Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut-△u-αut=vp,vt-△v-αvt=uqwith p,q 1 and pq1,where the viscous terms of third order are included.We first find the critical Fujita exponent,and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions.Moreover,time-decay profiles are obtained for the global solutions.It can be found that,diferent from those for the situations of general semilinear heat systems,we have to use distinctive techniques to treat the influence from the viscous terms of the highest order.To fix the non-global solutions,we exploit the test function method,instead of the general Kaplan method for heat systems.To obtain the global solutions,we apply the Lp-Lq technique to establish some uniform Lmtime-decay estimates.In particular,under a suitable classification for the nonlinear parameters and the initial data,various Lmtime-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system.It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing efect to establish the compactness of approximating solutions,which cannot be directly realized here due to absence of the smooth efect in the pseudo-parabolic system.     

Estimates of solutions of impulsive parabolic equations under Neumann boundary condition

In this paper, we prove the relation v(t)?u(t,x)?w(t), where u(t,x) is the solution of an impulsive parabolic equations under Neumann boundary condition ∂u(t,x)/∂ν=0, and v(t) and w(t) are solutions of two impulsive ordinary equations. We also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics, and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically.     

A convergent two-time method for periodic differential equations

Two timing, an ad hoc method for studying periodic evolution equations, can be given a rigorous justification when the problem is in standard form, u = ?f(t, u). First solve $\text{dw}\text{dσ}\text{= ?(I ? M) f(σ, w)}\text{for}\text{w(σ, v)}$, where M is the mean value operator and v is any initial value. Then w(σ, v) is periodic in σ but does not satisfy the original equation. Now, force a solution u(t), using nonlinear variation of constants, in the form w(σ, v(τ)), where σ = t is the fast time and τ = ?t is the slow time. With the resulting differential equation for v, one reads off from its nonconstant solutions thè approximate transient behavior of u(t) for times of order ?^?1. On the other hand, the equilibrium points (constant solutions) v₀ correspond to steady state (periodic solutions) of the original system. Interesting applications, such as to one-dimensional wave equations with cubic damping, can be given.     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t_∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T_∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)_‖L²(ω)=δ, then ‖(u−v)(⋅,T)_‖L²(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm).     

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Decay estimates for dissipative wave equations in exterior domains

Consider the initial boundary value problem for the linear dissipative wave equation (□+∂_t)u=0 in an exterior domain . Using the so-called cut-off method together with local energy decay and L² decays in the whole space, we study decay estimates of the solutions. In particular, when N?3, we derive L^p decays with p?1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+∂_t)u=f(u) with f(u)=|u|^α+1,|u|^αu in an exterior domain.     

Estimation of states and parameters in continuous non-linear systems with discrete observations

We consider a class of continuous non-linear systems defined by the ordinary differential equation x = f(x, t) + g(x, t)u, where u is an unknown input representing noise or disturbances. The object is to estimate states and parameters in these systems by means of a fixed number of discrete observations y_i = h(x(t_i), t_i) + v_i, 1 ? i ? m, where the v_i represents unknown errors in the measurements y_i. No statistical assumptions are made concerning the nature of the unknown input u or the unknown measurement errors v_i. A weighted least squares criterion is defined as a measure of the optimal estimate. A result concerning the existence of solutions of the differential equation which minimize the criterion is presented. The necessary conditions for an optimal estimate, a set of Euler-Lagrange equations and multi-point discontinuous non-linear boundary conditions, are given. The multi-point problem is converted to an equivalent continuous two-point boundary value problem of larger dimension in the case in which the observations are assumed to be linear functions of the state. A pair of equivalent quasilinearization algorithms is defined for the two-point system and the multi-point system. Quadratic convergence for these algorithms is proved.     

Stable viscosity matrices for systems of conservation laws

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u₁ + f(u)_x = 0, u?R^m, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u_t + f(u)_x = v(Du_x)_x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u₁ + f′(u₀) u_x = vDu_xx should be well posed in L² uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.     

