A toxin-mediated size-structured population model: Finite difference approximation and well-posedness by Qihua Huang, Hao Wang

“…Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. …”
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Existence and regularity of pullback attractors for a 3D non-autonomous Navier–Stokes–Voigt model with finite delay by Julia García-Luengo, Pedro Marín-Rubio

“…Under suitable assumptions on the function driving the delay time, the existence and uniqueness of weak solution are proved. Existence and relationships among pullback attractors in several phase-spaces are analyzed for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and another one given by a tempered condition. …”
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A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme by Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis

“…Existence and uniqueness results are established and convergence of the finite difference approximation to the unique bounded variation weak solution of the model is obtained. Numerical simulations demonstrating the accuracy of the method are presented. …”
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Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm by Xinfu Pang, Yang Yu, Haibo Li, Yuan Wang, Jinhui Zhao

“…Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. …”
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Nonlinear iterative approximation of steady incompressible chemically reacting flows by Gazca-Orozco, Pablo Alexei, Heid, Pascal, Süli, Endre

“…We prove that the weak solution, whose existence was already established in the literature, is unique, given some strengthened assumptions on the diffusive flux and the stress tensor, for small enough data. …”
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Simulation and Domain Identification of Sea Ice Thermodynamic System by Bing Tan, Peng Lu, Zhijun Li, Enmin Feng, Juan Wang

“…The unique existence of the weak solution of the thermodynamic system is proved. …”
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EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY by Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin

“…The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c \gt 0$ is also obtained.…”
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Stationary solutions to a hybrid viscous hydrodynamic model with classical boundaries by Federica Di Michele, Bruno Rubino, Rosella Sampalmieri, Kateryna Stiepanova

“…The well posedness of the problem is ensured by adding a viscous term necessary for the convergence of the hybrid limit to an appropriate weak solution. Some numerical tests are also performed for different values of the viscous coefficient, in order to evaluate the effects of the viscosity, especially on the boundaries of the device.…”
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Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems by Teffera M. Asfaw

“…In the last section, an example is provided addressing existence of weak solution in X=Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem of the type ut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t),  (x,t)∈Q; u(x,t)=0,  (x,t)∈∂Ω×(0,T); u(x,0)=0,  x∈Ω, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RN→R  (i=1,2,…,n) satisfies certain growth conditions, and f∈Lp′(Q), Q=Ω×(0,T), and p′ is the conjugate exponent of p.…”
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The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient by Angelo Alvino, Vincenzo Ferone, Anna Mercaldo

“…In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>β</mi><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mi>q</mi></msup><mo>+</mo><mi>f</mi><mo> </mo><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>0</mn><mo> </mo><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>∂</mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is a bounded open subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>N</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>div</mi><mfenced separators="" open="(" close=")"><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is a positive constant and <i>f</i> is a measurable function satisfying suitable summability conditions depending on <i>q</i> and a smallness condition.…”
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Finite and Infinte Time Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities by Milena Dimova, Natalia Kolkovska, Nikolai Kutev

“…Two new necessary and sufficient conditions for blow up of the weak solutions are established. The first one addresses the blow up of the global weak solutions at infinity. …”
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Energy Conservation for 3D Tropical Climate Model in Periodic Domains by Hui Zhang

“…In this paper, we prove the energy conservation for the weak solutions of the 3D tropical climate model under some sufficient conditions. …”
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Shinbrot’s energy conservation criterion for the 3D Navier–Stokes–Maxwell system by Ma, Dandan, Wu, Fan

“…This paper concerns the energy conservation for the weak solutions to the Navier–Stokes–Maxwell system. Although the Maxwell equation with hyperbolic nature, we still establish a $L^qL^p$ type condition guarantee validity of the energy equality for the weak solutions. …”
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Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group by Junli Zhang, Jialin Wang

“…In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by the A-harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition for x, ellipticity and growth condition with the growth index 1<p<2 for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.…”
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On the Boundary Value Condition of an Isotropic Parabolic Equation by Qitong Ou, Huashui Zhan

“…The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. …”
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Multiplicity Results for a p1x,p2x-Laplacian Equation via Variational Methods by A. Rezvani

“…We prove the existence and multiplicity of nontrivial weak solutions for the following p1x,p2x-Laplacian equation involving variable exponents: −div∇up1x−2∇u−div∇up2x−2∇u+up2x−2u=λhx,u,inΩ,u=0,on∂Ω. …”
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Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales by Xing Hu, Yongkun Li

“…Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.…”
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On Flows of Bingham-Type Fluids with Threshold Slippage by Evgenii S. Baranovskii

“…We establish sufficient conditions for the existence of weak solutions and provide their energy estimates. Moreover, it is shown that the set of weak solutions is sequentially weakly closed in a suitable functional space.…”
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On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term by Huashui Zhan, Yongping Li

“…After giving a result of the existence, if the diffusion coefficient is degenerate on the boundary, the local stability of the weak solutions is established without any boundary condition. …”
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Remarks on the existence and decay of the nonlinear beam equation by Jaime E. Mũnoz Rivera

“…We will consider a class of nonlinear beam equation and we will prove the existence and decay weak solution…”
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