Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_ξf(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. Further, we suppose that $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ is convex, and that $f$ satisfies a $p$-growth and -coercivity condition for some $p>\max \big\{ 1,\frac{n(q+1)}{n+q+1} \big\}$. Merely assuming that $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in L^\infty\big( 0,T;L^{q+1}(E,\mathbb{R}^N)\big)$. If $E$ does not shrink too fast, we show that for the solution $u$ constructed in the first step, $\vert u \vert^{q-1}u$ admits a distributional time derivative. Moreover, under suitable conditions on $E$ and the stricter lower bound $p \geq \frac{(n+1)(q+1)}{n+q+1}$, $u$ is continuous with respect to time.

This paper focuses on the existence, nonexistence, uniqueness and bifurcation phenomena of radial sign-changing solutions to an elliptic Dirichlet problem in a ball or an annulus. By providing a representation for the simple eigenvalues of $-\Delta$ in a ball or an annulus with Dirichlet boundary conditions, we prove that the bifurcation of radial sign-changing solutions with $n$ nodes originates from the $n$-th simple eigenvalue. Moreover, we obtain the uniqueness of radial sign-changing solutions with a given number of nodes to the Lane-Emden equation in a ball or an annulus.

In this paper, we investigate the following planar quasilinear Schrödinger equations characterized by distinct potentials $$ -\Delta u+\lambda V(x)u+\frac{\kappa}{2} \Delta(u^2)u = f(u),\,\ x\in\R^2, $$ where $\lambda > 0$ and $\kappa > 0$ are two parameters and the nonlinearity $f:\R\rightarrow\R$ has critical exponential growth. By employing the dual approach, we first establish the existence of nontrivial solutions for the equation featuring both periodic and radial potentials. Subsequently, we demonstrate the existence and concentration of nontrivial solutions for the equation with a steep potential well. A pivotal aspect of our methodology involves a meticulous analysis of the $L^\infty$-estimation. A significant aspect of the results derived in this research lies in the fact that the nonlinear term satisfies more generalized conditions than those typically required.

This study explores how vaccination, immune memory, and long-term immunity shape the transmission dynamics of measles by developing a fractional-order mathematical model. Caputo-Fabrizio fractional-order SEITR-VL model is formulated, in which population is divided into susceptible, exposed, infectious, treated, recovered, vaccinated, and lifelong immunity classes. The model incorporates memory effects so that past disease and immunity states can influence current transmission behavior. Basic mathematical properties such as positivity, boundedness, and the existence of solutions are verified. The effective reproduction number is derived using the next-generation matrix approach, and numerical solutions are obtained through the Laplace-Adomian Decomposition Method. Numerical experiments showed that increasing vaccination coverage, as well as improving recovery rates, leads to a clear decline in infection levels. In addition, the fractional-order structure introduces memory effects that moderate sharp epidemic peaks and slow the overall spread of the disease, resulting in smoother outbreak patterns. Since some parameters are not estimated from data, these findings are interpreted mainly at a qualitative level. The results emphasize the importance of vaccination and immune memory in controlling measles transmission. While the fractional-order framework provides a useful way to capture long-term and memory-dependent effects, further validation using real epidemiological data would be necessary for predictive or policy-oriented applications.

We investigate sharp-interface cohesive fracture models formulated as energy minimization problems. We argue that models with arbitrary cohesive interfaces are incompatible with linear bulk elasticity, in the sense that they cannot feature solutions in the form of a regular crack with a simple tip. To this end, we provide analytical and numerical solutions for a model problem consisting of a single straight crack under mode-III loading, where we show that the stress magnitude exceeds the cohesive yield threshold in a finite region around the crack tip. Our findings are consistent with the unavailability of existence results for such models, related to the lack of lower semicontinuity of the associated variational problem. In the mathematical literature, lower semicontinuity and existence of solutions is recovered by introducing a relaxed functional combining the cohesive surface energy on the crack set with a bulk behavior comparable to perfect plasticity, where the bulk strength is determined by the maximal allowable traction of the cohesive law. The relaxed energy provides a homogenised macroscopic model of the possible microscopic structuring of a dense distribution of cracks with vanishing displacement jumps. We report numerical simulations in antiplane shear that illustrate that the relaxed model admits an equilibrium solution in the form of straight cracks that capture both crack nucleation and propagation. Cracks emerging from pre-existing flaws and notches exhibit a smooth transition from classical crack tip plasticity solutions near the notch to a propagating cohesive crack accompanied by an elongated zone around the tip where the nonlinear bulk behavior is active and the stress is constant. We discuss how these observations can inform the development of mathematically consistent coupled models with a minimal number of constitutive parameters, highlighting the inconsistencies observed when arbitrarily combining models with different surface and bulk strengths.

