Narrow Results

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn–Hilliard (C-H) model (also Kuramoto–Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula is utilized to rigorously deduce the solution’s regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null noncontrollability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem’s solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness and asymptotics studies for nonlinear counterparts too.

In this paper, we establish some new functional inequalities of Adams’ type for radial Sobolev spaces of second order with logarithmic weights. The sharpness of these inequalities is also discussed. These inequalities (which can be singular) are new and complete the existing literature concerning this topic of research. Finally, we will end this work with a concrete application of these new inequalities to study some singular elliptic nonlinear equations involving new type of exponential growth condition at infinity.

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation $-{\mathrm{\Delta }}_{p}u+V(x)|u{|}^{p-2}u=g(x,u)$ in ${ℝ}^N$, where $1<p<+\infty$, the nonlinearity $g$ behaves as $|u{|}^{p-2}u$ at infinity and $V$ is a potential satisfying suitable assumptions so that an embedding theorem for weighted Sobolev spaces holds. Both the non-resonant and resonant cases are analyzed.

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole ${ℝ}^N$ with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev embeddings.

The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction–Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the tempered attractors when the stochastic equations are forced by periodic functions. We further prove the upper semicontinuity of these attractors when the intensity of stochastic perturbations approaches zero.

In this paper, we prove the existence of a pullback attractor for a strongly damped delay wave equation in ${ℝ}^n$. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions, so that some new methods to obtain the existence of pullback attractors for multi-valued processes on an unbounded domain are introduced.

We investigate the pathwise asymptotic behavior of the FitzHugh–Nagumo systems defined on unbounded domains driven by nonlinear colored noise. We prove the existence and uniqueness of tempered pullback random attractors of the systems with polynomial diffusion terms. The pullback asymptotic compactness of solutions is obtained by the uniform estimates on the tails of solutions outside a bounded domain. We also examine the limiting behavior of the FitzHugh–Nagumo systems driven by linear colored noise as the correlation time of the colored noise approaches zero. In this respect, we prove that the solutions and the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise.

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in $W^{2{\alpha }^{-},p}({ℝ}^2).$ In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in $W^{2{\alpha }^{-},p}({ℝ}^2).$ Here, we do not add any modifying factor on the nonlinear term.

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du=+\infty$ along with additional restrictions. For example, consider the forcing $f(u)=ulog(e^e+|u|)log(log(e^e+\left|u\right|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on ${ℝ}^n$. We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains.

Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshfree methods. In the SFEM-Q4, only the values of shape functions (not the derivatives) at the quadrature points are needed and the traditional requirement of coordinate transformation procedure is not necessary to implement the numerical integration. Consequently, no additional degrees of freedom are required as compared with the original FEM. In addition, the original “overly-stiff” FEM model for acoustic problems (governed by the Helmholtz equation) is properly softened due to the gradient smoothing operations implemented over the smoothing domains and the present SFEM-Q4 possesses a relatively appropriate stiffness of the continuous system. Therefore, the well-known numerical dispersion error for Helmholtz equation is decreased significantly and very accurate numerical solutions can be obtained by using relatively coarse meshes. In order to truncate the unbounded domains and employ the domain-based numerical method to tackle the acoustic radiation in unbounded domains, the Dirichlet-to-Neumann (DtN) map is used to ensure that there are no spurious reflections from the far field. The numerical results from several numerical examples demonstrate that the present SFEM-Q4 is quite effective to handle acoustic radiation problems and can produce more accurate numerical results than the standard FEM.

The finite element modeling of the dynamic and wave problems in unbounded media requires an artificial boundary condition to simulate the truncated infinite domain. The Dirichlet-to-Neumann boundary condition has been transformed from frequency to time domain by using the rational function approximation and auxiliary variable technique. It is extended to three-dimensional layer problem here. The resulting artificial boundary condition is stable itself in time domain, whereas the time-domain instability of the artificial boundary condition coupled with the finite element method is found for the foundation vibration recently and for the wave propagation here. A simple and effective method that introduces the damping proportional to the stiffness matrix in the finite element method is given to cure such coupling instability completely. The stabilized damping is so small that it does not affect the solution accuracy nearly. The numerical examples show the instability phenomenon and indicate the effectiveness of the damping method. The time-domain stability studies here can be a reference for the other artificial boundary conditions.

To improve the accuracy of the standard finite element (FE) solutions for acoustic radiation computation, this work presents the coupling of a radial point interpolation method (RPIM) with the standard FEM based on triangular (T3) mesh to give a coupled “FE-Meshfree” Trig3-RPIM element for two-dimensional acoustic radiation problems. In this coupled Trig3-RPIM element, the local approximation (LA) is represented by the polynomial-radial basis functions and the partition of unity (PU) concept is satisfied using the standard FEM shape functions. Incorporating the present coupled Trig3-RPIM element with the appropriate non-reflecting boundary condition, the two-dimensional acoustic radiation problems in exterior unbounded domain can be successfully solved. The numerical results demonstrate that the present coupled Trig3-RPIM have significant superiorities over the standard FEM and can be regarded as a competitive numerical techniques for exterior acoustic computation.

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic $p$-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain ${ℝ}^N$. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in $L^2({ℝ}^N)$. This attractor is further proved to be a bi-spatial $(L^2({ℝ}^N),L^r({ℝ}^N))$-attractor for any $r\in [2,\infty )$, which is compact, measurable in $L^r({ℝ}^N)$ and attracts all random subsets of $L^2({ℝ}^N)$ with respect to the norm of $L^r({ℝ}^N)$. Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in $L^r({ℝ}^N)$ for $r\in [2,\infty )$ in order to overcome the non-compactness of Sobolev embeddings on ${ℝ}^N$ and the nonlinearity of the fractional $p$-Laplace operator.

We prove the existence of bounded solutions of quasilinear degenerate parabolic equation of second order in unbounded domains and we study their asymptotic behavior near infinity.