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We consider the Cauchy problem for an evolution equation used to model bidirectional surface waves in convecting fluids. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial-value problem associated with this equation in ℝn. We also obtain some polynomial decay estimates for the energy.

This paper focuses on the following scalar field equation $$ \begin{cases} -\Delta u=f(u)&\mbox{in }\Omega,\\ u=0&\mbox{on }\partial\Omega, \end{cases} $$ where $$\Omega=\mathbb{R}^k\times D$$ , $$k\geq2$$ and $$D$$ is a bounded domain in $$\mathbb{R}^\ell$$ with $$\ell\geq1$$ . By using the variational method, we extend the results of Byeon and Tanaka in [6] and prove, in the zero mass case, the existence of a positive axially symmetric solution under suitable growth conditions on $$f$$ .

The well-posedness of the initial-value problem associated with the dissipative Kadomtsev–Petviashvili equation in the case of two-dimensional space is studied. It is proved by using a dyadic partition of unity in Fourier variables that the Cauchy problem associated with this equation is globally well posed in the anisotropic Sobolev space Hs0(ℝ2) for all s>−1/2. It is also shown that this result is sharp in a certain sense.

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and [Formula: see text]-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.

In this paper, we study the behavior of the solitary waves of the rotation-modified Kadomtsev-Petviashvili equation. By knowing the symmetry of solitary waves in the transverse direction, we improve the previous results and show that these solutions are also symmetric with respect to the direction of propagation.

In this paper, we consider a class of nonlinear nonlocal problem involving an anisortropic operator and an external potential. We show the existence of positive and sign-changing solutions of this problem via the variational methods.

In this paper we study the global well-posedness and the large-time behavior of solutions to the initial-value problem for the dissipative Ostrovsky equation. We show that the associated solutions decay faster than the solutions of the dissipative KdV equation.

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation ( u t − β u x x x − f ( u ) x ) x = γ u , where the nonlinearity f ( u ) satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.

This paper considers the one-dimensional Schrödinger equation with nonlocal nonlinearity that describes the interactions of nonlinear dispersive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the conditions under which the solutions are uniformly bounded in the energy space.

Studied here is the Ostrovsky, Stepanyams and Tsimring (OST) equation u t + u x x x − e H ( u + u x x ) x + u u x = 0 . It is showed that the associated initial value problem is locally analytically well-posed in the critical Sobolev spaces H − 3 ∕ 2 ( R ) . This improves the previous results on this equation in the Sobolev spaces.

Studied here is a coupled KdV system with a weak rotation. The existence of ground state solutions is proved, and the continuous dependence on the parameter and asymptotic behavior of ground state solutions are established. The orbital stability of ground states is also investigated.

In this article, we establish the existence of a sign-changing solution and two sign-constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.

In this article we consider the following integral equation associated to the BO–ZK operator in the half plane. By combining the lifting regularity and the moving planes method for integral forms, we demonstrate that there is no positive solution for this integral equation.

In this paper, we consider the modified Ostrovsky, Stepanyams and Tsimring equation u t + u x x x − η ( H u x + H u x x x ) + u 2 u x = 0 . We prove that the associated initial value problem is locally well-posed in Sobolev spaces H s ( R ) for s > − 1 ∕ 2 . We also prove that our result is sharp in the sense that the flow map of this equation fails to be C 3 in H s ( R ) for s − 1 ∕ 2 . Moreover, we prove that for any s > 1 ∕ 2 and T > 0 , its solution converges in C ( [ 0 , T ] ; H s ( R ) ) to that of the mKdV equation if η tends to 0.

The purpose of this article is twofold. First, we study the existence and asymptotic behavior of ground states of a fractional Schrodinger system with quadratic interaction. Second, we give some conditions, in terms of the mass and energy of the ground states, under which the solutions of the associated initial value problem have the uniform bound or may blow up in finite time. As a corollary, we show the strong instability of the ground states.

We consider the existence and stability of traveling waves of the Fifth-Order KdV equation for a general class of nonlinearities that satisfy power-like scaling relations. This class of nonlinearities includes sums and differences of powers. For such nonlinearities we use variational methods to show that there exist ground state traveling wave solutions and use the variational properties of the ground states to analyze their stability.

In this article, we establish the existence of a least-energy nodal solution and a positive ground state for nonlinear nonlocal problem involving the BO-ZK operator. By using constrained minimization on the Nehari-type manifolds, we prove the generalized BO-ZK equation has a least-energy nodal solution with its energy exceeding twice the least energy, and a positive ground state which both solutions are block-radial.

Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced.

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions.

In this paper we establish the best constant of an anisotropic Gagliardo–Nirenberg-type inequality related to the Benjamin–Ono–Zakharov–Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.

This paper focuses on the scalar field equation Dx2αu−νu−uyy=f(u), where α ∈ (0, 1), (x,y)∈RN×(−L,L)⊂RN+1, with N ≥ 1 and Dx2α stands for the fractional Laplacian. By using several variational methods, we establish the existence, long behavior, and multiplicity of solutions of this equation under the Dirichlet and Neumann boundary conditions.

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lesbesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.

Studied here is anisotropic Gagliardo–Nirenberg inequality in n-dimensional case. A fractional version of this inequality will be proved. Its sharp constant is also elicited in terms of the ground states of the associated nonlocal equation.

Considered here is the Schrodinger-improved Boussinesq system. First we prove local and global well-posedness in the energy space for the periodic initial-value problem. The proof combines a Strichartz-type estimate with the contraction mapping principle. Second we establish the existence and orbital stability of periodic and solitary traveling-wave solutions. The stability results are set out in the context of abstract Hamiltonian systems.

Background Teledermatopathology has the potential to link underserved areas to experts across the country and assist in making quick diagnoses, which may improve health care costs and delivery. Despite these potential benefits, teledermatopathology is not used routinely for primary diagnosis in the United States. Objective To assess the current status of and address the potential for improving health care by the use of teledermatolopathology for primary diagnosis. Methods Current available literature and online resources were reviewed to address 3 major variables that hinder the widespread use of teledermatopathology: diagnostic accuracy, licensure requirements, and reimbursement. Results Recent studies show similar diagnostic accuracy for this technology compared to conventional microscopy. State-to-state variation and ambiguity in laws serve as the biggest hurdles to the widespread use of teledermatopathology. More states are recognizing the importance of the implementation of specific laws regarding telemedicine. More studies are required to evaluate the systems that offer specific telemedicine licenses, in addition to those that pay for telemedicine services specifically. Limitations This study reviewed current legislation concerning teledermatopathology; these laws are subject to revision. Conclusion Improving diagnostic accuracy and limiting variations in policy and reimbursement may encourage more pathologists to use teledermatopathology technology.

We show that the Cauchy problem for the sixth-order Boussinesq equation with cubic nonlinearity is globally well-posed in H s ( R ) provided 3 / 2 s 2 .

In this work we study the generalized Boussinesq equation with a dissipation term. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive sufficient conditions for the blow-up of the solution to the problem. Furthermore, the instability of the stationary solutions of this equation is established.

In this paper, we study the ADMB–KdV equation. We show that the associated initial value problem is locally well posed in Sobolev spaces \(H^{s}(\mathbb{R}^{2})\) for s>−1/2. We also prove that our result is sharp in some sense.

Nut consumption has been found to decrease risk of coronary heart disease and diabetes and to promote healthy body weights possibly related to their favorable macronutrient profile. We therefore assessed the effect of pistachios on postprandial glucose and insulin levels, gut hormones related to satiety and endothelial function. In this randomized crossover study, 20 subjects with metabolic syndrome consumed five study meals over 5–10 weeks. The meals differed in fat type and quantity, but were matched according to available carbohydrates (CHOs). Three meals had 50 g available CHO: white bread (WB50g), white bread, butter and cheese (WB+B+Ch) and white bread and pistachios (WB+P). Two meals had 12 g available CHO: white bread (WB12g) and pistachios (P). Within each group of available CHO meals, postprandial glucose levels were the highest following the white bread-only meals, and glucose response was significantly attenuated when butter and cheese or pistachios were consumed (P

We consider the initial value problem associated to the nonlinear periodic sixth-order Boussinesq equation. For given initial data in Sobolev spaces H s ( T ) , the local well-posedness of this equation is established for s > − 1 / 2 .

In this paper we study the solitary waves of the Rosenau-RLW equation. By using some trigonometric function methods, a family of stable solitary wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed, for what should be an equation relevant to modeling in a number of fields.

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example.

This paper studies the ADM-BKdV equation. The associated initial value problem will be proved to be locally well-posed in anisotropic Sobolev spaces \(H^{s_1,s_2}\left( \mathbb R ^2\right) \) for \(s_1>-3/2,\,s_2>-1/2\) and \(2s_1+4s_2>-1\).