In this paper, we investigate a nonlinear Schrödinger equation involving mixed fractional $p$-Laplace operators, that is $$ (-\Delta)_p^{s_{1}}u + (-\Delta)_p^{s_{2}}u +V(x) | u | ^{p-2}u= f(u) \text { in } \mathbb R^N, $$ where $0 < s_{1} < s_{2} < 1$ and $2\leq p < \infty$, $(-\Delta)_p^{s_{i}}$ with $i\in \{1, 2\}$, is the fractional $p$-Laplace operator, $V(x)$ is a potential function that may change sign and satisfies mild regularity conditions, the nonlinearity $f \in C^1 (\mathbb R,\mathbb R)$ is a subcritical and superlinear function. We first establish a Struwe-type splitting lemma, then we obtain the existence of ground state solutions as well as sign-changing solutions based on variational techniques and this lemma. The main strategy of the proof is to locate the infimum of the relevant functional on a Nehari type sets.

Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Maz'ya type equation

Optimal control of Alzheimer's disease model via multi-target combination therapy.

In this study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we prove that the initial-boundary value problem admits a unique global weak solution for any F0-measurable L2(I)-valued random initial value with a finite second moment. We also utilize the properties of the fractional Laplacian and fractional Sobolev spaces to provide a proof of the existence of the uniqueness theorem. These results extend the analysis of the Ginzburg–Landau equations to models incorporating stochastic terms and the fractional Laplacian.

Abstract In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of (Jiao et al. in J Diff Dyn Syst 34:239-269, 2022) for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.

In this study, we examine the uniqueness conditions for solutions of fractal differential equations using the Krasnoselskii-Krein uniqueness theorem. The analysis establishes sufficient criteria that guarantee the existence of unique solutions. Additionally, we employ the successive midpoint method to numerically solve chaotic systems governed by both fractal and global derivatives. To evaluate the effectiveness of the proposed approach, graphical simulations are presented for various derivative orders. These results illustrate the method’s accuracy, stability, and reliability in capturing the intricate dynamics of the considered systems.

This article proposes a novel approach for dealing with the time-fractional Navier–Stokes equations via the natural generalized Laplace transform decomposition method (NGLTDM). This hybrid method utilizes both the natural generalized Laplace transform (NGLT) and a decomposition method. The method is correct because the series solutions become more accurate when more terms are added. We establish precise theorems that verify the existence of solutions and the convergence of the series. The analysis shows that the suggested method is more general than the Homotopy Perturbation Method (HPM) and the Adomian Decomposition Method (ADM). Also, this approach can be applied to handle difficult fluid dynamics problems governed by the Navier–Stokes equations. This study enhances analytical methodologies for fractional-order flow models.

