COERCIVE SOLVABILITY CONDITIONS OF THE THREE-ORDER DEGENERATE DIFFERENTIAL EQUATIONS

In this paper, we consider one class of the singular nonlinear third-order differential equations given on the entire axis. We show sufficient conditions for the existence of a solution to this equation and the satisfiability of the coercive estimate for solution. The considered equation has the following features. Its intermediate coefficient is not bounded and does not obey to a lower coefficient. In the literature, such equations are called the degenerate differential equations. Further, the corresponding differential operator is not semi-bounded: its energy space may not belong to the Sobolev classes. Previously, the solvability questions of the third-order singular differential equations was studied only in the case that their intermediate coefficients are equal to zero. The main result of this work is proved on the basis of one separability theorem for the linear third-order degenerate differential operators, Schauder's fixed point theorem and some Hardy type weighted integral inequalities.

Well-posedness of fractional degenerate differential equations in Banach spaces

We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.