Abstract:
We study a nonlocal abstract Ginzburg–Landau type equation. The equation includes variable coefficients with convolution terms and an abstract linear operator function $A$ in a Fourier-type Banach space $E$. For sufficiently smooth initial data, assuming growth conditions for the operator $A$ and the coefficient $a$, the existence and uniqueness of the solution and the $L^p$ -regularity properties are established. We obtain the existence and uniqueness of the solution, and the regularity of different classes of nonlocal Ginzburg–Landau-type equations by choosing the space $E$ and operator $A$ that occur in a wide variety of physical systems.
Keywords:diffusion equations, Ginzburg–Landau equation, dissipative operators, embedding in Sobolev and Besov spaces, $L^p$-regularity property of solutions, Fourier multipliers.
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