Degenerate Differential Equations Research Articles

Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ( $P_4$ ): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$ , where $A$ , $B$ , $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$ , $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$ ) into $X$ , $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$ . We completely characterize the well-posedness of ( $P_4$ ) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

Periodic solutions of second‐order degenerate differential equations with infinite delay in Banach spaces

AbstractWe consider the well‐posedness of the second‐order degenerate differential equations with infinite delay on [0, 2π] in Lebesgue–Bochner spaces and periodic Besov spaces , where , and M are closed linear operators in a Banach space X satisfying and the kernels . Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in and . These results are applied to examine some concrete examples.

Well‐posedness of second‐order degenerate differential equations with finite delay on Lp(ℝ;X)$$ {L}^p\left(\mathbb{R};X\right) $$

Using operator‐valued ‐Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well‐posedness of second‐order degenerate differential equations with finite delay on , where and are closed linear operators defined on a Banach space , the operators and are in for some fixed , and when and . These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations.

Features of constructing a solution clausen-type systems

We study the features of constructing solutions near a singularity of a regular homogeneous system consisting of two partial differential equations of the third order. A number of properties of the product of Clausen functions constructed near these singularities are proved. The features of the construction of the general solution of the Clausen system are investigated. Recently, special attention has begun to be removed on the solution of such systems, in connection with the research of multidimensional degenerate differential equations.

Degenerate Higher-Order Ordinary Differential Equations and Some of Their Applications

Degenerate Higher-Order Ordinary Differential Equations and Some of Their Applications

Construction of asymptotic solutions of some degenerate differential equations with a small parameter

Construction of asymptotic solutions of some degenerate differential equations with a small parameter

Strict Hölder regularity for fractional order abstract degenerate differential equations

Strict Hölder regularity for fractional order abstract degenerate differential equations

Well-posedness of second order degenerate differential equations with infinite delay in Hölder continuous function spaces

Using operator-valued Ċα-Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the Cα-well-posedness of second order degenerate differential equations with infinite delay: (Mu)′′(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t) and (Mu′)′(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t) on ℝ, where A:D(A)→X and M:D(M)→X are closed linear operators in a complex Banach space X, and a∈L1(ℝ+)∩L1(ℝ+;tαdt).

Well-Posedness of Third Order Degenerate Differential Equations with Finite Delay in Banach Spaces

In this paper, we give necessary and sufficient conditions for the $$L^p$$ -well-posedness (resp. $$B_{p,q}^s$$ -well-posedness) for the third order degenerate differential equation with finite delay: $$(Mu)'''(t) + (Nu)''(t)= Au(t) + Bu'(t) + Gu''_t + Fu'_t + Hu_t + f(t)$$ on $$[0,2\pi ])$$ with periodic boundary conditions $$(Mu)(0) = (Mu)(2\pi )$$ , $$(Mu)'(0) = (Mu)'(2\pi )$$ , $$(Mu)''(0) = (Mu)''(2\pi )$$ , where A, B, M and N are closed linear operators on a Banach space X satisfying $$D(A)\cap D(B)\subset D(M)\cap D(N)$$ , the operators G, F and H are bounded linear from $$L^p([-2\pi ,0];X)$$ (resp. $$B_{p,q}^s([-2\pi ,0];X)$$ ) into X.

Degenerate differential problems with fractional derivatives

We describe an extension of previous results on degenerate abstract equations described by Favini and Yagi in their monograph cited in the references.This allows us to study degenerate differential equations with fractional derivative in time. Both direct and relative inverse problems are studied. Finally, various applications to degenerate partial differential equations are described.

On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media

This paper is concerned with the global existence, uniqueness and homogenization of degenerate partial differential equations with integral conditions arising from coupled transport processes and chemical reactions in three-dimensional highly heterogeneous porous media. Existence of global weak solutions of the microscale problem is proved by means of semidiscretization in time deriving a priori estimates for discrete approximations needed for proofs of existence and convergence theorems. It is further shown that the solution of the microscale problem is two-scale convergent to that of the upscaled problem as the scale parameter goes to zero. In particular, we focus our efforts on the contribution of the so-called first order correctors in periodic homogenization. Finally, under additional assumptions, we consider the problem of the uniqueness of the solution to the homogenized problem.

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.

Fractional Integral Inequalities and Their Applications to Degenerate Differential Equations with the Caputo Fractional Derivative

We provide some new integral inequalities with the Caputo fractional derivative and consider degenerate ODEs with the Caputo derivative and a fractional diffusion equation as application. Also, we prove the existence and uniqueness theorems of generalized solutions under certain conditions.

Global boundedness and stability of solutions of nonautonomous degenerate differential equations

Global boundedness and stability of solutions of nonautonomous degenerate differential equations

Linear and nonlinear degenerate differential operators and applications

The boundary value and mixed value problems for linear and nonlinear singular degenerate abstract elliptic and parabolic equations are studied. The linear problems involve some parameters. The uniform Lp- separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems.

Periodic solutions of fractional degenerate differential equations with delay in Banach spaces

We characterize the Lp-well-posedness (resp. $$B_{p,q}^s$$ -well-posedness) for the fractional degenerate differential equations with finite delay: $$D^\alpha(Mu)(t)=Au(t)+Gu'_t+Fu_t+f(t),\;\;\;(t\in[0,2\pi]),$$ where α > 0 is fixed and A, M are closed linear operators in a Banach space X satisfying D(A) ∩ D(M) ≠ {0}, F and G are bounded linear operators from Lp([-2π,0];X) (resp. $$B_{p,q}^s$$ ([-2π,0];X)) into X. We also give a new example to which our abstract results may be applied.

Well-Posedness of Fractional Degenerate Differential Equations in Banach Spaces

Abstract 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.

Hölder Continuous Solutions of Second Order Degenerate Differential Equations with Finite Delay

In this paper, we characterize the $$C^\alpha $$ -well-posedness of the second order degenerate differential equation with finite delay $$(Mu)''(t) = Au(t) + Fu_t + f(t)$$ , ( $$t\in {\mathbb R}$$ ) by using known operator-valued Fourier multiplier results on $$C^\alpha ({\mathbb R}; X)$$ , where A, M are closed linear operators on a complex Banach space X satisfying $$D(A)\cap D(M) \not =\{0\}$$ , $$r > 0$$ is fixed and F is a bounded linear operator from $$C([-r, 0]; X)$$ into X.

Nonlocal Boundary Value Problems with Partially Integral Conditions for Degenerate Differential Equations with Multiple Characteristics

We study the solvability of new local and nonlocal boundary-value problems for degenerate differential equations with multiple characteristics. We establish the existence of regularsolutions and discuss possible generalizations and improvements of the obtained result.

Краевые задачи для вырождающихся и невырождающихся дифференциальных уравнений дробного порядка с нелокальным линейным источником и разностные методы их численной реализации

Краевые задачи для вырождающихся и невырождающихся дифференциальных уравнений дробного порядка с нелокальным линейным источником и разностные методы их численной реализации