Variable Exponent Spaces of Differential Forms on Riemannian Manifold by Yongqiang Fu, Lifeng Guo

“…After discussing the properties of these spaces, we obtain the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous 𝑝(𝑚)-harmonic equations with variable growth in 𝑊01,𝑝(𝑚)(Λ𝑘𝑀).…”
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The Dirichlet Problem for the Equation Δu−k2u=0 in the Exterior of Nonclosed Lipschitz Surfaces by P. A. Krutitskii

“…Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. …”
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On a Fractional Nonlinear Hyperbolic Equation Arising from Relative Theory by Zujin Zhang, Xiaofeng Wang, Zheng-an Yao

“…We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equation arising from relative theory by the Galerkin method. …”
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Nontrivial Solutions for a Class of p-Kirchhoff Dirichlet Problem by Xian Hu, Yong-Yi Lan

“…We show that the p-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.…”
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Rothe method for a mixed problem with an integral condition for the two-dimensional diffusion equation by Nabil Merazga, Abdelfatah Bouziani

“…Existence, uniqueness, and continuous dependence upon data of a weak solution of this latter are proved by means of the Rothe method. …”
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A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces by Xiang'ou Zhu

“…It is shown that if the pressure gradient belongs to 𝐿2/(2−𝑟)̇𝐻((0,𝑇);ℳ(𝑟(ℝ3̇𝐻)→−𝑟(ℝ3))), where ̇𝐻ℳ(𝑟(ℝ3̇𝐻)→−𝑟(ℝ3)) is the multipliers between Sobolev spaces whose definition is given later for 0<𝑟<1, then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.…”
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Extinction and Nonextinction for the Fast Diffusion Equation by Chunlai Mu, Li Yan, Yi-bin Xiao

“…For 0<m<1, under appropriate hypotheses, we show that m=p is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of -Δ with homogeneous Dirichlet boundary.…”
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On Solutions of a Parabolic Equation with Nonstandard Growth Condition by Huashui Zhan

“…The stability of weak solutions is based on a natural partial boundary value condition. …”
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On the Navier-Stokes equations with temperature-dependent transport coefficients by Eduard Feireisl, Josef Málek

“…We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. …”
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Solvability of the Brinkman-Forchheimer-Darcy Equation by Piotr Skrzypacz, Dongming Wei

“…The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. …”
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The Stability of the Solutions for a Porous Medium Equation with a Convection Term by Huashui Zhan, Miao Ouyang

“…The existence of the weak solution is proved by the monotone convergent method. …”
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Flow of electrorheological fluid under conditions of slip on the boundary by R. H. W. Hoppe, M. Y. Kuzmin, W. G. Litvinov, V. G. Zvyagin

“…A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.…”
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A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces by TianLi LI, Wen Wang, Lei Liu

“…Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. …”
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The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation by Huashui Zhan

“…The uniqueness of weak solution can be proved without the boundary value condition on Γ2.…”
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On the surjectivity of linear transformations by M. Damlakhi, V. Anandam

“…This result is used to discuss the existence of an Lp-weak solution of Du=v where D is a differential operator with smooth coefficients and v∈Lp.…”
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Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises by Jianhua Huang, Yuhong Li, Jinqiao Duan

“…Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.…”
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Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory by Jin-soo Hwang

“…We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.…”
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The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term by Ying Wang, YunXi Guo

“…Provided that (1-∂x2)u0∈M+(R), u0∈H1(R), and u0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.…”
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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. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. …”
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WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition by Ouadie Koubaiti, Ahmed Elkhalfi, Jaouad El-mekkaoui

“…Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. …”
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