σ-Moderate solutions of Lu = uα and fine trace on the boundary

We investigate the class of all positive solutions of the equation Lu = u^α, 1 < α ≤ 2, in a smooth domain E ⊂ℝ^d. We define the fine trace tr(u) of a solution u as a pair (Γ, v), where Γ is a set of singular boundary points of u, and v is a certain σ-finite measure on the complement of Γ. We describe all possible traces and we construct the minimal solution with the given trace.     

Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.     

Gradient estimates for ut=ΔF(u) on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations:$u_t=\mathrm{\Delta }F(u),$ with $F^{\prime }(u)>0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME):$u_t=\mathrm{\Delta }\left(u^p\right),p>0,$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau?s celebrated Liouville theorem for positive harmonic functions.     

Maple packages for computing Hirota’s bilinear equation and multisoliton solutions of nonlinear evolution equations

The Hirota method for generating Hirota’s bilinear equation and constructing soliton solutions of nonlinear evolution equations is discussed and illustrated. Two Maple programs Bilinearization and Multisoliton are presented to automatically calculate Hirota’s bilinear equations for nonlinear evolution equations and to compute their N-soliton solutions for N = 1, 2 or 3, respectively. Different kinds of examples are used to demonstrate the effectiveness of the packages.     

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.     

Riemann–Hilbert approach and $$N$$ -soliton solutions of the generalized mixed nonlinear Schrödinger equation with nonzero boundary conditions

Theoretical and Mathematical Physics - We apply the inverse scattering transformation to the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition at infinity. The...     

Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs

Archiv der Mathematik - Let $$G=(V, E)$$ be a locally finite connected graph and $$\Delta $$ be the usual graph Laplacian operator. According to Lin and Yang (Rev. Mat. Complut., 2022), using...     

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.     

Correction to «Holder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions»

Summary In the proof of [1, Theorem 1.2] an incorrect assertion was made. Using the notation of that work, we claimed that [D(u)–v] is finite, which may not be true. We give here a correct, simpler proof of the theorem.     

Viscosity solutions of the first boundary value problem for the second order fully nonlinear parabolic equation under natural structure conditions ∗

We study the first boundary value problem for the second-order fully nonlinear parabolic equation under natural structure conditions. The 1 , Q solution has a priori W^1,0 _infinite bound. And moreover we prove the esistence of viscosity solution by using the accretive operator. This is the extension of the method used in [ I ] . Our method has the advantage that the existence of solution does not depend on the esistence of super- and subsolutions such as perron's method. Finally the uniqueness of viscosity solution is proved by using the method developed in [2] and [3].     

Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with the Cahn–Hilliard equation subject to the boundary conditions and the initial condition ψ(0,x) = ψ₀(x) where J = (0,∞), and Ω ⊂ ℝ ⁿ is a bounded domain with smooth boundary Γ = ∂ G, n≤ 3, and Γ _s ,σ _s ,g _s > 0, h are constants. This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal L _p -regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor. Mathematics Subject Classification (2000) 82C26, 35B40, 35B65, 35Q99     

Asymptotics and existence of bounded solutions of a nonlinear Schrödinger equation on the half-line

We study the asymptotics and existence of nonzero bounded solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear second-order ordinary differential equation. We prove the existence of countably many nonzero bounded solutions on the half-line and derive asymptotic formulas at infinity for these solutions.     

Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞

Here, all solutions of the form u=r^kf() to the p-harmonic equation, div(|u|^p–2u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=r^kf() of the limit equation for p=, u _x ² u_xx+2u_xu_yu_xy+u _y ² u_yy=2, are constructed, some of which have a strong singularity at the origin (Theorem 2).     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

The behavior of bounded solutions to the equation Δu−c(x)u=0 on riemannian manifolds of special type

In the paper we consider solutions of the equation Δu−c(x)u=0,c(x)≥0, on complete Riemannian manifolds constituted as follows: the exterior of some compact set is isometric to the direct product of the semiaxis by some compact manifold with the metricds ²=h ²(r)dr ²+g ²(r)dθ². Necessary and sufficient conditions under which bounded solutions of the equation have a limit independent of θ asr→∞ are obtained and also conditions under which the two-sided Liouville theorem is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 215–221, February, 1999.     

Interior and boundary continuity of the solution of the singular equation (β(u))t=Lu

We extend some results of DiBenedetto and Vespri (Arch. Rational Mech. Anal 132(3) (1995) 247) proving the interior and boundary continuity of bounded solutions of the singular equation $\text{(β(u))}{}_{\mathrm{t}}\text{=}\text{L}\text{u}$ where $\text{L}$ is a second order elliptic operator with bounded measurable coefficients that depend both on space and time in a proper way.     

Existence and uniqueness of positive solutions to?m-point boundary value problem for?nonlinear fractional differential equation

In this paper, we consider the following nonlinear fractional m-point boundary value problem where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. By the properties of the Green function, the lower and upper solution method and fixed-point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As applications, examples are presented to illustrate the main results.     

Large solutions to the Monge–Ampère equations with nonlinear gradient terms: Existence and boundary behavior

In this paper, we obtain conditions about the existence and boundary behavior of (strictly) convex solutions to the Monge–Ampère equations with boundary blow-up

$\det ?D^{2}u(x)=b(x)f(u(x))\pm |\mathrm{?}u{|}^q,x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

and

$\det ?D^{2}u(x)=b(x)f(u(x))(1+|\mathrm{?}u{|}^q),x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

where Ω is a strictly convex, bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$, $q\in [0,N]$ (or $q\in [0,N)$), $b\in C^{\infty }(\mathrm{\Omega })$ which is positive in Ω, but may vanish or blow up on the boundary, $f\in C[0,\infty )$, $f(0)=0$, and f is strictly increasing on $[0,\infty )$ (or $f\in C(\mathbb{R})$, $f(s)>0,?s\in \mathbb{R}$, and f is strictly increasing on $\mathbb{R}$).