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We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted L p spaces. This technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal control problem, aimed at identifying a suitable therapy, guiding the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems. [ABSTRACT FROM AUTHOR]

In this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non‐exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds. [ABSTRACT FROM AUTHOR]

The global dynamics of a nonlocal predator–prey model with mutation are investigated in this paper. First, by building a new comparison principle and constructing monotone iterative sequences, we give the existence of the solution for the corresponding initial boundary value problem. Then the uniqueness and uniformly boundedness of the solution are established by using the fundamental solution and quasi-fundamental solution of the heat equation, Grönwall's inequality and auxiliary functions. Finally, under the truncation function method, we discuss the asymptotic behavior of the solution. It is worth mentioning that the asymptotic behavior here is different from the conclusion of the classical model and we use more refined analysis and estimation in the research process. [ABSTRACT FROM AUTHOR]

Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal Lp$$ {L}^p $$ regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness for a class of semilinear superdiffusion equations with spatial nonlocal operators. We first establish the Gagliardo–Nirenberg inequality in ψ -Bessel potential spaces. Based on this, the well-posedness results of local and global mild solution for corresponding linear problem are obtained via apriori estimates. We also obtain the well-posedness results for the nonlinear problem under different conditions. These conclusions are mainly based on the Mihlin–Hörmander’s multiplier estimates, embedding theorem and fixed point theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and long-time dynamics for a class of wave equations with a nonlocal structural energy damping. The main result establishes that for each exponent β ∈ (0 , 1 / 2 + ϵ) of the structural term, for a small ϵ > 0 , the corresponding problem has a compact global attractor A β , which coincides with the unstable manifold M β (N) emanating from the set N of stationary points. This class of problems whose dissipation intensity depends on the energy of the system has connection with flight structure models, see NASA-AirForce reports (Balakrishnan in A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988; Balakrishnan and Taylor in Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). [ABSTRACT FROM AUTHOR]

AbstractIn this work, we focus on a one-dimensional initial-boundary value problem of a thermoelastic piezoelectric beam in the presence of magnetic effects with distributed delay acting in the mechanical equation. The well-posedness of the system is initially demonstrated by applying semigroup theory (Hille-Yosida theorem). Also, based on the construction of a Lyapunov functional, which is equivalent to the energy functional of the problem through multiplier techniques, we show that a unique dissipation through frictional damping is strong enough to ensure exponential stabilization of the model. Next, the results are compared to those of the electrostatic or quasi-static approaches. Moreover, we consider the fully dynamic and electrostatic or quasi-static piezoelectric beams with thermal effects, boundary feedback controllers and boundary distributed delays and using the multipliers technique, we establish an exponential stability result of the solution. Our results are related to the distributed delay weights. Finally, we give some numerical tests to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

Each year, millions of people die from the airborne infectious illness tuberculosis (TB). Several drug-susceptible (DS) and drug-resistant (DR) forms of the causative agent, Mycobacterium tuberculosis (MTB), are currently common in the majority of affluent and developing nations, particularly in Bangladesh, and completely drug-resistant strains are beginning to arise. The main purpose of this research is to develop and examine a non-integer-order mathematical model for the dynamics of tuberculosis transmission using the fractal fractional operator. By demonstrating characteristics such as the boundedness of solutions, positivity, and reliance of the solution on the original data, the biological well-posedness of the mathematical model formulation was investigated for TB cases from 2002 to 2017 in KPK Pakistan. Ulam-Hyres stability is also used to assess both local and global aspects of TB bacterial infection. Sensitivity analysis of the TB model with therapy was also examined. The advanced numerical technique is used to find the solution of the fractional-order system to check the impact of fractional parameters. Simulation highlights that all classes have converging qualities and retain established positions with time, which shows the actual behavior of bacterial infection with TB. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of p -evolution equations of arbitrary order with variable coefficients depending on time and space variables (t , x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H 2 -strong solution of the assimilation system converges exponentially to the reference solution in the sense of L 2 as t → ∞. We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation. [ABSTRACT FROM AUTHOR]

The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided. [ABSTRACT FROM AUTHOR]

