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Hyperbolic equations in uniform spaces

Cholewa J. W., Dlotko T.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 3

249-263

The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of R^n. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space H[...]^R^n) x L[...](R^n). Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.

Asymptotics of pseudodifferential parabolic equations

Cholewa J. W., Dłotko T., Turski A. W.

Demonstratio Mathematica

2002

Vol. 35, nr 1

74-91

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.