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Problem with integral condition for evolution equation

Kalenyuk P. I., Kuduk G., Kohut I. V., Nytrebych Z. M.

Journal of Mathematics and Applications

2015

Vol. 38

71--76

We propose a method of solving the problem with non-homogeneous integral condition for homogeneous evolution equation with abstract operator in a linear space H. For right-hand side of the integral condition which belongs to the special subspace H ⊆ L, in which the vectors are represented using Stieltjes integrals over a certain measure, the solution of the problem is represented in the form of Stieltjes integral over the same measure.

Some theorems on fractional semilinear evolution equations

Tidke H. L.

Journal of Applied Analysis

2012

Vol. 18, nr 2

209-224

In this paper, we investigate the existence, uniqueness and other properties of solutions of fractional semilinear evolution equations in Banach spaces. The results are obtained by using fractional calculus, the well-known Banach fixed point theorem coupled with Bielecki type norm and the integral inequality established by E. Hernandez.

Non-isothermal phase-field models and evolution equation

Morro A.

Archives of Mechanics

2006

Vol. 58, nr 3

257-271

Phase transitions between two phases are modelled as space regions where a phase field, or order parameter, changes smoothly. The literature shows a seeming contradiction in that some papers lead to the use of the reduced chemical potential through the temperature, others do not. The paper has a threefold purpose. First, to revise the arguments of known approaches and possibly generalize the associated schemes. Secondly, to show that a further approach is possible which involves the phase field as an internal variable. Thirdly, to contrast the various schemes and the corresponding results. It follows that differences arise because different fields enter the models and different forms are considered for the balance of energy and the second law of thermodynamics.