Arbitrary Polynomial Growth Research Articles

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Mean attractors of stochastic delay lattice [formula omitted]-Laplacian equations driven by superlinear noise in high-order product Bochner spaces

This paper is concerned with the existence and uniqueness of mean random attractors of a class non-autonomous stochastic delay p-Laplacian lattice systems defined on a high-dimensional integer set Zd driven by a family infinite-dimensional superlinear noise. We first establish the global-in-time existence and uniqueness of the solutions in C([τ,∞),L2k(Ω,ℓ2(Zd)))∩Lq(Ω,Llocq((τ,∞),ℓq(Zd))) for any k⩾1 when the draft term has an arbitrary polynomial growth rate q>2 and the coefficient of the noise admits a superlinear growth order q̃∈[2,q). We then show that the mean random dynamical system generated by the solution operators has a unique weakly compact and weakly attracting mean random attractor in the high-order product Bochner space L2k(Ω,F;ℓ2(Zd))×L2k(Ω,F;L2k((−ϱ,0),ℓ2(Zd))), where ϱ is the time delay parameter. The dissipative property of the draft term is employed to carefully controlling the superlinear growth diffusion term. When k=1, our results are new even in the product Hilbert space L2(Ω,F;ℓ2(Zd))×L2(Ω,F;L2((−ϱ,0),ℓ2(Zd))). This work can be regard as a further study of mean attractors of stochastic p-Laplacian lattice systems in the works of Wang and Wang (2020) and Chen et al. (2023).

Well-Posedness and Global Attractors for Viscous Fractional Cahn–Hilliard Equations with Memory

We examine a viscous Cahn–Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.

Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on R<sup>n</sup>

This paper is concerned with the existence and upper semi-continuity of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity and multiplicative noise in H1(Rn). First, we study the existence and uniqueness of solutions by a noise arising in a continuous random dynamical system and the asymptotic compactness is established by using uniform tail estimate technique, and then the existence of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity. As a motivation of our results, we prove an existence and upper semi-continuity of random attractors with respect to the nonlinearity that enters the system together with the noise.

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

Global Solvability of the Cauchy-Dirichlet Problem for a Class of Strongly Nonlinear Parabolic Systems

We consider a class of nonlinear parabolic systems for elliptic operators of variational structure with nondiagonal principal matrices. Additional terms in the systems can have quadratic growth with respect to the gradient and arbitrary polynomial growth with respect to solutions. We obtain sufficient conditions for the time-global weak solvability of the Cauchy–Dirichlet problem and study the regularity of the solution. The case of two spatial variables is considered.

Boundedness for reaction–diffusion systems with Lyapunov functions and intermediate sum conditions

We study the uniform boundedness of solutions to reaction–diffusion systems possessing a Lyapunov-like function and satisfying an intermediate sum condition. This significantly generalizes the mass dissipation condition in the literature and thus allows the nonlinearities to have arbitrary polynomial growth. We show that two dimensional reaction–diffusion systems, with quadratic intermediate sum conditions, have global solutions which are bounded uniformly in time. In higher dimensions, bounded solutions are obtained under the condition that the diffusion coefficients are quasi-uniform, i.e. they are close to each other. Applications include boundedness of solutions to chemical reaction networks with diffusion.

Random attractors for stochastic semilinear degenerate parabolic equations with delay

In this paper, we consider a stochastic semilinear degenerate parabolic equation with delay in a bounded domain in RN and the nonlinearity satisfying an arbitrary polynomial growth condition. The random dynamical system generated by the equation is shown to have a random attractor in C([−τ,0],Lp(O)∩D01(O,σ)), which is a compact and invariant tempered set and attracts every tempered random subset of C([−τ,0],L2(O)) in the topology of C([−τ,0],Lp(O)). In a particular case, the random attractor consists of singleton sets (i.e., a random fixed point), which generates an exponentially stable non-trivial stationary solution. This theoretical result improves some recent ones for stochastic semilinear degenerate parabolic equations.

Asymptotic behavior for the semilinear reaction\u2013diffusion equations with memory

In this paper, we consider the dynamics of a reaction–diffusion equation with fading memory and nonlinearity satisfying arbitrary polynomial growth condition. Firstly, we prove a criterion in a general setting as an alternative method (or technique) to the existence of the bi-spaces attractors for the nonlinear evolutionary equations (see Theorem 2.14). Secondly, we prove the asymptotic compactness of the semigroup on L^{2}(varOmega )times L_{mu }^{2}(mathbb{R}; H_{0}^{1}( varOmega )) by using the contractive function, and the global attractor is confirmed. Finally, the bi-spaces global attractor is obtained by verifying the asymptotic compactness of the semigroup on L^{p}( varOmega )times L_{mu }^{2}(mathbb{R}; H_{0}^{1}(varOmega )) with initial data in L^{2}(varOmega )times L_{mu }^{2}(mathbb{R}; H_{0} ^{1}(varOmega )).

