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The Klein–Gordon equation with quadratic nonlinearity is frequently used to model various problems of physics. On the other hand, it was shown that in many situations, the use of fractional derivatives instead of ordinary derivatives provides more realistic models. In this work, a one-dimensional sequential time-fractional Klein–Gordon equation posed on a bounded interval, subject to certain initial and boundary conditions, is investigated. The fractional derivatives are considered in the Caputo sense and a weight function h > 0 is allowed in front of the quadratic nonlinearity. Namely, we first establish general sufficient conditions for the nonexistence of solutions. Next, some particular cases of weight functions h are discussed. In our approach, we make use of the test function method. This method requires an appropriate choice of a family of test functions. In this paper, we make a judicious choice of test functions taking into consideration the nonlocal property of the Caputo fractional derivative, the initial and boundary conditions, and the geometry of the domain. [ABSTRACT FROM AUTHOR]

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in R n , n ≥ 2 . The global existence and L ∞ -bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) b > 1 - α n , for any initial value and birth rate; (ii) 0 < b ≤ 1 - α n , for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the L ∞ -bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigate a boundary value problem governed by a complex system of partial differential equations given by: - div [ L ϑ z , ϑ , ∇ ϑ ] + M ^ | ϑ | p - 2 ϑ = L σ (z , ħ ∗ σ (ϑ) , ∇ (ħ ∗ σ (ϑ))) , under the following non-homogeneous Neumann boundary condition: L ϑ z , ϑ , ∇ ϑ. ν = Θ (z , ϑ) . Here, the functions L ϑ , L σ , and Θ exhibit Carathéodory characteristics. The operator σ : W 1 , p (B) → W 1 , p R N operates as an extension operator correlated with the domain B , while ħ denotes an integrable function defined over R N . Notably, the problem introduces a nonlocal operator, manifesting as the convolution ħ ∗ σ (ϑ) of ħ with σ (ϑ) , associated with the variable ϑ . Under conditions necessitating thorough examination, we establish the existence of a weak solution to the mentioned problem by employing the topological degree method. [ABSTRACT FROM AUTHOR]

Let $ R_i $, $ i = 1,2 $, be a root system in $ {\mathbb{R}}^{N_i}\backslash\{0\} $, $ N_i\geq 1 $, $ W_i = W(R_i) $ be the associated finite reflection group, and $ k_i: R_i\to [0,\infty) $ be a multiplicity function, i.e. $ k_i $ is $ W_i $-invariant. For $ \ell\geq 0 $, we introduce the operator $ L_{\ell,k_1,k_2} $ defined by$ L_{\ell,k_1,k_2}u(x,y) = \Delta_{k_1}u(x,y)+|x|^{2\ell}\Delta_{k_2}u(x,y),\,\,(x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}, $where $ \Delta_{k_i} $ is the Dunkl Laplacian operator associated with $ R_i $ and $ k_i $. Our goal in this paper is to establish a Liouville-type result for the semilinear inequality$ -L_{\ell,k_1,k_2}u\geq |u|^p,\,\, (x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}, $where $ u = u(x,y) $ and $ p>1 $. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of bounded weak solutions for a class of p(u)-problems characterized by degenerate coercivity and Dirichlet-type boundary conditions. The main techniques employed include the regularization approach, combined with the variational method, and the theory of Sobolev spaces with variable exponents. [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]

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively. [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 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]

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 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]

For differential inclusions and hemivariational inequalities driven by anisotropic differential operators, we establish the existence of generalized variational solutions and weak solutions. The main novelty consists in allowing that the driving operators might not satisfy any ellipticity condition, which is achieved for the first time in the anisotropic and nonsmooth context. The approach is based on a finite dimensional approximation process. [ABSTRACT FROM AUTHOR]

We study the existence of weak solutions for a system and a coupled system of Λ -Hilfer fractional differential equations on compact domains using the Lax–Milgram and Minty–Browder theorems. Furthermore, we provide an illustrative example, and a regularity result to imply that the obtained solution is classical. [ABSTRACT FROM AUTHOR]

In this investigation, we study the existence of weak solution for a class of nonhomogeneous anisotropic elliptic problem. These problems involve the anisotropic operators with variable exponents. Based on the topological degree theory concerning a specific subset of demicontinuous operators under generalized (S +) and the theory of anisotropic Sobolev spaces with variable exponent, we demonstrate the existence of weak solution for this problem. Our findings are applicable to problems with no-flux boundary conditions, and they extend and generalize several results previously reported in the literature. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the existence and multiplicity of solutions for a second order Kirchhoff-type problem. Under appropriate conditions, we prove the existence of an open interval of positive parameters under which the problem admits at least three distinct weak solutions. Three weak solutions follow from a recent G. Bonanno variational principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity results for a second-order Hamiltonian system generated by impulsive effects. Our approach is based on variational methods and critical point theory. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDE’s) are, we believe, little known in our country, especially at a medium-advanced level, in particular, their historical roots and methods developed in their evolution. The objective of this article is to contribute to a knowledge in this direction especially to colleagues and students who are interested in studying and researching in this central Branch of mathematics. As a model we have chosen H¨ormander’s book, [1], whose study and teaching will be a sign of great progress in this direction in our country. In this article we only give an introduction to part I dedicated to functional analysis and which includes chapters I and II that deal with the theory of distributions and some special spaces of distributions, respectively.

