Semilinear Cauchy problems with almost sectorial operators

• Autorzy: Dlotko, T.
• Języki publikacji: EN

Abstrakty

EN

Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Holder spaces illustrates the abstract result.

Słowa kluczowe

PL

zagadnienie Cauchy'ego

EN

Cauchy problem; integrated semigroup; almost sectorial operator; mild solution; classical solution

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 4
• Strony: 333-346
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Dlotko, T.

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

Bibliografia

• [A-Z] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Univ. Press, Cambridge, 1990.
• [Al] W. Arendt, Contributions to the theory of semigroup distributions, Ph.D. Dissertation, Univ. Antwerpen, 1983.
• [A2] —, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
• [A-C-L] J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, in preparation.
• [FR] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
• [H-P] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc., Providence RI, 1957.
• [H] L. Hormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93-139.
• [K-H] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), 160-180.
• [N-S] J. M. A. M. van Neerven and B. Straub, On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvent, Houston J. Math. 24 (1998), 137-171.
• [PA] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
• [P] F. Periago, Global existence, uniqueness, and continuous dependence for a semi-linear initial value problem, J. Math. Anal. Appl. 280 (2003), 413-423.
• [P-S] F. Periago and B. Straub, A functional calculus for almost sectorial operators and applications to abstract evolution equations, 3. Evol. Equations 2 (2002), 41-68.
• [T] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990), 416-447.