Smoothing properties for a coupled system of nonlinear evolution dispersive equations

We study the smoothness properties of solutions to the coupled system of equations of Korteweg—de Vries type. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u₀, v₀ possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t). v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution.     

On the continuation and boundedness of solutions of a nonlinear differential equation

Sufficient conditions are given so that all solutions of the nonlinear differential equation u″ + φ(t, u, u′)u′ + p(t) gf(u) g(u′) = h(t, u, u′) are continuable to the right of an initial t-value t₀ ? 0. These conditions are then extended so that all solutions u of the equation in question together with their derivative u′ are bounded for t ? t₀ .     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Error analysis of a mixed finite element approximation of the semilinear Sobolev equations

Based on a mixed finite element method, we construct semidiscrete approximations of the solution u and the flux term ?u+?u _t of the semilinear Sobolev equations. The existence and uniqueness of the semidiscrete approximations are demonstrated and the error estimates of optimal rate in L ² normed space are derived. And also we construct the fully discrete approximations of u and ?u+?u _t and analyze the convergence of optimal rate in L ² normed space.     

Positive solutions for systems of nonlinear eigenvalue problems for functional differential equations

Values of?λ?are determined for which there exist positive solutions of the system of functional differential equations, u″?+?λa(t)f(v _t )?=?0,v″?+?λb(t)g(u _t )?=?0, for 0?t?u(s)?=?v(s)?=?φ(s), ?r?≤?s?≤?0, and the boundary conditions u(0)?=?v(0)?=?φ(0)?=?u(1)?=?v(1)?=?0. A Guo–Krasnosel'skii fixed point theorem is applied.     

On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations

Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ?_t²u ? Δu + m²u + f(u) = 0 (NLKG) $\text{u¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{u¦}{}_0\text{= Ψ}$, are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ?_t²v ? Δv + m²v = 0$\text{v¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{v¦}{}_0\text{= Ψ}$, (KG) In particular, the finiteness of time-integrals in L_q over R₊ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L₂¹ × L₂ for n = 3.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

Rate of convergence in a class of singular perturbations

Let V_?, W_?, W and X be Hilbert spaces (0 < ? ? 1) with V_? ? W_? ? W ? X algebraically and topologically, each space being dense in the one that follows it. For each t? [0, T] let a_?(t; u, v), b_?(t; u, v) and b(t; u, v) be continuous sesqui-linear forms on V_?, W_? and W, respectively, which satisfy certain ellipticity conditions. Consider the two equations a_?(t; u_?, v) + b_?(t; u_?, v) = 〈f_?, v〉 (v?V_?) and (u′, v)_x + b(t; u, v) = 〈f, v〉 (v?W). Estimates are obtained on the rate of convergence of u_? to u, assuming a_?(t; u, v) → (u, v)_x and b_?(t; u, v) → b(t; u, v) in an appropriate sense. These results are then applied to singular perturbation of a class of parabolic boundary value problems.     

Obstructions to Existence in Fast-Diffusion Equations

The study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory. Thus, in the sub-field of very fast diffusion it is known that the Cauchy problem can be ill-posed, either because of non-uniqueness, or because of non-existence of solutions with small data. The equations we consider take the general form u_t=(D(u,u_x)u_x)_x or its several-dimension analogue. Fast diffusion means that D→∞ at some values of the arguments, typically as u→0 or u_x→0. Here, we describe two different types of non-existence phenomena. Some fast-diffusion equations with very singular D do not allow for solutions with sign changes, while other equations admit only monotone solutions, no oscillations being allowed. The examples we give for both types of anomaly are closely related. The most typical examples are v_t=(v_x/∣v∣)_x and u_t=u_xx/∣u_x∣. For these equations, we investigate what happens to the Cauchy problem when we take incompatible initial data and perform a standard regularization. It is shown that the limit gives rise to an initial layer where the data become admissible (positive or monotone, respectively), followed by a standard evolution for all t>0, once the obstruction has been removed.