Blow-up and global existence to a degenerate reaction-diffusion equation with nonlinear memory

In this paper, we consider a degenerate reaction-diffusion equation

     

Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

In this paper we consider the wave equation with nonlinear damping and source terms. We are interested in the interaction between the boundary damping −|y_t(L,t)|^m−1y_t(L,t) and the interior source |y(t)|^p−1y(t). We find a sufficient condition for obtaining the blow-up solution of the problem. Furthermore, we also obtain that the solution may blow up even if m≥p.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions

This paper is concerned with the existence of positive solutions of the singular nonlinear elliptic equation with a Dirichlet boundary condition

     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.     

Global existence for rough solutions of a fourth-order nonlinear wave equation

In this paper, we prove that the cubic fourth-order wave equation is globally well-posed in Hs(Rn) for by following the Bourgain's Fourier truncation idea in Bourgain (1998) [2]. To avoid some troubles, we technically make use of the Strichartz estimate for low frequency part and high frequency part, respectively. As far as we know, this is the first result on the low regularity behavior of the fourth-order wave equation.     

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.     

Polynomial stabiization of the wave equation with Ventcel's boundary conditions

We consider the wave equation on the unit square of the plane with Ventcel boundary conditions on a part of the boundary. It was shown by A. Heminna [8] that this problem is not exponentially stable. Here using a Fourier analysis and a careful analysis of the 1‐d problem with respect to the Fourier parameter l, we show a polynomial stability of this system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Global existence and nonexistence for some degenerate and quasilinear parabolic systems

The author discusses the degenerate and quasilinear parabolic system

     

Solvability and stability of a semilinear wave equation with nonlinear boundary conditions

We discuss solvability for the semilinear equation of the vibrating string x_tt(t,y)−Δx(t,y)+f(t,y,x(t,y))=0 in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Energy decay of solutions of a degenerate Kirchhoff equation with a weak nonlinear dissipation

In this paper we study decay properties of the solutions to the degenerate Kirchhoff equation with a weak nonlinear dissipative term.     

Removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain

The removability of singularities of solutions for the Dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied. A method based on a priori energetic estimates of solutions to elliptic boundary value problems is used. The growth in the vicinity of a boundary point (finite or at infinity) for generalized solutions is studied.     

On the convective Cahn-Hilliard equation with degenerate mobility

In this paper, we study the existence of weak solutions for the convective Cahn-Hilliard equation with degenerate mobility. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05