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In this paper, we establish a local higher integrability result for the gradient of a weak solution to a parabolic multi-phase equation. To achieve this, we prove parabolic Poincaré type inequalities and reverse Hölder type inequalities for the gradient of a weak solution in each of difference types of intrinsic cylinders. In particular, we formulate a delicate plan of alternatives and stopping time arguments to address the presence of two different transitions. [ABSTRACT FROM AUTHOR]

A weak formulation is devised for the K(m,n)$K(m,n)$ equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions. [ABSTRACT FROM AUTHOR]

In this paper we analyze the existence of solution to the homogeneous Dirichlet for a nonlinear elliptic singular problem, $$\begin{align*} \begin{cases} -\Delta u=\dfrac{|\nabla u|^q}{|u|^\alpha}+\lambda f & {\rm in}\ \Omega, \\ u=0 & {\rm on}\ \partial\Omega,\\ \end{cases} \end{align*} $$ { − Δ u = | ∇ u | q | u | α + λf in Ω , u = 0 on ∂Ω , where $ N\ge 2 $ N ≥ 2 , $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N is a bounded regular domain, q>2, $ \alpha α < q 2 , $ \lambda >0 $ λ > 0 and f is a nonnegative function belonging to a suitable Lebesgue space. Under additional hypotheses on the data, we are also able to show the uniqueness of the 'good' solution. [ABSTRACT FROM AUTHOR]

We examine a time-fractional pseudo-hyperbolic equation involving positive operators. We explore the determination of initial velocity and perturbation. It is demonstrated that these initial inverse problems are ill posed. Additionally, we prove that under certain conditions, the inverse problems exhibit well-posedness properties. Our focus is on developing a theoretical framework for these initial inverse problems associated with time-fractional pseudo-hyperbolic equations, laying the groundwork for future studies on numerical algorithms to solve these problems. This investigation is crucial for understanding the fundamental behavior of the equations under various initial conditions and perturbations. By establishing a rigorous theoretical framework, we pave the way for future research to focus on practical numerical methods and simulations. Our results provide a deeper insight into the mathematical structure of time-fractional pseudo-hyperbolic equations, ensuring that future computational approaches are built on a solid theoretical foundation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of weak solutions of the p (x) -Laplacian problems that are subject to Dirichlet boundary conditions, employing variational techniques as our primary analytical framework. [ABSTRACT FROM AUTHOR]

We consider a mathematical model which describes the equilibrium of two elastic membranes fixed on their boundary and attached to an adhesive body, say a glue. The variational formulation of the model is in a form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the model, and then we state and prove a convergence result, for which we provide the corresponding mechanical interpretation. Next, we consider two associated optimization problems for which we provide existence results. Finally, we the present numerical simulation which validates our convergence result. We end this paper with some concluding remarks and an Appendix, in which we present the preliminary material needed in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with a chemotaxis-Navier–Stokes model with indirect signal production involving Dirichlet signal boundary condition in a bounded domain with smooth boundary. A recent literature has asserted that for all reasonably regular initial data, the associated no-flux/saturation/no-flux/no-slip problem possesses at least one globally defined weak solution in the logistic-type degradation here is weaker than quadratic case. But the knowledge on regularity properties of solution has not yet exceeded some information on fairly basic integrability features. The present study reveals that each of these weak solutions becomes eventually classical and bounded under some suitably strong sub-quadratic degradation assumption and an explicit smallness condition. Furthermore, in comparison with the related contributions in the case of the direct signal production, our findings inter alia rigorously reveal that the indirect signal production mechanism genuinely contributes to the global solvability and eventual smoothness of the chemotaxis-Navier–Stokes system. [ABSTRACT FROM AUTHOR]

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position-dependent mass and dissipation terms, in particular, allowing for δ -like coefficients and prove that the problem has a very weak solution. Furthermore, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory. A special case of the model considered here is the so-called telegraph equation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying a fractional Schrödinger equation of the form { (− Δ) s u + V (x) u = f (x , u) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , where 0 < s < 1 , n > 2 s , Ω is a smooth bounded domain in R n , (− Δ) s denotes the fractional Laplacian of order s, f (x , t) is a function in C (Ω ‾ × R) , and f (x , t) / t is nondecreasing in t and converges uniformly to an L ∞ function q (x) as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a "mountain pass" approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f (x , t) exhibits superlinearity in t at infinity, represented by q (x) ≡ + ∞ . [ABSTRACT FROM AUTHOR]

