Question

what is mild solution and semi classical solution?

Answer 1

Abstract Cauchy problems

Consider the abstract Cauchy problem:

where A is a closed operator on a Banach space X and x∈X. There are two concepts of solution of this problem:
a continuously differentiable function u:[0,∞)→X is called a classical solution of the Cauchy problem if u(t) ∈ D(A) for all t ≥ 0 and it satisfies the initial value problem,
a continuous function u:[0,∞) → X is called a mild solution of the Cauchy problem if

Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable.[5]
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem[6] Let A be a closed operator on a Banach space X. The following assertions are equivalent:
for all x∈X there exists a unique mild solution of the abstract Cauchy problem,
the operator A generates a strongly continuous semigroup,
the resolvent set of A is nonempty and for all x ∈ D(A) there exists a unique classical solution of the Cauchy problem.
When these assertions hold, the solution of the Cauchy problem is given by u(t) = T(t)x with T the strongly continuous semigroup generated by A.