On the existence of classical solutions for differential-functional IBVP by Krzysztof A. Topolski

“…By using Leray-Schauder theorem we prove the existence of classical solutions. Our formulation and results cover a large class of parabolic problems both with a deviated argument and integro-differential equations.…”
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On a global classical solution of a quasilinear hyperbolic equation by Y. Ebihara, D. C. Pereira

“…In this paper we establish the existence and uniqueness of global classical solutions for the equation which arises in the study of the extensional vibrations of thin rod, or torsional vibrations of thin rod.…”
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Integrability of derivations of classical solutions of Dirichlet's problem for an elliptic equation by M. I. Hassan

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On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations by Jiafa Xu, Lishan Liu

“…It is proved that for general initial data u0∈Hs with s>3, the life span of the classical solutions satisfies T>Cln  N3/4.…”
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Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity by Jincheng Shi, Shengzhong Xiao

“…We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. …”
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Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source by Baghaei, Khadijeh

“…$We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $ k\,(1-\alpha )<\frac{4}{n+5}$ and the initial value $ v_{0}$ and $\mu $ satisfy the following conditions: \begin{align*} 0<\Vert v_{0}\Vert _{L^{\infty }(\Omega )}\le \Bigg [\frac{4\big [1-k \, \big (1-\alpha \big )\big ]}{k\, (n+1)\,(1-\alpha )}\Bigg ]^{\frac{1}{k}}-1, \end{align*} and \begin{equation*} \mu > \frac{k\,n\,(1-\alpha ) \Vert v_{0}\Vert _{L^{\infty }(\Omega )}}{(n+1)(1+\Vert v_{0}\Vert _{L^{\infty }(\Omega )})}. …”
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Existence and boundedness of solutions for a parabolic-parabolic predator-prey model by Fengxiang Zhao, Haotian Tang, Jiashan Zheng, Kaiqiang Li

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Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model by Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang

“…Astraightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitroglioblastoma growth is also presented.…”
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The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations by Cong Sun, Lixia Li

“…Through applying Galerkin method, we establish the approximating solution for one-dimension Klein-Gordon-Zakharov equations and obtain the local classical solution. By applying integral estimates, we also obtain the existence and uniqueness of the global classical solution of Klein-Gordon-Zakharov equations.…”
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Properties of the solution of some multipoint problems for integro-differential equation of Barbashin type by Nurmuhammet Gurbanmammedov, Parahat Gurbanmammedov, Ogulhan Haydarova

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The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks by P. A. Krutitskii

“…The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. …”
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Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem by Ya-Ning Li, Hong-Rui Sun

“…We show that A generates an exponentially bounded (α,β)-ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L loc 1(ℝ+,X). …”
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Regularizing Model for the 2D MHD Equations with Zero Viscosity by Ensil Kang, Jihoon Lee

“…We provide an approximating system to the equations and prove global-in-time existence of classical solution to this approximating system. By using approximating system, a priori estimates for the equations can be justified.…”
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New solvability condition of 2-d nonlocal boundary value problem for Poisson’s operator on rectangle by Dovlet M. Dovletov

“…New solvability conditions are obtained in respect of existence, uniqueness and a priori estimate of the classical solution. Second order of accuracy difference scheme is presented.…”
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Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media by Kuan, Jeffrey, Čanić, Sunčica, Muha, Boris

“…This regularized problem is consistent with the original, nonregularized problem in the sense that the weak solutions constructed here, converge, as the regularization parameter tends to zero, to a classical solution of the original, nonregularized problem when such a classical solution exists, assuming viscoelasticity in the Biot poroelastic matrix [1]. …”
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Nonlocal boundary value problem in terms of flow for Sturm-Liouville operator in differential and difference statements by Dovlet M. Dovletov

“…Sturm-Liouville operator with second kind of nonlocal boundary value conditions is considered. For the classical solution, a priori estimate is established and unique existence is proved. …”
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The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges by P. A. Krutitskii

“…The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. …”
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Pricing Option with Stochastic Interest Rates and Transaction Costs in Fractional Brownian Markets by Lina Song, Kele Li

“…Pricing formulas are derived by the variable substitution and the classical solution of the heat conduction equation. By the mathematical software and the parameter estimation methods, the results are reported and compared with the data from the financial market.…”
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On the Modified Jump Problem for the Laplace Equation in the Exterior of Cracks in a Plane by P. A. Krutitskii, A. Sasamoto

“…The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtained. …”
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Dissipative measure-valued solutions to the magnetohydrodynamic equations by Jianwei Yang, Huimin Wang, Qihong Shi

“… In this paper, we study the dissipative measure-valued solution to the magnetohydrodynamic equations of 3D compressible isentropic flows with the adiabatic exponent γ > 1 and prove that a dissipative measure-valued solution is the same as the standard smooth classical solution as long as the latter exists, provided they emanate from the same initial data (weak–strong) uniqueness principle. …”
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