L^p-estimates and regularity for SPDEs with monotone semilinearity

Semilinear stochastic partial differential equations on bounded domains {mathscr {D}} are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are L^p-estimates for such equations. The L^p-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space H^2({mathscr {D}}') and ell ^2-integrable with values in H^3({mathscr {D}}'), for any compact {mathscr {D}}' subset {mathscr {D}}. Using results from L^p-theory of SPDEs obtained by Kim (Stoch Proc Appl 112:261–283, 2004) we get analogous results in weighted Sobolev spaces on the whole {mathscr {D}}. Finally it is shown that the solution is Hölder continuous in time of order frac{1}{2} - frac{2}{q} as a process with values in a weighted L^q-space, where q arises from the integrability assumptions imposed on the initial condition and forcing terms.

Multiple solutions for a class of singular quasilinear problems

In this paper we use sub-supersolution and minimax methods to show the existence and multiplicity of solutions for the following class of singular quasilinear problems:{−Δu−Δ(u2)u=a(x)u−β+h(x,u)inΩ,u>0inΩ,u=0on∂Ω, where Ω is a bounded smooth domain of RN(N≥3), the function a(x) is nonnegative, β>0 is a constant and the nonlinearity h(x,u) is continuous. In our first result, the nonlinearity h has an arbitrary polynomial growth and we obtain the existence of a solution for the problem via sub-supersolution method. For the second result, h has subcritical growth and we show the existence of a second solution by applying the Mountain Pass Theorem.

Three solutions for parametric problems with nonhomogeneous (a,2)-type differential operators and reaction terms sublinear at zero

We consider parametric Dirichlet problems driven by the sum of a Laplacian and a nonhomogeneous differential operator ((a,2)-type equation) and with a reaction term which exhibits arbitrary polynomial growth and a nonlinear dependence on the parameter. We prove the existence of three distinct nontrivial smooth solutions for small values of the parameter, providing sign information for them: one is positive, one is negative and the third one is nodal.

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR STRONGLY DAMPED WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS AND BALANCED POTENTIALS

We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. The associated linear operator is$(-\unicode[STIX]{x1D6E5}_{W})^{\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{t}u$, where$\unicode[STIX]{x1D703}\in [\frac{1}{2},1)$and$\unicode[STIX]{x1D6E5}_{W}$is the Wentzell–Laplacian. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth of each potential, as well as mixed dissipative/antidissipative behaviour.

Strong global attractors for nonclassical diffusion equation with fading memory

Based on semigroup theory and the method of contractive function, we prove the existence of global attractors for a nonclassical diffusion equation with fading memory in the space H^{2}(Omega)cap H_{0}^{1}(Omega) times L_{mu}^{2}(mathbb{R}^{+};H^{2}(Omega)cap H_{0}^{1}(Omega)), when the nonlinearity satisfies arbitrary polynomial growth.

Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

In this paper, using a new method (or technique) called “Asymptotic Contractive Semigroup Method” (see Theorem 2.3) we prove the existence of global attractor for the nonclassical diffusion equations in H1(Rn). The nonlinearity satisfies the arbitrary order polynomial growth conditions and the forcing term g only belongs to H−1(Rn). The results obtained essentially improve and complement in some previously published papers.

Stabilization of a class of semilinear degenerate parabolic equations by Ito noise

AbstractWe investigate the effect of Ito noise on the stability of stationary solutions to a class of semilinear degenerate parabolic equations with the nonlinearity satisfying an arbitrary polynomial growth condition. We will show that an Ito noise of sufficient intensity will stabilize the unstable stationary solution.

Bi-space Global Attractors for a Class of Nonclassical Parabolic Equations with Arbitrary Polynomial Growth in Unbounded Domain

In this article, we consider the dynamical behavior of the nonclassical diffusion equation in unbounded domain while the nonlinearity satisfy the arbitrary order polynomial growth conditions. Using the tail-estimated method and the asymptotic a priori estimate method, we obtain the existence of \({(H^1(\Omega)\cap L^{p}(\Omega),L^{2}(\Omega))}\)-global attractor, \({(H^1(\Omega)\cap L^{p}(\Omega),L^{p}(\Omega))}\)-global attractor, \({(H^1(\Omega)\cap L^{p}(\Omega),H^{1}(\Omega))}\)-global attractor and \({(H^1(\Omega)\cap L^{p}(\Omega),H^1(\Omega)\cap L^{p}(\Omega))}\)-global attractor.

Pullback Attractors for a Non-Autonomous Semilinear Degenerate Parabolic Equation on ℝ N

In this paper, we prove the existence of pullback attractors for the following non-autonomous semilinear degenerate parabolic equation on ℝN: $$\frac{\partial u}{\partial t} - \textup{div} (\sigma (x)\nabla u) + \lambda u+ f(x,u) = g(x,t), $$ under a new condition concerning a variable non-negative diffusivity σ(⋅), an arbitrary polynomial growth order of the nonlinearity f and an exponent growth of the non-autonomous external force g.