A system of time‐fractional diffusion equations posed in an exterior domain of ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$( N≥3$$ N\ge 3 $$) under homogeneous Dirichlet boundary conditions is investigated in this paper. The time‐fractional derivatives are considered in the Caputo sense. Using nonlinear capacity estimates specifically adapted to the nonlocal properties of the Caputo fractional derivative, the geometry of the domain, and the boundary conditions, we obtain sufficient conditions for the nonexistence of a weak solution to the considered system. [ABSTRACT FROM AUTHOR]

We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality. [ABSTRACT FROM AUTHOR]

This paper mainly deals with a self-consistent chemotaxis-Navier–Stokes system with porous medium diffusion in a three-dimensional (3D) bounded and smooth domain. The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered, which leads to stronger coupling than the usual chemotaxis-fluid model studied in most existing literatures. It is proved that for any suitably regular initial data, the associated no-flux/no-flux/Dirichlet problem possesses at least one global weak solution or global very weak solution. To the best of our knowledge, this is the first result on the global solvability of the 3D self-consistent chemotaxis-Navier–Stokes system with porous medium diffusion. Our results inter alia provide a more in-depth understanding on the chemotaxis-Navier–Stokes system, and significantly improve previously known ones. [ABSTRACT FROM AUTHOR]

A stationary heat transfer model generalizing the Boussinesq approximation is considered. For the corresponding boundary value problem the property of a global existence of its weak solution is proved. We establish special requirements on the model data under which the maximum principle for temperature is valid. Moreover, the solvability property of the control problem for the examined heat transfer model is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following quasilinear chemotaxis–consumption system with degenerate signal-dependent motility u t = ∇ · (v α ∇ u m) + α ∇ · (v α - 1 u ∇ v) + r u - μ u 2 , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under the homogeneous Neumann boundary condition in Ω ⊂ R n (n ≥ 1) with α , r , μ > 0 and m > 1 . It is shown that above system admits at least one global weak solution fulfilling the following boundedness property ‖ u (· , t) ‖ L p (Ω) + ‖ v (· , t) ‖ W 1 , ∞ (Ω) ≤ C for all t > 0 , p > 2 and m > 1 . This result gives an evident of the regularized effect of porous-medium-type diffusion. [ABSTRACT FROM AUTHOR]

The paper deals with the steady MHD equations for a viscous incompressible fluid in a bounded and generally multiply connected domain Ω⊂ℝ3$$ \Omega \subset {\mathrm{\mathbb{R}}}&amp;#x0005E;3 $$ with the no‐slip boundary condition for the velocity u$$ \mathbf{u} $$ and inhomogeneous generalized impermeability boundary conditions for the magnetic field b$$ \mathbf{b} $$. The main results concern an appropriate definition of the weak solution (u,b)$$ \left(\mathbf{u},\mathbf{b}\right) $$, its existence, continuous dependence on the data when the data tend to zero, and uniqueness in a "small" neighborhood of the zero solution (0,0)$$ \left(\mathbf{0},\mathbf{0}\right) $$. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of weak solutions for the nonlinear parabolic problem associated with the following fourth‐order equation, under Navier boundary conditions: ∂u∂t+Δp(x)2u+g(x,t,u,∇u)=f(x,t)inQ:=Ω×(0,T),$$ \frac{\partial u}{\partial t}&#x0002B;{\Delta}_{p(x)}&#x0005E;2u&#x0002B;g\left(x,t,u,\nabla u\right)&#x0003D;f\left(x,t\right)\kern3.0235pt \mathrm{in}\kern0.30em \mathcal{Q}:&#x0003D; \Omega \times \left(0,T\right), $$where Ω⊂ℝn(N≥2)$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;n\kern3.0235pt \left(N\ge 2\right) $$ is a bounded domain. Under some appropriate assumptions, we obtain the existence of a weak solution for the above problem by using the topological degree theory for operators of the type L+S+C$$ \mathcal{L}&#x0002B;\mathcal{S}&#x0002B;\mathcal{C} $$, in the framework space Lp−(0,T;V)$$ {L}&#x0005E;{p-}\left(0,T;\mathcal{V}\right) $$. [ABSTRACT FROM AUTHOR]

