptotics of pseudodifferential parabolic equations

• Autorzy: Cholewa J. W. , Dłotko T. , Turski A. W.
• Języki publikacji: EN

Abstrakty

EN

The paper provides new type examples covered by the general theory of global attractors for abstract parabolic equations presented in the monograph [C-D 1]. Inside the class of sectorial equations of the form (1) u+Au = F(u), t > 0, u(0) = uo, we cover pseudodifferential parabolic problems (2) m = -(-A)u + f(u), a należy (0,1), studied with suitable initial-boundary conditions and also their generalizations to problems with the main part being a finite sum of the fractional powers.

Słowa kluczowe

EN

fractional parabolic equation; global solution; global attractor

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2002
• Tom: Vol. 35, nr 1
• Strony: 74--91
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Cholewa J. W.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

autor

Dłotko T.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

autor

Turski A. W.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

Bibliografia

• [AM] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhàuser, Basel, 1995.
• [B-P-F-S] C. B ardos, P. Penel, U. Frisch and P.-L. Sulem, Modified dissipativity for a nonlinear evolution equations arising in turbulence, Arch. Rational Mech. Anal. 71 (1979), 237-256.
• [B-K-W 1] P. Biler, G. Karch and W. A. Woyczynski, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252.
• [B-K-W 2] P. Biler, G. Karch and W. A. Woyczynski, Critical nonlinearity and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré - Analyse non Lineaire 18 (2001), 613-637.
• [C-C-D] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Examples of global attractors in parabolic problems, Hokkaido Math. J. 27 (1998), 77-103.
• [C-D 1] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.
• [C-D 2] J. W. Cholewa and T. Dlotko, Abstract parabolic problem with non-Lipschitz nonlinearity, in: Evolution equations: existence, regularity and singularities, Banach Center Publications, 52, PWN, Warsaw, 2000, 73-81.
• [DA] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
• [FR] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
• [F-S-Z] U. Frisch, M. F. Shlesinger and G. M. Zaslavsky (Eds.), Lévy Flights and Related Topics in Physics, Springer-Verlag, Berlin, 1995.
• [G-G-S] M. Giga, Y. Giga and H. Sohr, Lp estimates for the Stokes system, in: Functional Analysis and Related Topics, H. Komatsu (Ed.), Springer-Verlag, Berlin, 1993, 55-67.
• [GU 1] D. Guidetti, On interpolation with boundary conditions, Math. Z. 207 (1991), 439-460.
• [GU 2] D. Guidetti, On elliptic systems in L1, Osaka J. Math. 30 (1993), 397-429.
• [HA] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, R. I., 1988.
• [HE] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
• [KA] T. Kato, Schrodinger operators with singular potentials, Israel J. Math. 13 (1973), 135-148.
• [KO] H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285-346.
• [L-M] J. H. Lightbourne, R. H. Martin, Relatively continuous nonlinear perturbations of analytic semigroups, Nonlinear Analysis TMA 1 (1977), 277-292.
• [PA] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
• [P-S] J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J. 23 (1993), 161-192.
• [TA] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, Inc., New York, 1997.
• [TR] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher Verlag, Berlin, 1978.
• [VA] V. Varlamov, Nonlinear heat equation with a fractional Laplacian in a disk, Colloq. Math. 81 (1999), 101-122.
• [YO] K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1971.