This paper investigates a Cahn–Hilliard–Swift–Hohenberg system, focusing on a three-species chemical mixture subject to physical constraints on volume fractions. The resulting system leads to complex patterns involving a separation into phases as typical of the Cahn–Hilliard equation and small scale stripes and dots as seen in the Swift–Hohenberg equation. We introduce singular potentials of logarithmic type to enhance the model's accuracy in adhering to essential physical constraints. The paper establishes the existence and uniqueness of weak solutions within this extended framework. The insights gained contribute to a deeper understanding of phase separation in complex systems, with potential applications in materials science and related fields. We introduce a stable finite element approximation based on an obstacle formulation. Subsequent numerical simulations demonstrate that the model allows for complex structures as seen in pigment patterns of animals and in porous polymeric materials. [ABSTRACT FROM AUTHOR]

This work is concerned with the fractional-in-time parabolic–parabolic Keller–Segel system in a bounded domain Ω ⊂ R d ( d ≥ 2 ), for distinct fractional diffusions of the cells and chemoattractant. We prove results on existence, uniqueness, continuous dependence on the initial data and its robustness, continuation, and a blow-up alternative of solutions in Lebesgue spaces. Then, we use those results to show the existence of global solutions to the problem, when the chemoattractant diffusion is not slower than the cell diffusion. [ABSTRACT FROM AUTHOR]

We consider a multi-component version of the conserved Allen-Cahn equation proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for phase separation. In our case, the free energy is characterized by a mixing entropy density which belongs to a large class of physically relevant entropies like, for example, the Boltzmann-Gibbs entropy. We establish the well-posedness of the Cauchy-Neumann problem with respect to a natural notion of (finite) energy solution which is more regular under appropriate assumptions and is strictly separated from pure phases if the initial datum is. We then prove that the energy solution becomes more regular and strictly separated instantaneously. Also, we show that any finite energy solution converges to a unique equilibrium. The validity of a dissipative inequality (identity for strong solutions) allows us to analyze the problem within the theory of infinite-dimensional dissipative dynamical systems. On account of the obtained results, we can associate to our problem a dissipative dynamical system and we can prove that it has a global attractor as well as an exponential attractor. [ABSTRACT FROM AUTHOR]

This article is intended to present a qualitative and numerical analysis of well-posedness and boundary stabilization problems of the well-known Korteweg-de Vries-Burgers equation. Assuming that the boundary control is of memory type, the history approach is adopted in order to deal with the memory term. Under sufficient conditions on the physical parameters of the system and the memory kernel of the control, the system is shown to be well-posed by combining the semigroups approach of linear operators and the fixed point theory. Then, energy decay estimates are provided by applying the multiplier method. An application to the Kuramoto-Sivashinsky equation will be also given. Lastly, we present a numerical analysis based on a finite difference method and provide numerical examples illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze a space-time spectral method for the subdiffusion equation. Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and the new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials (GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity. [ABSTRACT FROM AUTHOR]

We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the blow-up solutions of the fourth-order Schrödinger equation with combined power-type nonlinearities i u t + α Δ 2 u + β Δ u + λ 1 u σ 1 u + λ 2 u σ 2 u = 0 , where 4 < n < 8 , β = 0 , 1 , α , λ 1 ∈ R and λ 2 < 0 . Firstly, using Banach’s fixed point theorem, iterative method and nonlinear estimates, we establish the local well-posedness of solutions with the initial data u 0 ∈ H 2 (R n) . Then, based on variational analysis theory for dynamical system, using localized Virial identity, we establish a new Morawetz estimates and upper bound estimates to prove the existence of blow-up solutions in finite time. Finally, applying the local well-posedness above, we demonstrate the blow-up criteria of solutions and prove it by contradiction method. [ABSTRACT FROM AUTHOR]

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, a modified pseudostress tensor, which depends on the pressure as well as the diffusive and convective terms, is introduced as an auxiliary unknown, and the incompressibility condition is then used to eliminate the pressure, which is subsequently computed using a postprocessing formula. Consequently, to discretize the resulting mixed formulation, it is sufficient to provide a tensorial weak Galerkin space for the pseudostress and a space of piecewise polynomial vectors of total degree at most ’k’ for the velocity. Moreover, the weak gradient/divergence operator is utilized to propose the weak discrete bilinear forms, whose continuous version involves the classical gradient/divergence operators. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method’s good performance and confirming the theoretical rates of convergence are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability of the two‐dimensional stationary Navier–Stokes equations on the whole plane R2$\mathbb {R}^2$. In Fujii [Ann. PDE, 10 (2024), no. 1. Paper No. 10], it was proved that the stationary Navier–Stokes equations on R2$\mathbb {R}^2$ is ill‐posed for solutions around zero. In contrast, considering solutions around the nonzero constant flow, the perturbed system has a better regularity in the linear part, which enables us to prove the unique existence of solutions in the scaling critical spaces of the Besov type. [ABSTRACT FROM AUTHOR]