The main crux of this manuscript is to study the existence of weak solutions for the following fractional double phase elliptic problem with Dirichlet boundary conditions (- Δ) p σ w + (- Δ) q σ w = ψ (z) + g (z , w) in Θ , w = 0 , on R N \ Θ , where 1 < p < q < N σ , σ ∈ (0 , 1) and (- Δ) p σ , (- Δ) q σ are fractional Laplacian operators defined in the principle value sense. The existence results are obtained by combining the Galerkin approximation, Lebesgue and Sobolev spaces, the Young measures approach, and the assumption of growth conditions on a given datum g . Our results expand upon and broaden several recent studies in this area of literature. [ABSTRACT FROM AUTHOR]

In this work, we consider a more general version of the nonlocal thermistor problem, which describes the temperature diffusion produced when an electric current passes through a material. We investigate the doubly nonlinear problem where the nonlocal term is present on the right-hand side of the equation that describes the temperature evolution. Specifically, we employ topological degree theory to establish the existence of a solution to the considered problem. Furthermore, we separately address the uniqueness of the obtained solution. Additionally, we establish a priori estimates to demonstrate the convergence of a developed finite volume scheme used for the discretization of the continuous parabolic problem. Finally, to numerically simulate the proposed finite volume scheme, we use the Picard-type iteration process for the fully implicit scheme and approximate the nonlocal term represented by the integral with Simpson's rule to validate the efficiency and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

The article discusses the existence of weak solutions of the Kelvin–Voigt model in the context of the motion of water with added polymers. The model considers the delay in establishing equilibrium states due to relaxation time and internal reorganization processes. The study aims to prove the weak solvability of the initial–boundary value problem by introducing regular Lagrangian flows and defining weak solutions based on a priori estimates and the theory of the topological degree of contracting vector fields. The research was funded by the Russian Science Foundation and the author declares no conflicts of interest. [Extracted from the article]

We consider fully parabolic two‐species chemotaxis system with p$$ p $$‐Laplacian diffusion in a smooth bounded domain Ω⊂ℝn,n≥2$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;n,n\ge 2 $$ with 1

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods. • A novel nonlinear parabolic system is introduced. • The theoretical results are applied to the image denoising and restoration. • Various numerical implementations on some grayscale images are carried out. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence result of weak solutions to a class fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type LKp(x)$$ {\mathcal{L}}_K&#x0005E;{p(x)} $$. The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results. [ABSTRACT FROM AUTHOR]

Conservation laws with two-parameter hysteretic fluxes are introduced. The total variation of the flux of a viscous solution does not increase as time progresses. The maximum principle for the flux is also established. Viscous shock waves are identified, and solutions to the Riemann problem are obtained. The one with the least total variation of flux is preferred. A definition of weak solutions for the hysteretic flow model, a non-conservative hyperbolic system, is provided. Approximate solutions of the system are constructed using the front tracking method with randomly adjusted discretization locations. These approximate solutions have uniformly bounded total variation, demonstrating their compactness. [ABSTRACT FROM AUTHOR]

In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $ (u,b) $ (u , b) of the non-resistive MHD equations for the initial data $ u_0\in \dot B^{\frac {d}{p}-1}_{p,1}(\mathbb {R}^{d}) $ u 0 ∈ B ˙ p , 1 d p − 1 (R d) and $ b_0\in \dot B^{\frac {d}{p}}_{p,1}(\mathbb {R}^{d}) $ b 0 ∈ B ˙ p , 1 d p (R d) with $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ from $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d yet. [ABSTRACT FROM AUTHOR]

We study a boundary value problem for a mathematical model describing the nonisothermal steady-state flow of a viscous fluid in a 3D (or 2D) bounded domain with locally Lipschitz boundary. The heat and mass transfer model considered here has the feature that a regularized Rayleigh dissipation function is used in the energy balance equation. This permits taking into account the energy dissipation due to the viscous friction effect. A theorem on the existence of a weak solution is proved under natural assumptions on the model data. Moreover, we establish extra conditions guaranteeing that the weak solution is unique and/or strong. [ABSTRACT FROM AUTHOR]

