Inverse random source and Cauchy problems for semi-discrete stochastic parabolic equations in arbitrary dimensions

• The fractional type KdV-BBM equation is derived as a pyhsical model. • The long time existence result for the Cauchy problem for the fractional KdV-BBM equation is established. • The maximal existence time is extended beyond hyperbolic time scale by using a modified energy technique. • A Fourier pseudospectral method is proposed for the numerical investigations of solutions to the fractional KdV-BBM equation. We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data ∥ u 0 ∥ H N + α / 2 = ϵ from 1 ϵ to 1 ϵ 2 . The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.

In this paper, we solve the Cauchy problem for a loaded fractional diffusion equation in an infinite strip. The loaded term is defined as the trace of the fractional derivative of the desired solution on a continuous curve lying inside the domain. We consider all three cases of possible distribution of the order of differentiation in the loaded term (µ) and the order of the time-fractional derivative in the principal differential part of the equation (α). In the first case considered (α > µ), the problem under study is reduced to an integral equation. In the second case (α = µ), we obtain a functional equation. In the third case (α < µ), we are dealing with a differential equation. We show that the condition α > µ ensures the unique solvability of the problem under consideration. In the case of an essentially loaded equation (α ≤ µ), the problem may lose both uniqueness and solvability. In particular, it is shown that if α < µ, then the problem under consideration ceases to be uniquely solvable, and the corresponding homogeneous problem has infinitely many nontrivial solutions. Moreover, in this case, the solvability requires additional conditions that narrow the set of admissible input data.

The aim of this paper is to construct stochastic processes allowing to obtain probabilistic representations of classical, weak or viscosity solutions of the forward Cauchy problem for several types of systems of nonlinear PDEs arising as viscous conservation and balance laws in various applications. The required stochastic processes are constructed as solutions of corresponding stochastic differential equations (SDEs) both forward and backward in time. Due to non-linearity of PDE systems under consideration additional relations must be added to the SDEs in order to obtain closed systems that can be studied independently. These relations are proved to generate probabilistic representations of the required solutions of the Cauchy problem for the original nonlinear PDE systems. Probabilistic representations are used to develop new numerical algorithms for approximation of classical and viscosity solutions to nonlinear PDEs.

A posteriori-driven adaptive strategy for solving inverse Cauchy problems in diffusion-reaction models

Cauchy problem for stochastic regularized nonlinear dispersive wave equations

We consider controlled initial-boundary value problems for semilinear partial differential equations with first- and second-order time derivatives, represented as a Cauchy problem for a semilinear evolution equation in a Hilbert space with an unbounded skew-adjoint operator and an incoming linear control function. Based on the author's previously obtained results for this Cauchy problem, sufficient conditions are established for the exact controllability of the equations under consideration to a given final state (as well as to given intermediate states at intermediate times) over an arbitrarily fixed (without additional conditions) time interval. Furthermore, a theorem on the stability of the control process is proved for an abstract evolution equation.

We study the measure transition problem for bilateral Laplace transforms of meromorphic functions on vertical strips. Given a meromorphic function F admitting Laplace representations on two adjacent strips separated by a vertical line, we investigate how the corresponding determining measures are related. Our first result shows that in the absence of poles on the separatrix the determining measures coincide. We next derive explicit transition formulas for the case of finitely many poles and obtain sufficient conditions under which these formulas remain valid for infinitely many poles. Applications are given to the analytic continuation of the Riemann ζ function, periodic and almost periodic functions, and quotients of Γ functions related to the confluent hypergeometric function. Finally, using generalized Cauchy integrals, we construct an entire function admitting distinct Laplace representations on the right and left half-planes, thereby producing a ghost transition. This provides a counterexample to uniqueness of solutions of the Cauchy problem for the heat equation.

This article investigates the spreading properties of yellows viruses within sugar beet agro-ecosystems using reaction-diffusion systems, with spreading speeds and traveling wave solutions serving as key analytical tools. In these systems, each unknown function corresponds to a distinct ecological variable: infected hosts (i.e., sugar beets), infected vectors, susceptible vectors, and vector predators. In the absence of predators, we analyze the spreading characteristics of infected hosts and infected vectors, with susceptible vectors treated as the native population. The spreading speed of yellows viruses is given, which equals the minimal wave speed of monotonic traveling wave solutions modeling disease spreading and prevalence. When predators are introduced for biological control, the existence and nonexistence of traveling wave solutions starting from the disease-free and predator-free steady state are studied. For the corresponding Cauchy problem, the invasion speed of predators is established. Specifically, this speed is derived under the assumption that vectors are native species, and it remains independent of disease prevalence. Subsequently, we numerically compare the observed viral prevalence under the presence of predators with different expansion capabilities. We then present two distinct scenarios focusing on the spread or extinction of yellows viruses. In the context of virus prevention and control, these results deepen our understanding of the importance of the predation rate, predator mobility, and biting rate.

