on the rectangular domain G = (0, a)× (0, b) . Several conditions for the uniform separability an... more on the rectangular domain G = (0, a)× (0, b) . Several conditions for the uniform separability and the resolvent estimates for the corresponding linear problem are given in abstract Lp-spaces. Especially, we prove that the linear differential operator is positive and is a generator of an analytic semigroup. Moreover, the existence and uniqueness of maximal regular solution of the above nonlinear problem are obtained. One of the important characteristics of these DOEs are that the degeneration process are taking place at different speeds at boundary, in general. Maximal regularity properties of regular degenerated nonlinear DOEs are studied e.g. in [1, 8, 10] . Unlike to these we consider here the singular degenerate DOEs. In applications maximal regularity properties of infinite systems of singular degenerate PDE are studied. Let γ = γ (x) , x = (x1, x2, ..., xn) be a positive measurable function on a domain Ω ⊂ R. Let Lp,γ (Ω;E) denote the space of strongly measurable E-valued func...
Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary... more Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.
Electronic Journal of Qualitative Theory of Differential Equations, 2021
In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations tha... more In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here, L p -estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness, L p -regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated wi...
In this paper, the integral initial value problems for Boussinesq type equations are studied. The... more In this paper, the integral initial value problems for Boussinesq type equations are studied. The equation include the general differential operators. The existence, uniqueness and regularity properties of solution of these problems are obtained. By choosing differential operators including in the equation, the regularity properties of the Cauchy problem for different type of Boussinesg equations are studied.
In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equations are studi... more In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equations are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term
The boundary value problems for the degenerate differential-operator equations with small paramet... more The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued Lp-spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.
We presents the study the separability properties for differential-operator equations in Morrey s... more We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation is obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied... more In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied.The equation involves convolution terms with a general kernel functions whose Fourier transform are operator functions defined in a Banach space E together with some growth conditions. Here, assuming enough smoothness on the initial data and the operator functions, the local, global existence, uniqueness and regularity properties of solutions are established in terms of fractional powers of given sectorial operator functon. Furthermore, conditions for finite time blow-up are provided. By choosing the space E and the operators, the regularity properties the wide class of nonlocal wave equations in the field of physics are obtained.
The abstract elliptic and parabolic equations on exterior domain are considered. The equations ha... more The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
The abstract elliptic and parabolic equations on exterior domain are considered. The equations ha... more The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
on the rectangular domain G = (0, a)× (0, b) . Several conditions for the uniform separability an... more on the rectangular domain G = (0, a)× (0, b) . Several conditions for the uniform separability and the resolvent estimates for the corresponding linear problem are given in abstract Lp-spaces. Especially, we prove that the linear differential operator is positive and is a generator of an analytic semigroup. Moreover, the existence and uniqueness of maximal regular solution of the above nonlinear problem are obtained. One of the important characteristics of these DOEs are that the degeneration process are taking place at different speeds at boundary, in general. Maximal regularity properties of regular degenerated nonlinear DOEs are studied e.g. in [1, 8, 10] . Unlike to these we consider here the singular degenerate DOEs. In applications maximal regularity properties of infinite systems of singular degenerate PDE are studied. Let γ = γ (x) , x = (x1, x2, ..., xn) be a positive measurable function on a domain Ω ⊂ R. Let Lp,γ (Ω;E) denote the space of strongly measurable E-valued func...
Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary... more Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.
Electronic Journal of Qualitative Theory of Differential Equations, 2021
In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations tha... more In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here, L p -estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness, L p -regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated wi...
In this paper, the integral initial value problems for Boussinesq type equations are studied. The... more In this paper, the integral initial value problems for Boussinesq type equations are studied. The equation include the general differential operators. The existence, uniqueness and regularity properties of solution of these problems are obtained. By choosing differential operators including in the equation, the regularity properties of the Cauchy problem for different type of Boussinesg equations are studied.
In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equations are studi... more In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equations are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term
The boundary value problems for the degenerate differential-operator equations with small paramet... more The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued Lp-spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.
We presents the study the separability properties for differential-operator equations in Morrey s... more We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation is obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied... more In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied.The equation involves convolution terms with a general kernel functions whose Fourier transform are operator functions defined in a Banach space E together with some growth conditions. Here, assuming enough smoothness on the initial data and the operator functions, the local, global existence, uniqueness and regularity properties of solutions are established in terms of fractional powers of given sectorial operator functon. Furthermore, conditions for finite time blow-up are provided. By choosing the space E and the operators, the regularity properties the wide class of nonlocal wave equations in the field of physics are obtained.
The abstract elliptic and parabolic equations on exterior domain are considered. The equations ha... more The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
The abstract elliptic and parabolic equations on exterior domain are considered. The equations ha... more The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
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