This work proposes a novel nonlinear parabolic equation with p(x)$$ p(x) $$‐growth conditions for image restoration and enhancement. Based on the generalized Lebesgue and Sobolev spaces with variable exponent, we demonstrate the well‐posedness of the proposed model. As a first result, we prove the existence of a weak solution to our model when the reaction term is bounded by a suitable function. Secondly, we use the approximations method to establish the existence of a nonnegative weak SOLA (Solution Obtained as Limit of Approximations) solution to the proposed model. We illustrate our theoretical results in the context of image denoising and enhancement. Specifically, we present various numerical implementations on a set of grayscale images. To further enrich these simulations, we assess the efficiency of the proposed model on several color images. The obtained numerical results strongly suggest that our model outperforms existing state‐of‐the‐art methods in terms of both visual and quantitative comparison, demonstrating superior efficiency and robustness in noise removal and contrast enhancement. [ABSTRACT FROM AUTHOR]

The system of Brusselator-type reaction-diffusion equations (RDs) on open bounded convex domains 풟 ⊂ ℝ d (d ≤ 3) with Robin boundary conditions (Rbcs) has been mathematically analyzed. The Faedo–Galerkin approach is used to demonstrate the global existence and uniqueness of a weak solution to the system. The weak solution's higher regularity findings are constructed under more regular conditions on the initial data. In addition, continuous dependence on the initial conditions has been proved. [ABSTRACT FROM AUTHOR]

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h (t , x) | u | p in (0 , ∞) × (0 , 1) subject to the initial conditions u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) and the Navier boundary conditions u (t , 1) = u x x (t , 1) = 0 , where α ∈ (0 , 1) , β ∈ (1 , 2) , ∂ t α (resp. ∂ t β ) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, p > 1 and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂ 4 ∂ x 4 , the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed. [ABSTRACT FROM AUTHOR]

We consider 3D stochastic Landau-Lifshitz-Gilbert equations coupled with the Maxwell equations with the full energy. We have proved the existence of the weak solution as well as some further regularities of it. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples. [ABSTRACT FROM AUTHOR]

We study porous medium equations with a divergence form of drift terms in a bounded domain with no-flux lateral boundary conditions. We establish L q -weak solutions for 1 ≤ q < ∞ in Wasserstein space under appropriate conditions on the drift, which is an extension of authors' previous works done in the whole space into the case of bounded domains. Applying existence results to a certain Keller-Segel equation of consumption type, construction of L q -weak solutions is also made, in case that the equation of a biological organism is of porous medium type. [ABSTRACT FROM AUTHOR]

This paper delves into the global existence of weak solutions for a three-dimensional magnetoelastic interaction model. This model combines a fractional harmonic map heat flow with an evolution equation for displacement. By using the Faedo-Galerkin method, we successfully establish the global existence of weak solutions for this coupled system. [ABSTRACT FROM AUTHOR]

In this paper, we show that a weak solution to 3D shear-thinning flows becomes strong provided that the gradient of one component of velocity belongs to a certain Lebesgue space class. [ABSTRACT FROM AUTHOR]

This paper proves existence of a global weak solution to the inhomogeneous (i.e., non-constant density) incompressible Navier–Stokes system with mass diffusion. The system is well-known as the Kazhikhov–Smagulov model. The major novelty of the paper is to deal with the Kazhikhov–Smagulov model possessing the non-constant viscosity without any simplification of higher order nonlinearity. Every global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of Aubin–Lions–Simon type. [ABSTRACT FROM AUTHOR]

We consider a mathematical model of the initiation and development of atherosclerosis in the early stage, defined in different domains. The existence of nonnegative weak solutions is proved via studying an auxiliary system with absolute value terms and applying a novel semi-discrete, operator splitting numerical scheme. Numerical simulations were performed in an idealized two-dimensional geometry in order to verify our assumptions in model building and theoretical analysis. The framework we developed can also be used for nonlinear reaction–diffusion systems defined in other different domains with more complex coupling conditions. [ABSTRACT FROM AUTHOR]

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games. [ABSTRACT FROM AUTHOR]

We prove that there exists a large-data and global-in-time weak solution to a system of partial differential equations describing the unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a mechanically and thermally isolated container of any dimension. To overcome the principal difficulties connected with ill-posedness of the diffusive Oldroyd-B model in three dimensions, we assume that the fluid admits a strengthened dissipation mechanism, at least for excessive elastic deformations. All the relevant material coefficients are allowed to depend continuously on the temperature, whose evolution is captured by a thermodynamically consistent equation. In fact, the studied model is derived from scratch using only the balance equations for linear momentum and energy, the formulation of the second law of thermodynamics and the constitutive equation for the internal energy. The latter is assumed to be a linear function of temperature, which simplifies the model. The concept of our weak solution incorporates both the temperature and entropy inequalities, and also the local balance of total energy provided that the pressure function exists. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of weak solutions for full compressible magnetohydrodynamic flows in a 3D bounded domain, with non-homogeneous boundary condition for the velocity, absolute temperature and density on the inflow part, with Navier-type slip boundary condition for magnetic field. We show the weak–strong uniqueness principle as well as the global existence under a new notion of weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results. [ABSTRACT FROM AUTHOR]

