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In this paper, new classes of functions are defined. These spaces generalize Morrey spaces and give a refinement of Lebesgue spaces. Some embeddings between these new classes are also proved. Finally, the authors apply these classes of functions to obtain regularity results for solutions of partial differential equations of parabolic type. [ABSTRACT FROM AUTHOR]

In many fields of science and engineering, partial differential equation (PDE) constrained optimal control problems are widely used. We mainly solve the optimization problem constrained by the time-periodic eddy current equation in this paper. We propose the three-block splitting (TBS) iterative method and proved that it is unconditionally convergent. At the same time, the corresponding TBS preconditioner is derived from the TBS iteration method, and we studied the spectral properties of the preconditioned matrix. Finally, numerical examples in two-dimensions is applied to demonstrate the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method. [ABSTRACT FROM AUTHOR]

The aim of this paper was to propose a systematic study of a (1 + 1) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis and here present exact solutions of the equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show some regularity properties of solutions to Lu=f, where L is a second order nondivergence form operator and f belongs to a homogeneous Herz space K˙q(⋅)α,p(⋅) with two variable exponents. [ABSTRACT FROM AUTHOR]