Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials by Dumitru Motreanu, Mircea Sofonea

“…For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.…”
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Flow of electrorheological fluid under conditions of slip on the boundary

“…A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.…”
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A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid by Le Thi Phuong Ngoc, Truong Thi Nhan, Nguyen Thanh Long

“…In Part 1, we use the Galerkin method and compactness method to prove the existence of a unique weak solution of the problem above on (0,T), for every T>0. …”
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On a Class of Schrödinger System Problem in Orlicz–Sobolev Spaces by H. El-Houari, L. S. Chadli, H. Moussa

“…Using the mountain pass theorem, we obtain the existence of a nontrivial and nonnegative weak solution of a quasi-linear Schrödinger system driven by the ω⋅-Laplacian operator in Orlicz–Sobolev spaces.…”
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Optimal Convergence Rates for Solutions of the Monopolar Non-Newtonian Flows by Zhi-qiang Luo

“…By using the energy methods, the perturbed weak solution of perturbed system asymptotically converges to the solution of the original system with the optimal rates (1+t)-1/4.…”
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Remarks on the Pressure Regularity Criterion of the Micropolar Fluid Equations in Multiplier Spaces by Fengjun Guo

“…This study is devoted to investigating the regularity criterion of weak solutions of the micropolar fluid equations in . …”
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The Existence of Solutions to the Nonhomogeneous A-Harmonic Equations with Variable Exponent by Haiyu Wen

“…We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous A-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation d⋆A(x,dω)=B(x,dω) in the weighted variable exponent Sobolev space Wdp(x)(Ω,Λl,μ).…”
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A Capacity Associated with the Weighted Lebesgue Space and Its Applications by Guoliang Li, Guanglan Wang, Lei Zhang

“…More properties of that capacity are explored, and applications to a trace inequality for the weak solution of the equation are considered.…”
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Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity by Baishun Lai

“…In the extremal case λ=λ*, we prove the existence of a weak solution which is the unique solution even in a very weak sense. …”
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On the Stochastic 3D Navier-Stokes-α Model of Fluids Turbulence by Gabriel Deugoue, Mamadou Sango

“…The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. …”
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Existence Theorem for Noninstantaneous Impulsive Evolution Equations by Gul I Hina Aslam, Amjad Ali, Maimona Rafiq

“…The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.…”
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The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid by Jae-Myoung Kim

“…In this paper, we establish the global existence and uniqueness of a weak solution u,b depending on a number q in ℝ2. Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1+t−1/2.…”
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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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