Equations In Unbounded Domains Research Articles

Existence of time-dependent attractor of wave equation in locally uniform space

In this article, we study non-autonomous dynamical behavior of weakly damped wave equation in unbounded domain. First of all, we introduce the time-dependent locally uniform space. After that, the pullback asymptotical compactness is proved by applying the contractive function method. Eventually, we obtain the existence of ( H lu 1 ( R 3 ) × L lu 2 ( R 3 ) , H ρ 1 ( R 3 ) × L ρ 1 ( R 3 ) ) -time-dependent attractor of wave equation.

On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains

On Solutions of the Navier Problem for a Polyharmonic Equation in Unbounded Domains

Unconditionally energy-stable mapped Gegenbauer spectral Galerkin method for diffusive–viscous wave equation in unbounded domains

The diffusive–viscous wave equation plays a significant role in investigating the attenuation of seismic wave propagating in fluid-saturated medium. This paper focuses on the design of unconditionally energy-stable spectral method for the diffusive–viscous wave equation defined in unbounded domains Rd(d=2,3). We develop such a spectral method by using mapped Gegenbauer functions for the spatial approximation and Crank–Nicolson scheme for the temporal discretization. Then we show that the fully-discrete method satisfies the discrete energy-dissipation law without any restriction on the time step size. To achieve an efficient implementation, the matrix diagonalization procedure is employed to solve the linear systems for d=2,3. Finally, we carry out numerical examples in 3D case to demonstrate the good behavior of our method.

$\mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains

$\mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains

On uniqueness of mild $$L^{3,\infty }$$-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

AbstractThis paper is concerned with the uniqueness of bounded continuous $$L^{3,\infty }$$ L 3 , ∞ -solutions on the whole time axis $$\mathbb R$$ R or the half-line $$(-\infty ,T)$$ ( - ∞ , T ) to the Navier–Stokes equations in 3-dimensional unbounded domains. When $$\Omega $$ Ω is an unbounded domain, it is known that a small solution in $$BC(\mathbb R;L^{3,\infty })$$ B C ( R ; L 3 , ∞ ) is unique within the class of solutions which have sufficiently small $$L^{\infty }(\mathbb R; L^{3,\infty })$$ L ∞ ( R ; L 3 , ∞ ) -norm; i.e., if two solutions u and v exist for the same force f, bothu and v are small, then the two solutions coincide. There is another type of uniqueness theorem. Farwig et al. (Commun Partial Differ Equ 40:1884–1904, 2015) showed that if two solutions u and v exist for the same force f, u is small and if v has a precompact range $${\mathscr {R}}(v):=\{v(t); -\infty<t<T\}$$ R ( v ) : = { v ( t ) ; - ∞ < t < T } in $$L^{3,\infty },$$ L 3 , ∞ , then the two solutions coincide. However, there exist many solutions which do not have precompact range. In this paper, instead of the precompact range condition, by assuming some decay property of v(x, t) with respect to the spatial variable x near $$t=-\infty ,$$ t = - ∞ , we show a modified version of the above-mentioned uniqueness theorem. As a by-product, in the half-space$$\mathbb R^3_+,$$ R + 3 , we obtain a non-existence result of backward self-similar $$L^{3,\infty }$$ L 3 , ∞ -solutions sufficiently close to some homogeneous function Q(x/|x|)/|x| in a certain sense.

Averaged controllability of heat equation in unbounded domains with random geometry and location of controls: The Green’s function approach

Averaged controllability of heat equation in unbounded domains with random geometry and location of controls: The Green’s function approach

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

Exponential attractors for modified Swift-Hohenberg equation in $\mathbb R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $\mathbb R^{N}$ with a mildly integrable potential is considered. Existence of exponential attractors containing a finite-dimensional global attractor in $H^2(\mathbb R^{N})$ is shown under the dissipative mechanism of fourth order parabolic equations in unbounded domains.

Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity

Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|\to +\infty$ (for example, growth restriction of solution as $|x|\to +\infty$, or the solution to belong to some functional spaces).Note, that we need some restrictions on the data-in behavior as$|x|\to +\infty$ for the initial-boundary value problemsfor equations considered above to be solvable.
 However, there are nonlinear parabolic equations for whichthe corresponding initial-boundary value problems are uniquely solvable withoutany conditions at infinity.
 We prove the unique solvability of the initial-boundary value problemwithout conditions at infinity for some of the higher-orders anisotropic parabolic equationswith variable exponents of the nonlinearity. A priori estimate of the weak solutionsof this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.

Analysis and Hermite Spectral Approximation of Diffusive-Viscous Wave Equations in Unbounded Domains Arising in Geophysics

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.

On the Oscillation Properties of Elliptic Equations in Unbounded Domains

On the Oscillation Properties of Elliptic Equations in Unbounded Domains

Propagation of KPP equations with advection in one-dimensional almost periodic media and its symmetry

Reaction-diffusion equations in unbounded domain are used to study the propagation phenomena of biological species. When propagation can happen in different directions, an interesting question arises: In which direction is propagation the fastest?For the one-dimensional KPP equation in almost periodic media with advection:(⁎){ut=(a(x)ux)x+b(x)ux+f(x,u)t>0,x∈R,u(0,x)=u0(x)∈[0,1] is nonzero with compact support, let ω+ and ω− be the spreading speeds of (⁎) in the positive and negative directions respectively. The above question becomes this: Which is larger, ω+ or ω−? In this paper, after establishing the existence of ω+ and ω−, we give a complete answer to this question: sgn(ω−−ω+)=sgn(limx→∞⁡1x∫0xb(s)a(s)ds).

Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

In this paper we prove the monotonicity of positive solutions to \( -\Delta _p u = f(u) \) in half-spaces under zero Dirichlet boundary conditions, for \((2N+2)/(N+2)< p < 2\) and for a general class of regular changing-sign nonlinearities f. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity

We deal with the initial-boundary value problems with some restrictions at infinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions.

On the Global Well-Posedness and Decay of a Free Boundary Problem of the Navier–Stokes Equation in Unbounded Domains

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in Lp in time and Lq in space framework in a uniformly H∞2 domain Ω⊂RN for N≥4. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the Lq-Lr estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction N≥4 is required to deduce an estimate for the nonlinear term G(u) arising from divv=0. However, we establish the results in the half space R+N for N≥3 by reducing the linearized problem to the problem with G=0, where G is the right member corresponding to G(u).

Global attractor for a class of inhomogeneous reaction–diffusion equation in unbounded domain

We consider the existence of the global attractor for the following inhomogeneous reaction–diffusion equation ut−Δu−V(x)u+|u|p−2u=g,inRn×R+,u(0)=u0∈L2(Rn)∩Lq(Rn),where n≥3, p>2 and the function V(x) satisfies some suitable assumptions. Since the primary operator −Δ−V is not positive definite in H1(Rn), so we cannot obtain that the corresponding semigroup has a bounded absorbing set in L2(Rn) by the Gronwall inequality. Thus, through the cutting function technique and the method of iteration, we prove that for any q>p, the equation has a bi-spaces global attractor A, which attracts any bounded subset B∈L2(Rn)∩Lq(Rn) in the topology of Lq(Rn).

INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY

Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|\to +\infty$ (for example, solution's growth restriction as $|x|\to +\infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|\to +\infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.

Dissipative mechanism and global attractor for modified Swift-Hohenberg equation in $R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $R^{N}$ with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in $H^2(R^{N})$ in particular.

The Hopf Bifurcation Theorem in Hilbert Spaces for Abstract Semilinear Equations

We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of \(\mathbb {R}^n\).