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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]

Using Mountain Pass Theorem, we consider the existence of weak solutions of weighted Robin problem involving p. -biharmonic operator a x Δ p x 2 u = λ b (x) u q (x) - 2 u , i n Ω a (x) Δ u p (x) - 2 ∂ u ∂ υ + β (x) u p (x) - 2 u = 0 , o n ∂ Ω under some conditions in the space W a , b 2 , p (.) Ω. [ABSTRACT FROM AUTHOR]

This paper investigates a doubly haptotactic cross-diffusion system with $ p $-Laplacian diffusion. Under Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^2 $ with smooth boundary, it is proved that if $ p>2 $ for all sufficiently smooth initial data, the model possesses at least one global weak solution. [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]

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, T] provided that either the norm π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 2 and 3 2 < β < ∞ or ∇ π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 3 and 1 < β < ∞ is sufficiently small. [ABSTRACT FROM AUTHOR]

A nonlinear time-fractional cable equation posed on the interval (0, 1) under a homogeneous Dirichlet boundary condition is investigated in this work. The considered equation reflects the anomalous electro-diffusion in nerve cells. Using nonlinear capacity estimates specifically adapted to the considered problem, we establish sufficient conditions for the nonexistence of weak solutions. [ABSTRACT FROM AUTHOR]

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

Abstract 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 , $$ \textstyle\begin{cases}(-\Delta )^{s}u+V(x)u = f(x,u) &\text{in }\Omega , \\ u>0 &\text{in }\Omega , \\ u =0 &\text{in }\mathbb{R}^{n}\setminus \Omega , \end{cases}$$ where 0 < s < 1 $0< s 2 s $n>2s$ , Ω is a smooth bounded domain in R n $\mathbb{R}^{n}$ , ( − Δ ) s $(-\Delta )^{s}$ denotes the fractional Laplacian of order s, f ( x , t ) $f(x, t)$ is a function in C ( Ω ‾ × R ) $C(\overline{\Omega}\times \mathbb{R})$ , and f ( x , t ) / t $f(x, t)/t $ is nondecreasing in t and converges uniformly to an L ∞ $L^{\infty}$ function q ( x ) $q(x)$ 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 ) $f(x,t)$ exhibits superlinearity in t at infinity, represented by q ( x ) ≡ + ∞ $q(x)\equiv +\infty $ .

In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element ap-proximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semi-discrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the sys-tem emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the results.

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]

A nonlinear time-fractional cable equation posed on the interval $ (0, 1) $ under a homogeneous Dirichlet boundary condition is investigated in this work. The considered equation reflects the anomalous electro-diffusion in nerve cells. Using nonlinear capacity estimates specifically adapted to the considered problem, we establish sufficient conditions for the nonexistence of weak solutions.

In the present paper, we study the nonexistence of nontrivial weak solutions to a class of fractional $p$-Laplacian equation in two cases which are $sp > N$ and $sp < N$. In each of these cases, by using fractional Laplacian theory and inequality techniques, we obtain concrete range of parameter for which nontrivial weak solution of the problem does not exist. Our work complements the known nonexistence results in this direction.

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]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

We prove partial regularity of solutions u of the nonlinear quasimonotone system div A x , u , D u + B x , u , D u = 0 under natural polynomial growth of its coefficient functions A and B. We propose a new direct method based on an L p estimate with low exponent p > 1 for a linear elliptic system with constant coefficient. [ABSTRACT FROM AUTHOR]

In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element approximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semidiscrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the system emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the 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]

The main goal of this paper is to show that if u ∈ W m , p (R n) is a weak solution of Q u = f where f ∈ X p , k r , q (R n) , then u ∈ X p , k m + r , q (R n) with 1 < p , q < ∞ , 0 < r < 1 , k is a temperate weight function in the Hörmander sense, Q = ∑ | β | ≤ m c β ∂ β is a linear partial differential operator of order m ≥ 0 with non-zero constant coefficients c β , and where X p , k r , q (R n) is either the weighted Triebel-Lizorkin or the weighted Besov space. The way to prove this result is based on the boundedness of the continuous wavelet transform with rotations. [ABSTRACT FROM AUTHOR]

In this paper, we shall study the weak solution to the supercritical deformed Hermitian–Yang–Mills equation in the boundary case. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence of solutions and energy estimates for a specific type of periodic boundary value problem. By precisely determining the parameter's localization, we demonstrate the existence of non-zero solutions and infer the presence of multiple solutions for positive parameter values. This analysis necessitates the sublinearity of the nonlinear component at both the origin and infinity. Lastly, we present a multiplicity result along with an example. The proof relies on a local minimum theorem for differentiable functionals. [ABSTRACT FROM AUTHOR]

In this paper, we consider a time-fractional wave equation for positive operators, including the classical Laplacian with the Dirichlet boundary condition. Determinations of initial velocity and perturbation are investigated. It is also shown that these inverse problems of determining the initial data are ill-posed. Moreover, under some conditions of well-posedness properties of the inverse problems are proved. As an appendix, we also provide some proof of the direct problems. Here, we develop the theoretical part of the inverse problems of finding the initial data for the time-fractional wave equations. [ABSTRACT FROM AUTHOR]