MaximalLp-regularity for a second-order differential equation with unbounded intermediate coefficient

Correctness conditions for high-order differential equations with unbounded coefficients

“…In the case that the intermediate coefficients do not depend on the potential and the diffusion coefficient and can grow as a linear function, the correctness of the secondorder singular elliptic equations was studied in [7][8][9][10]. The correctness conditions for the second-order and third-order one-dimensional differential equations with rapidly growing intermediate coefficients were obtained in [11][12][13][14][15][16]. However, in [11][12][13][14][15][16] the condition of weak oscillation is imposed on the intermediate and senior coefficients.…”

“…The correctness conditions for the second-order and third-order one-dimensional differential equations with rapidly growing intermediate coefficients were obtained in [11][12][13][14][15][16]. However, in [11][12][13][14][15][16] the condition of weak oscillation is imposed on the intermediate and senior coefficients. In this paper, sufficient conditions for the existence and uniqueness of a solution y(x) of (1) are obtained.…”

Solvability and maximal regularity results for a differential equation with diffusion coefficien

“…The question arises whether there exists a unique solution of equation 1.1, if |r(x)| grows more rapidly than |x| ln |x| (|x| 1) and cannot be controlled by the coefficient s(x) . It is also interesting to consider the case when the coefficient ρ(x) in the leading term of the equation ( In contrast with [13,14], in the current paper we consider the equation (1.1) with the coefficient ρ(x) in the leading term. The study of (1.1) is not only of theoretical interest.…”

“…It is known that an operator l above arises as generator of the transition semigroup of a stochastic Ornstein-Uhlenbeck process that determines a Brownian motion with a variable covariance matrix connected with ρ(x) . Studying (1.1) with the coefficient ρ(x), we overcame new difficulties compared to [14], such as the choice and estimation of the linear functional in Theorem 3.1, as well as the construction of the operators B λ and M λ , and the estimation of the norm B λ in Theorem 3.4. Furthermore, if ρ(x) tends to zero at infinity, then we may consider the degeneracy case.…”

Regularity for Semilinear Neutral Hyperbolic Equations with Cosine Families

“…It is worth giving several examples of the use of such and other classes of differential equations in engineering and scientific tasks, for instance, almost automorphic mild solutions of hyperbolic evolution equations with stepanov-like almost automorphic forcing term have studied in [13], and the local well-posedness to the Cauchy problem of the 2D compressible Navier-Stokes-Smoluchowski equations with vacuum was considered in [14]. Recently, regularity problems of second order differential equations have been discussed in [15][16][17]. The fixed point of a locally asymptotically nonexpansive cosine family is introduced in [17,18].…”