The paper deals with proving the weak solvability of an initial–boundary value problem for the modified Kelvin–Voigt model taking into account memory along the trajectories of motion of fluid particles. To this end, we consider an approximation problem whose solvability is established with the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, we show that the sequence of solutions of the approximation problem has a subsequence that weakly converges to the solution of the original problem as the approximation parameter tends to zero. [ABSTRACT FROM AUTHOR]

The authors analyze continuity equations with Stratonovich stochasticity, defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ1(t), ..., ẆN(t) driven by smooth spatially dependent vector fields a1(x), ..., aN(x) on M. The velocity u belongs to Lt1Wx1,2 with divhu bounded in Lt,xp for p > d + 2, where d is the dimension of M (they do not assume divhu ∈ Lt,x∞). For carefully chosen noise vector fields ai (and the number N of them), they show that the initial-value problem is well-posed in the class of weak L2 solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this "regularization by noise" result is based on a L2 estimate, which is obtained by a duality method, and a weak compactness argument. [ABSTRACT FROM AUTHOR]

In this article, we modify the results obtained by Mitidieri and Pohozhaev on sufficient conditions for the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of the Laplace operator and with a nonlinear term of the form a(x) |∇(Δmu)|q + b(x)|∇u|s. We obtain optimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of test functions. As a result, we prove the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction. [ABSTRACT FROM AUTHOR]

We study a time-fractional parabolic inequality posed on a bounded interval and involving a wight function W, where the fractional derivative is considered in the Caputo sense. We establish a general condition ensuring that the set of weak solutions is empty. Next, some particular cases of the weight function W are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Riemann-type hydrodynamical equation and establish the existence of a weak solution to the equation in a lower order Sobolev space (Hs(ℝ))N with 1 < s ≤ 3/2 by using a modified pseudo-parabolic regularization method. Although the absence of some important conservation laws leads to the failure of global existence of the approximate solutions, it has been addressed by making good use of the structure of the system. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to study the global existence of weak solution of magneto elastic interactions, which is modeled by a three-dimensional mathematical framework. A fractional generalization of the harmonic map heat flow is coupled to an evolution equation for displacement to characterize the model. Faedo-Galerkin method with some commutator is used to estimate and prove global existence of weak solutions for the proposed model. [ABSTRACT FROM AUTHOR]

A distributed renewable resource of any nature is considered on a two-dimensional sphere. Its dynamics is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource is carried out by constant or periodic impulse harvesting. It is shown that, after choosing an admissible exploitation strategy, the dynamics of the resource tend to limiting dynamics corresponding to this strategy and there is an admissible harvesting strategy that maximizes the time-averaged harvesting of the resource. [ABSTRACT FROM AUTHOR]

In this paper, we show that a weak solution for unsteady flows of 3D shear‐thinning fluids is strong under certain integrability assumptions about two components of the vorticity. In particular, if two components of the vorticity belong to the marginal critical space L1(0,T;BMO)$$ {L}^1\left(0,T; BMO\right) $$, then the solution is strong. [ABSTRACT FROM AUTHOR]

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level. [ABSTRACT FROM AUTHOR]

Here, the existence and approximation results for degenerated anisotropic (p , q) -Laplacian with weights − ∑ i = 1 N ∂ ∂ x i ((a (x) | ∂ u ∂ x i | p i − 2 + b (x) | ∂ u ∂ x i | q i − 2) ∂ u ∂ x i ) = f (x , u , ∇ u) and a competing anisotropic (p , q) -Laplacian with weights − ∑ i = 1 N ∂ ∂ x i ((a (x) | ∂ u ∂ x i | p i − 2 − b (x) | ∂ u ∂ x i | q i − 2) ∂ u ∂ x i ) = f (x , u , ∇ u) are studied with Dirichlet boundary condition and on a bounded smooth domain in R N , N ≥ 3 where f : Ω × R × R N → R is a Carathéodory function. The proofs are based on weighted antitropic Sobolev spaces, Nemytskij operators and finite dimensional approximation. [ABSTRACT FROM AUTHOR]