Cauchy Problems and Applications

Of concern is the Cauchy problem du dt ∈ Au, u(0) = u0, t > 0, where u : [0,∞) → X, X is a real Banach space, and A : D(A) ⊂ X → X is nonlinear and multi-valued. It is showed by the method of lines, combined with the Crandall–Liggett theorem that this problem has a limit solution, and that the limit solution is a unique strong one if A is what is called embeddedly quasi-demi-closed. In the case of linear, single-valued A, further results are given. An application to nonlinear partial differential equations in non-reflexive X is given.