This study models trachoma transmission in Cameroon using a deterministic approach with integer and fractional-order derivatives, incorporating direct, fly-mediated, and environmental transmission routes. Fitting disease data from 1990–2019, the model forecasts trachoma prevalence until 2035. The research confirms the solution existence and uniqueness, calculates the basic reproduction number R0λ where λ∈(0,1] represents the fractional-order parameter, and analyzes equilibrium stability. A stable trachoma-free equilibrium exists when R0λ&lt;1, while an endemic equilibrium is proven stable for R0λ&gt;1 under specific conditions. Calibration of a fractional model with Cameroon data yielded an R0 of 1.169 (indicating endemicity) and identified an optimal fractional order of λ=0.98. By calculating the strength number, we found that another epidemic wave could occur in 50 years. Global sensitivity analysis highlighted key parameters affecting trachoma dynamics. A numerical scheme of the model based on the Adams–Bashforth–Moulton method is constructed and its stability demonstrated. It is then used to perform several numerical simulations, first to validate the theoretical results obtained, and then to compare the different models (statistical and deterministic). The conclusion is reached that the disease will persist in the population (R0&gt;1), although the statistical model shows that it could disappear by 2030. This proves that, for trachoma dynamics in Cameroon, it is advisable to use a deterministic model.

This paper studies positive ω-periodic solutions for a class of neutral-type impulsive neural networks with distributed delays and two parameters. Using cone-theoretic fixed point methods, Green's function techniques, a Green-weighted norm, and a Lyapunov-Razumikhin approach, we establish sufficient conditions for the existence of positive periodic solutions. We further derive explicit small-gain conditions under which the positive periodic solution is unique and globally exponentially stable. These conditions reflect the combined effects of leakage, distributed delays, neutral feedback, and impulsive perturbations. In addition, computable estimates are obtained for the contraction constant of the Poincaré map and the exponential convergence rate to the periodic attractor. The main contribution is a unified framework linking cone-based existence theory with weighted small-gain stability analysis for neutral-type impulsive neural networks with distributed delays. Numerical examples illustrate the applicability of the results.

Neurodegenerative diseases (NDs), such as Alzheimer's, Parkinson's, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein ( ), produced by neurons, misfolds into a toxic form, known as scrapie prion protein ( ). induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial spread of toxic proteins over the brain connectome. Our first objective is to establish the existence and uniqueness of a global positive solution to the prion disease models. Afterwards, we analyze the asymptotic behavior of the models by identifying regimes of persistence and extinction of toxic proteins. For the deterministic delayed systems, we perform a stability analysis for the persistence and demonstrate that the system undergoes a Hopf bifurcation. We also study the intensity of fluctuations of the equilibrium state of the stochastic model. Additionally, we present numerical simulations to illustrate the model dynamics using biologically relevant parameters.

Investigation the Epidemiological Transition of Ebola Virus under Contagious Population with Sustainable Extended Fractional Operator.

ABSTRACT This article aims to establish an averaging principle for conformable delay stochastic differential equations involving non‐Lipschitz coefficients. First, we derive Duhamel's formula utilizing standard cosine family of linear operators. Subsequently, by virtue of Picard iteration technique and contradiction method, we demonstrate the existence and uniqueness of mild solution for the considered system, respectively. Then under suitable averaging conditions, we prove that the solutions to the original equations can be approximated by the solutions to the averaged equations both in the sense of mean square and probability. It means that Khasminskii classical approach can be extended to stochastic differential equations of conformable type. As verification, an example is provided to illustrate the theoretical results.

In this paper, we study an inverse problem for the nonstationary Stokes system in a bounded domain Ω with a nonlinear integral overdetermination condition, describing the kinetic energy E(t) of the fluid. We construct two classes of solutions: weak and very weak. In the case where the kinetic energy E belongs to W21(0,T), we construct weak solutions. If E belongs only to L2(0,T), we construct very weak solutions.

ABSTRACT This paper addresses the existence and large‐time asymptotic behavior of strong solutions to the viscous liquid–gas two‐phase flow model subject to slip boundary conditions in a three‐dimensional, simply connected bounded domain with a smooth boundary consisting of finite number 2D connected components. Compared to the Cauchy problem studied in Yu [ Journal of Differential Equations 272 (2021): 732–759] and Guo et al. [ Journal of Mathematical Physics 52 (2011): 9], the main advancement lies in overcoming key difficulties involving boundary integral estimates. We establish the global existence and uniqueness of strong solutions for the system provided that the initial energy is sufficiently small. Moreover, we characterize the large‐time decay of these solutions. Notably, our analysis allows for initial densities exhibiting large oscillations and including vacuum states.