The present article deals with the existence of weak solutions to a class of p(z)p\left(z)-Kirchhoff-type problems. To address these problems, we employ a variational approach in conjunction with the theory of variable exponent Sobolev spaces, while imposing suitable assumptions on the source term. Furthermore, we utilize the theory of Young measures.

This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M(\int_{\Omega}\phi u {\rm d}x){\rm div} (A(x,t,u)\nabla u)=g(x,t,u)$ in $\Omega\times(0,T)$, where $\Omega$ is a bounded domain of $\mathbb{R}^n$ $(n\geq1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M \mathbb{R}\rightarrow\mathbb{R}$, $\phi \Omega\rightarrow\mathbb{R}$, $g \Omega\times(0,T)\times\mathbb{R}\rightarrow\mathbb{R}$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on the variable $(x,t)$, we investigate two uniqueness theorems and give a continuity result depending on the initial data.

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.

In this paper we establish uniqueness of the weak solutions of nonlinear stochastic functional differential equations of neutral type in Hilbert spaces. We also study the continuous dependence of such solutions on the initial data

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.

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 study a two‐dimensional (2D) bidirectional fluid–solid coupling problem in blood vessels and prove the existence of weak solutions to this kind of problem. When studying the fluid motion of arterial blood flow in arterial vascular tissue, where the density of the vessel wall is roughly equal to that of the blood, the coupling between the blood and the vessel wall is highly nonlinear. In this case, the fluid space where the arterial blood flow is located is no longer fixed in the fluid–structure interaction (FSI) model, and its boundary position is determined by the position of the elastic vessel wall. The contact force exerted by the elastic vascular structure on the blood flow also affects the movement of the blood flow, while the position of the blood vessel is determined by the contact force exerted by the blood flow on the blood vessel; that is, two‐way coupling occurs between the vascular diameter and the blood flow space. MSC2020 Classification: 35D30, 35M31 [ABSTRACT FROM AUTHOR]

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]

In this study, we explore the theoretical and numerical aspects of the generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) incorporating weakly singular kernels in a d -dimensional domain, where d ∈ { 2 , 3 }. For the continuous problem, we provide an in-depth discussion on the existence and the uniqueness of weak solution using the Faedo-Galerkin approximation technique. Further, regularity results for the weak solution are derived based on assumptions of smoothness for both the initial data and the external forcing. Using the regularity of the solution, the uniqueness of weak solutions has been established. In terms of numerical approximation, we introduce a semi-discrete scheme using the conforming finite element method (CFEM) for space discretization and derive optimal error estimates. Subsequently, we present a fully discrete approximation scheme that employs backward Euler discretization in time and CFEM in space. A priori error estimates for both the semi-discrete and fully discrete schemes are discussed under minimal regularity assumptions. To validate our theoretical findings, we provide computational results that lend support to the derived conclusions. [ABSTRACT FROM AUTHOR]

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coefficient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a finite number of variables. A similar result is true for Stefan's problem. In the limit, when the number of phases tends to infinity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence and uniqueness of solutions to non-local problems involving a discontinuous Kirchhoff-type function via a global minimum principle of Ricceri. More precisely, we first obtain the uniqueness result of weak solutions to nonlinear fractional Laplacian problems of Brézis–Oswald type. We then demonstrate the existence of a unique positive solution to Kirchhoff-type problems driven by the non-local fractional Laplacian as its application. The main features of the present paper are the lack of the continuity of the Kirchhoff function in [ 0 , ∞) and the localization of a positive solution. [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]

The paper considers the weak solvability of the initial–boundary value problem describing the motion of weakly concentrated aqueous polymer solutions taking into account the memory of the fluid. In this model, the memory is considered along the trajectory of fluid particles determined by the velocity field. The topological approximation approach and the theory of regular Lagrangian flows are used. [ABSTRACT FROM AUTHOR]

The article discusses the criteria for the existence of weak solutions of the Dirichlet problem for nonlinear degenerate elliptic equations with any -right-hand side. It highlights the conditions under which the Dirichlet problem has a weak solution and presents results for a larger family of weight functions. The text also includes theorems and examples illustrating the necessary and sufficient conditions for the nonemptiness of certain sets, along with acknowledgments of research funding and a declaration of no conflicts of interest. [Extracted from the article]

We use recent approximation results in BV space to derive L∞ bounds for u, L∞ bounds in time for the BV seminorm ∫|Du| of u, and L² bounds for ut for the weak solution u ∈ C([0, ∞); L²(Ω) ∩ BV (Ω)), Ω ⊂ ℝN open and bounded, of the time flow .... We assume Neumann boundary condition and φ(x, p) is in a class of linear growth functions in p. Importantly, φ(·, p) ∈ L¹(Ω) in contrast to the classical results stated in [1] where φ has a continuity assumption in the x variable. We also use the convergence of the solution above to derive an L∞ bound for the solution u* to the corresponding stationary problem, since u(t) → u* in L¹(Ω). [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]

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]

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]

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]

This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation ut -- M (∫ Ω φu dx) div (A(x, t, u) ▽u) = g(x, t, u) in Ω x (0, T), where Ω is a bounded domain of Rn (n ≥ 1), T > 0 is a positive number, A(x, t, u) is an n x n matrix of variable coefficients depending on u and M: R → R, φ: Ω → R, g: Ωx(0, T) x R → R are given functions. We consider two different assumptions on g. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if A(x, t, u) = a(x, t) depends only on the variable (x, t), we investigate two uniqueness theorems and give a continuity result depending on the initial data. [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]

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]

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]