In the article, we establish the global well‐posedness in W1,2,2(ℝ×ℝ+)$$ {W}&#x0005E;{1,2,2}\left(\mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}&#x0005E;{&#x0002B;}\right) $$ of the integro‐differential equation in the case of anomalous diffusion when the one‐dimensional negative Laplace operator is raised to a fractional power in the presence of the transport term. The model is relevant to the cell population dynamics in the mathematical biology. Our proof relies on a fixed point technique. [ABSTRACT FROM AUTHOR]

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the L2$$ {L}&#x0005E;2 $$ norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a pseudo‐parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data u0 ∈ L2. In the case of initial data u0 ∈ Lq, q ≠ 2, we obtain the local existence result. Our main tool here is using fundamental tools, namely, Banach fixed point theorem and Sobolev embeddings. In addition, we give an example to illustrate the effectiveness of the method has been proposed. [ABSTRACT FROM AUTHOR]

The availability of cancer measurements over time enables the personalized assessment of tumor growth and therapeutic response dynamics. However, many tumors are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumor states by using a single spatial tumor dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumor dynamics with a phase field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fréchet differentiable. We then analyze the backward inverse problem concerning the reconstruction of earlier tumor states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate. [ABSTRACT FROM AUTHOR]

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general C 0 -semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers' equation with nonlinear local dynamics and both distributed and boundary disturbances. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments. [ABSTRACT FROM AUTHOR]

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ ∈ [ 0 , 1 ] . The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. [ABSTRACT FROM AUTHOR]

We demonstrate that the presence of summands with involution in the argument in an ordinary differential equation can significantly affect the well-posedness of the Cauchy and other problems. Furthermore, we show that the above effects can influence the well-posedness of classical boundary value problems for partial differential equations, particularly in the case of parabolic and pseudoparabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $ \alpha -\frac {1}{2} $ α − 1 2 with $ \frac {1}{2} \lt \alpha \lt 1 $ 1 2 < α < 1. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method. [ABSTRACT FROM AUTHOR]

α This studies the existence and stability of α-core of games with discontinuous vector payoffs. By proposing the conditions of minimum values of games with vector payoffs and coalitional C-security, this gives two kinds of sufficient conditions to guarantee the existence of α-core of games with discontinuous vector payoffs. Furthermore, by using the lemma for generalized Hadmard well-posedness, the well-posedness of α-core is proven for a kind of game with discontinuous vector payoffs. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis system with Lotka–Volterra competitive kinetics, $ \begin{align*} \begin{cases} u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1 - u - a_1 v), & x\in\Omega, \ t>0, \\ v_t = d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v (1 - a_2 u - v), & x\in\Omega, \ t>0, \\ 0 = d_3 \Delta w + \alpha u + \beta v - \gamma w, & x\in\Omega, \ t>0, \end{cases} \end{align*} $under homogeneous Neumann boundary conditions and suitable initial conditions, where $ \Omega \subset \mathbb{R}^n $ $ (n \in \mathbb{N}) $ is a bounded domain with smooth boundary, $ d_1, d_2, d_3, \chi_1, \chi_2, \mu_1, \mu_2 > 0 $, $ a_1, a_2 \ge 0 $ and $ \alpha, \beta, \gamma > 0 $. Under largeness conditions on $ \chi_1 $ and $ \chi_2 $, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends previous results of [38] to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic–hyperbolic–elliptic limit system with $ d_1 = d_2 = 0 $. [ABSTRACT FROM AUTHOR]

The main object of the present paper is the study of thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier's law and acting on both the rotation angle and the transverse displacements. We establish an exponential stability result for the considered problem in case of equal wave speeds. In the opposite case, we show the lack of exponential stability. Furthermore, a polynomial stability result is established. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity. [ABSTRACT FROM AUTHOR]

We prove the global existence and uniqueness of solutions to the Vlasov–Riesz–Fokker–Planck system around the global Maxwellian in the periodic spatial domain. Depending on the order of Riesz potential, we present two frameworks for the construction of global-in-time solutions with Sobolev and analytic regularity. The analytic function framework covers the Vlasov–Dirac–Benney–Fokker–Planck system. Furthermore, we show the exponential decay of solutions toward the global Maxwellian. Our result is generalized to the whole space case in which the decay rate of convergence is algebraic. [ABSTRACT FROM AUTHOR]

