THE EXISTENCE OF A WEAK SOLUTION OF THE SEMILINEAR FIRST-ORDER DIFFERENTIAL EQUATION IN A BANACH SPACE

Mariusz Jużyniec

Abstrakt

This paper is devoted to the investigation of the existence and uniqueness of a suitably defined weak solution of the abstract semilinear value problem u_ (t) = Au(t) + f(t; u(t)); u(0) = x with x 2 X; where X is a Banach space. We are concerned with two types of solutions: weak and mild. Under the assumption that A is the generator of a strongly continuous semigroup of linear, bounded operators, we also establish sufficient conditions such that if u is a weak (mild) solution of the initial value problem, then u is a mild (weak) solution of that problem.

Słowa kluczowe: operator, semigroup, weak solution
References

Ball J.M., Strongly continuous semigroups, weak solutions, and the variation of constant formula, Proc. Amer. Math. Soc., 1977, 370–373.

Hundertmark D., Meyries M., Machinek L., Schnaubelt R., Operator Semigroups and Dispersive Equations, 16th Internet Seminar on Evolution Equations, 2013.

Goldstein J., Semigroups of Linear Operators and Applications, Oxford U. Press, New York 1985.

Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, 1983.