ABSTRACT The relative arbitrage portfolio outperforms a benchmark portfolio over a given time‐horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi‐agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well‐posed market dynamical system of McKean–Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guaranty relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.

Abstract We investigate the Cauchy problem for the Hunter–Saxton equation with a general continuous forcing term . For weak solutions satisfying , we determine as the sharp critical exponent governing energy conservation and uniqueness. When , every weak solution conserves the ‐energy and is unique for arbitrary continuous . Conversely, for , we construct continuous forcings that admit non‐conservative weak solutions and exhibit failure of uniqueness. Our counterexample further shows that dissipative solutions with identical initial data may be non‐unique, thereby distinguishing the conservative and dissipative frameworks. The analysis relies on Littlewood–Paley theory, Besov embeddings, and delicate commutator estimates.

In this paper, we study the Cauchy problem for the compressible Navier–Stokes system of non-Newtonian fluids governed by the Power Law model. We establish the global existence and uniqueness of classical solutions for a particular class of compressible non–Newtonian fluid equations. Furthermore, employing Hausdorff decomposition and a low-frequency cancelation technique, we derive optimal time-decay rates for these classical solutions. Crucially, our analysis extends to the highest-order derivatives, demonstrating the robustness of these results in the non–Newtonian framework.

In this paper, we will investigate the global existence of solutions for the parabolic-elliptic-ODE system 0,\\\\ 0 & = \\Delta v+w, \\quad x\\in\\mathbb R^d, \\ t>0,\\\\ \ au w_t & = -w+u, \\quad x\\in\\mathbb R^d, \\ t>0,\\\\ u(x, 0) & = u_0(x), \\quad w(x,0)=w_0(x), \\quad x\\in\\mathbb R^d \\end{aligned}\\right. \\end{align*}$$]]> { u t = Δ u − ∇ ⋅ ( u ∇v ) , x ∈ R d , t > 0 , 0 = Δv + w , x ∈ R d , t > 0 , τ w t = − w + u , x ∈ R d , t > 0 , u ( x , 0 ) = u 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ R d under initial data conditions in (generalized) pseudomeasure spaces in spatial dimensions d ≥ 3 . This approach enables us to relax the regularity assumptions on the initial conditions.

Abstract This paper deals with an age-structured HIV model with antiretrowviral therapy and two infection routes (virus-to-cell and cell-to-cell). The model is first formulated as an abstract non-densely defined Cauchy problem and the existence of the equilibria is obtained under some conditions. Based on the existence of equilibria, the global asymptotical stability of the disease-free equilibrium is studied by applying theory of operator semigroups and spectral analysis. The stability and Hopf bifurcation results with two delays around the endemic equilibrium are also well described under some conditions by the geometric stability switch criteria. Finally some numerical examples are presented to illustrate the obtained results.

This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in H s H^s , with s > d 2 + 2 − μ s > \frac {d}{2} + 2 - \mu , with μ = 3 14 \mu = \frac {3}{14} and μ = 3 7 \mu = \frac 37 in the cases d = 1 d = 1 and d ≥ 2 d \geq 2 respectively. This represents an improvement over the state-of-the-art low regularity theory in d ≥ 2 d \geq 2 dimensions.

Abstract In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction–diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.

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.

ABSTRACT The Runge approximation is one of the interesting properties of partial differential equations. Studies have demonstrated its significance for sampling methods of inverse problems and learning‐based numerical methods for differential equations. In this paper, we discuss both qualitative and quantitative Runge approximation properties for the Lamé system. Our analysis relies on conditional stability of the Cauchy problem for the Lamé system and duality arguments, with optimal regularity conditions for the Lamé coefficients being taken into account. Under the fundamental assumption, the uniqueness result for the Cauchy problem leads to the qualitative Runge approximation property in the ‐norm. Under the strong assumption, employing the conditional stability estimates for the Cauchy problem and the truncated singular value decomposition, we derive quantitative Runge approximation estimates.

In this paper, an initial-boundary value problem for an inhomogeneous heat equation with a piecewise constant argument and Dirichlet boundary conditions is considered. The Fourier method is used to investigate the problem. By expanding the solution in terms of eigenfunctions, the initial-boundary value problem is reduced to the Cauchy problem for an ordinary differential equation with respect to the expansion coefficients with a piecewise continuous argument. The existence and uniqueness of the solution to this problem are proved. As a result, it is shown that the original problem has a unique solution, which is constructed in explicit form.

The Cauchy Problems for the 2D Compressible Euler Equations and Ideal MHD System are Ill-Posed in $$H^\frac{7}{4}(\mathbb {R}^2)$$