Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.     

Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δ_ut$-\mathrm{\Delta }{u}_t$ and the linear dissipative term _ut$u_t$ by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.     

Global attractor for a nonlinear wave equation arising in elastic waveguide model

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model _utt−Δu−Δ_utt+Δ²u−Δ_ut−Δg(u)=f(x)$u_{tt}-\Delta u-\Delta u_{tt}+{\Delta }^{2}u-\Delta u_t-\Delta g\left(u\right)=f\left(x\right)$. It proves that when the space dimension N≤5$N\le 5$, under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension.     

Boundedness of global solutions of a porous medium equation with a localized source

In this paper a localized porous medium equation _ut=u^r(Δu+af(u(_x0,t)))$u_t=u^r(\mathrm{\Delta }u+\mathit{af}(u(x_0,t)))$ is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices

In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L^∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case. Received: November 18, 2003     

Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms

We consider the Cauchy problem for systems of semilinear hyperbolic equations. Using the L_p→L_q type estimation for the corresponding linear parts, the existence and uniqueness of weak global solutions are investigated. We also established the behavior of solutions and their derivatives as t→+∞. Using the method of test functions developed in the works (Mitidieri and Pokhozhaev, 2001 [11], Veron and Pohozaev, 2001 [12] and Caristi, 2000 [23]) we obtain the analogue of the Fujita-Hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping. It follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense.     

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions u(t,x)?0 to the fast diffusion equation, t_∂u=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m?mc=(d−2)/d. The boundedness statements are true even for m?0, while the positivity ones cannot be true in that range.     

Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d?5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)_‖2<‖ΔW_‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.     

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions, provided that the C¹ norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

欧拉 - 玻尔兹曼方程整体光滑解

本文证明了R^d 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 H^s(R^d)x L²(R₊ x S^d-1;H^s(R^d))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.     

Phase transitions with semi-diffuse interfaces

In this paper, we examine new “phase-field” models with semi-diffuse interfaces. These models have the property that the −1/+1 planar phase transitions take place over a finite interval. The models also support multiple interface solutions with interfaces centered at arbitrary points L₁<L₂<?<L_N. These solutions correspond to local minima of an entropy functional (see (3.3) and (3.7)) rather than saddle points and are dynamically stable. The classical models have no such exact solutions but they do support solutions with N equally spaced transition points where the order parameter transitions between values p_min(N) and p_max(N) satisfying −1<p_min(N)<0<p_max(N)<1. These solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

Kirchhoff systems with dynamic boundary conditions

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p>p_n with p_n<2. The systems involve nonlinear external forces and may be affected by a perturbation of the type |u|^p−2u. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q.     

A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ₁(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ₁(Ω)), in the case Ω=²(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ₁(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.     

On the breakdown of classical solutions of the planar motion of an elastic string

We prove a result of formation of singularities for the classical solutions of the planar motion of a nonlinear elastic string. In a particular, but physically relevant, case we give a characterization of the global C¹ solutions with positive tension.     

Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index

In this paper we study a quasilinear elliptic problem with the Dirichlet boundary condition in a bounded domain involving the operator Au=−Δ_pu−Δ_qu. Assuming that the nonlinearity has a concave-convex behavior we obtain some multiplicity results. More precisely, we obtain five nontrivial solutions; two by a minimization argument, two by the mountain pass theorem, and the other by a cohomological linking theorem.     

The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors

In this paper we study the asymptotic behavior of globally smooth solutions of the Cauchy problem for the multidimensional isentropic hydrodynamic model for semiconductors in R^d. We prove that smooth solutions (close to equilibrium) of the problem converge to a stationary solution exponentially fast as t→+∞.