Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

A blow up result for a fractionally damped wave equation

In this paper we prove a blow up result for solutions of the wave equation with damping of fractional order and in presence of a polynomial source. This result improves a previous result in [5]. There we showed that the classical energy is unbounded provided that the initial data are large enough.     

General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source

In this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous Dirichlet boundary condition. We prove that, for certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. This result improves earlier ones obtained by Messaoudi and Tatar [S.A. Messaoudi, N.-E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci. 30 (2007) 665-680] in which only the exponential and polynomial decay rates are considered. Conversely, for certain initial data in the unstable set, there are solutions that blow up in finite time. The last result is new, since it allows a larger class of initial energy which may take positive values.     

On the asymptotic behavior of nonlinear waves in the presence of a short-range potential

Consider the nonlinear wave equation with zero mass and a time-independent potential in three space dimensions. When it comes to the associated Cauchy problem, it is already known that short-range potentials do not affect the existence of small-amplitude solutions. In this paper, we focus on the associated scattering problem and we show that the situation is quite different there. In particular, we show that even arbitrarily small and rapidly decaying potentials may affect the asymptotic behavior of solutions.     

Multiple solutions for quasilinear elliptic equations with a finite potential well

We consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past, including the so called Modified Nonlinear Schrödinger Equations. By combining a new variational approach via q-Laplacian regularization and the compactness arguments from [4] we establish infinitely many bound state solutions for the quasilinear Schrödinger type equations, extending the earlier work of [4] for semilinear equations.     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping

In this paper we consider the so-called p-system with linear damping, and we will prove an optimal decay estimates without any smallness conditions on the initial error. More precisely, if we restrict the initial data (V₀,U₀) in the space H³(R⁺)∩L1,γ(R⁺)×H²(R⁺)∩L1,γ(R⁺), then we can derive faster decay estimates than those given in [P. Marcati, M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2) (2005) 224-240; H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations 174 (1) (2001) 200-236] and [M. Jian, C. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations 246 (1) (2009) 50-77].     

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms

We focus on the global well-posedness of the system of nonlinear wave equations

     

一个具有非线性边界条件的抛物系统整体存在性和爆破

In this paper the nonnegative classical solutions of a parabolic system with nonlinear boundary conditions are discussed. The existence and uniqueness of a nonnegative classical solution are proved. And some sufficient conditions to ensure the global existence and nonexistence of nonnegative classical solution to this problem are given.     

Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order

In this paper, we investigate the existence and uniqueness of the solution to the Cauchy problem for a class of nonlinear wave equations of higher order and prove the existence and nonexistence of global solutions to this problem by a potential well method.     

Time-periodic solutions of a nonlinear wave equation

The existence of a time-periodic solution of an n-dimensional nonlinear wave equation is established with n=2 and 3.     

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

We state a Wiener criterion for the regularity of points with respect to a relaxed Dirichlet problem for a p-homogeneous Riemannian Dirichlet form.     

Quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic Cauchy problems

In the weakly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultradifferentiable functions and functionals. We find a sharp condition on the modulus of continuity assuring the well-posedness in nonquasianalytic classes.     

An integro-differential equation for surface ocean waves with finite depth

An explicit integro-differential equation formulation is derived for surface ocean waves with finite depth. The equation involves only 2D surface variables. For this equation, we establish the stability and existence of solutions, and explain the effect of depth on surface wave properties.