In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like | u | α or | u | α − 1 u (α > 1) and the initial data u 0 belongs to the Sobolev spaces L s p (G) for 1 < p < ∞ and 0 < s < N / p. Since stratified Lie groups G include the Euclidean space R n as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on R n to G. It should be noted that our proof is very different from it given by Ribaud on R n. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on R n. By using the generalized fractional chain rule on G , we can avoid the discussion of Fourier analysis on G and make the proof more simple. [ABSTRACT FROM AUTHOR]

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on R n , where n ⩾ 3 . In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from [ 0 , ∞) to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates, Hölder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces. [ABSTRACT FROM AUTHOR]

This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third-grade fluid equations in bounded domain D ⊂ R d , d = 2 , 3 , in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stability result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using “Krylov-Bogoliubov Theorem,” we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of L 2 , which leads to the existence of an ergodic invariant measure. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the well-posedness and stabilization of the generalized Rao-Nakra beam equation. The system consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. First, we prove that the system is well-posed. Second, we show that the analytic stability holds when all the displacements are globally damped through Kelvin-Voigt damping. Third, we establish the case where the damping acts only on the shear angle displacements of the top and bottom layers. In this case, we prove, by using frequency domain arguments combined with the multiplier method, that the energy of the system decays polynomially with a decay rate $ t^{-\frac{1}{2}} $. [ABSTRACT FROM AUTHOR]

This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation. [ABSTRACT FROM AUTHOR]

We study the Cahn–Hilliard–Biot model with respect to its mathematical well-posedness. The system models flow through deformable porous media in which the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg–Landau energy functional, with additional influence from both viscoelastic and fluid effects. The flow-deformation coupling in the system is governed by Biot's theory. This results in a three-way coupled system that can be viewed as an extension of the Cahn–Larché equations by adding a fluid flowing through the medium. We distinguish the cases between a spatially dependent and a state-dependent Biot–Willis function. In the latter case, we consider a regularized system. In both cases, we use a Galerkin approximation to discretize the system and derive suitable energy estimates. Moreover, we apply compactness methods to pass to the limit in the discretized system. In the case of Vegard's law and homogeneous elasticity, we show that the weak solution depends continuously on the data and is unique. Lastly, we present some numerical simulations to highlight the features of the system as a tumor growth model. [ABSTRACT FROM AUTHOR]

We investigate the well-posedness of the 3d co-rotational Beris-Edwards system for incompressible nematic liquid crystal flows with the Landau-De Gennes bulk potential. The system under consideration consists of the Navier–Stokes equations for the fluid velocity $ \boldsymbol u $, and an evolution equation for the $ Q $-tensor order parameter. We prove the existence of solutions to the Cauchy problem of the system with initial data $ (\boldsymbol u_0,Q_0) $ having small $ L_{uloc}^3(\mathbb{R}^3) $-norm of $ (\boldsymbol u_0,\nabla Q_0) $. Moreover, the uniqueness of $ L^3_{uloc} $-solutions is obtained. [ABSTRACT FROM AUTHOR]

For the Fornberg–Whitham equation, the local well-posedness in the critical Besov space B p , 1 1 + 1 p (R) with 1 ≤ p < ∞ has been studied in Guo (Nonlinear Anal RWA 70:103791, 2023). However, for the endpoint case p = ∞ , whether it is locally well-posed or ill-posed in B ∞ , 1 1 (R) is still open. In this paper, we prove that the Fornberg–Whitham equation is well-posed in the critical Besov space B ∞ , 1 1 (R) with solutions depending continuously on initial data, which is different from that of the Camassa–Holm equation (Guo et al. in J Differ Equ 327:127, 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data. [ABSTRACT FROM AUTHOR]

A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions. [ABSTRACT FROM AUTHOR]

This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.

In this article, we prove the well-posedness and stability of a one-dimensional thermo-porous-elastic system, considering a thermal coupling of the Green and Naghdi types III. In this scenario, we propose a new model where the temperature of the material directly affects the stress tensor of displacements and the equilibrated stress tensor on the volume fractions in the porous medium, thereby generalizing some existing results in the literature on the subject. Additionally, the exponential decay of energy is proven when the evolution equation corresponding to the dynamics of the elastic skeletal structure exhibits frictional-type damping.

In this paper we study the well-posedness for the periodic Cauchy problem and the internal controllability of a one-dimensional system that describes the propagation of long water waves with small amplitude in the presence of surface tension. The well-posedness is proved by using the Fourier transform restriction method and the controllability is proved by using the moment method.