Applied Mathematics and Optimization最新文献

This article studies subcritical elliptic problems driven by a polyharmonic double phase operator and establishes the existence of an unbounded sequence of weak solutions. Our approach relies on the symmetric mountain pass theorem of Ambrosetti and Rabinowitz and successfully treats the delicate degenerate regime of the operator. The results appear to be the first in the literature to address polyharmonic double phase problems within this framework.

This article is concerned with the asymptotic behavior of solutions for the 3D non-autonomous stochastic Kelvin–Voigt–Brinkman–Forchheimer equations driven by additive white noise on unbounded domains. The existence and uniqueness of tempered random attractors are proved for the equations. We also establish that the tempered random attractors are periodic when the non-autonomous external term is periodic in time. The energy equation method is employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non-compactness of Sobolev embeddings on unbounded domains.

We study an optimal control problem for semilinear parabolic equations with infinite horizon, pointwise state contraints and two different types of control constraints (pointwise in space and time, and pointwise in time and (L^2) in space). First we prove first-order necessary conditions, then we provide a second-order sufficient condition for local optimality. The second-order condition is formulated using an extended cone that considers the infinite horizon and the control and state constraints. This condition is sufficient for strict local optimality in the (L^2)-sense. Finally, we address the approximation of the infinite horizon control problem by finite horizon problems. We analyze the convergence of these approximations and provide error estimates.

In this work, we investigate the McKean-Vlasov stochastic partial differential equations driven by Poisson random measure. By adapting the variational framework, we prove the well-posedness and large deviation principle for a class of McKean-Vlasov stochastic partial differential equations with monotone coefficients. The main results can be applied to quasi-linear McKean-Vlasov equations such as distribution dependent stochastic porous media equation and stochastic p-Laplace equation. Our proof is based on the improved weak convergence approach proposed in (Liu, W.et al.: Potential Anal. 59, 1141–1190 (2023)), which is specifically developed to handle the large deviation principle for distribution-dependent stochastic systems. Furthermore, by using the methodological strategy in (Wu, W. and Zhai, J.: SIAM J. Math. Anal. 56, 1–42 (2024)), we employ the time discretization procedure and relative entropy estimates to successfully drop the compactness assumption of embedding in the Gelfand triple, enabling us to address both bounded and unbounded domains in applications.

In this paper, we are devoted to studying the following nonlocal elliptic-parabolic equations involving the fractional (p, 2)-Laplacian

where (Omega subset mathbb {R}^N) is a bounded domain with Lipschitz boundary, ((-Delta )_{p}^{alpha }+(-Delta )_{2,a}^{iota }) is the fractional (p, 2)-Laplacian with (0<{iota }<alpha <1), (p,qge 2), (a:mathbb {R}^Ntimes mathbb {R}^Nrightarrow [0,infty )) is a bounded function, (partial _t^{beta }) is the Riemann-Liouville time fractional derivative with (0<beta <1), (lambda ) is a parameter, and (gin L^infty (0,infty ;L^2(Omega ))). The existence theory of solutions is established by applying the Galerkin method combined with fractional calculus theory. Then, by the comparison theorem, the uniqueness of the global weak solution is derived. Moreover, under some suitable assumptions, we also give a decay estimate of solutions. There are two main features of this paper. First, our problem is the combination of both the Riemann-Liouville time fractional derivative and the fractional (p, 2)-Laplacian operator. In particular, the fractional Laplacian ((-Delta )_{2,a}^{iota }) has a weight function (a(cdot )) which plays a role in transforming between two states. If (pne 2), the presence of two fractional operators with different growth, which generates a double phase anisotropic energy. Second, by the Lax-Milgram theorem, the above problem presents a Choquard nonlinear term, which also leads to non-local characteristics.

The aim of this paper is to establish the limiting behavior of random attractors for non-autonomous Benjamin-Bona-Mahony equations with nonlinear colored noise and time-delay defined on unbounded channels. A suitable condition to control the time-delay term is given and then several necessary uniform estimates on the solutions of the problem are established. The pullback asymptotical compactness of the non-autonomous cocycle associated with non-autonomous BBM equation with nonlinear colored noise and time-delay in (C([-rho ,0],H_0^1(mathcal {O}))) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains and weak dissipative structure of the equation. Then the existence of tempered pullback random attractors of the equation is established in (C([-rho ,0],H_0^1(mathcal {O}))) for nonlinear growth diffusion term as well as Lipschitz time-delay term. At last, the upper semi-continuity of two random attractors is investigated by utilizing the strategy of showing that the solution of BBM equation with time-delay convergent to the solution of non-delay BBM equation as the time delay approaches zero. This work extended our previous work [8, Mathematische Annalen, 2023] to non-autonomous BBM equation with time-delay and further considered the limiting behavior of random attractors.

This article primarily investigates the convergence rates of solutions to the chemotaxis-Stokes system, which consists of two distinct species represented by the following system

subject to homogeneous Neumann boundary conditions within a smooth bounded domain (Omega subset mathbb {R}^3). By formulating a suitable energy functional, the following results are established: (bullet ) If (lambda _{2}mu _{1}>blambda _{1}) and both (mu _{1}) and (mu _{2}) are sufficiently large, it is demonstrated that for any global bounded solution originating from adequately regular initial data with (u_{10}, u_{20}not equiv 0),

where ((u_{1star },u_{2star },u_{2star },u_{1star },0)) denotes the unique positive spatially homogeneous equilibrium of this system. (bullet ) If (lambda _{2}mu _{1}le blambda _{1}) and (mu _{1}) is sufficiently large, then all global bounded solutions with reasonably smooth initial data satisfying (u_{10}not equiv 0) exhibit the following behavior:

Many applications involve nonsmooth optimization problems that often exhibit a low-dimensional structure in their optimal solutions. The projection gradient method (PG), the alternating direction method of multipliers (ADMM), and the accelerated projection gradient method (APG) are particularly effective for solving nonconvex composite programming problems and are known to determine the optimal sparsity pattern after a finite number of iterations. However, the exact number of iterations required to identify the final sparsity pattern remains an open problem. In this work, we develop a novel analytical framework to characterize the complexity of determining the active manifold and provide a rigorous proof. Using this framework, we show that PG, ADMM, and APG satisfy the necessary assumptions, enabling us to characterize the complexity of identifying the final active manifold for composite programs with nonsmooth, nonconvex regularizers, such as the (L_0) and (L_p) norms, without requiring nondegeneracy conditions. Finally, we present numerical validation for the derived theoretical complexity bound.

In this paper, we study nonlinear elliptic equations on finite weighted graphs from two complementary perspectives. First, we investigate a discrete analogue of the classical Yamabe-type problem, focusing on the existence and nonexistence of spectral-threshold solutions. Secondly, we consider a Schrödinger-type equation with prescribed mass and establish the existence and nonexistence of normalized solutions in every (L^2)-growth regime of the nonlinearity, without any restriction on the mass.

For a class of nonlinear systems in Banach spaces we provide feedback laws which lead to null controllability and to good estimates for the associated minimum time function.