The existence of ground states for fourth-order wave equations

Focusing on the fourth-order wave equation u_tt+Δ²u+f(u)=0, we prove the existence of ground state solutions u=u(x+ct) for an optimal range of speeds c∈Rⁿ and a variety of nonlinearities f.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation

The nonlinear wave equation utt=(c²x(u)ux) arises in various physical applications. Ames et al. [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] did the complete group classification for its admitted point symmetries with respect to the wave speed function c(u) and as a consequence constructed explicit invariant solutions for some specific cases. By considering conservation laws for arbitrary c(u), we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations. We use existing work on such related linear systems to extend the known symmetry classification in [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] to include nonlocal symmetries. Moreover, we find sets of c(u) for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation.     

Double reduction of a nonlinear (2+1) wave equation via conservation laws

Conservation laws of a nonlinear (2+1) wave equation u_tt = (f(u)u_x)_x +  (g(u)u_y)_y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f′(u) and g′(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f′(u) and g′(u) are linearly independent.     

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.     

The slow wave solution to the FitzHugh-Nagumo equations

This paper gives a new existence proof for a travelling wave solution to the FitzHugh-Nagumo equations, u_t = u_xx +f(u)?w, w _t = ? (u?γw). The proof uses a contraction mapping argument, and also shows that the solution (u, c, w) to the travelling wave equations, where c is the wave speed, converges as ? → 0⁺ to the solution to the equations having ?=0, c=0, and w=0.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

Free and forced vibrations of nonlinear wave equations in ball

First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation (∂_t²−△)u=f(r,u) in n-dimensional ball B_a with center at the origin and radius a, where f is smooth, monotone decreasing in u, and satisfies f(r,0)=0. f(r,u) has asymptotic properties . For n=1,3 we shall show the existence of infinitely many radially symmetric time-periodic solutions with different periods of irrational multiple of a. Second, we shall deal with BVP for a forced nonlinear wave equation (∂_t²−△)u=εg(r,t,u), where g is T-periodic in t and ε is a small parameter. Under some Diophantine condition on a/T we shall show the existence of time-periodic solutions of the BVP. For 1?n?5 we shall construct infinitely many T satisfying the above Diophantine inequality, using asymptotic expansions of the zero points of the Bessel functions.     

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.     

General Sobolev Orthogonal Polynomials

In this paper, we study orthogonal polynomials with respect to the inner product (f, g)_S^(N) =〈u, fg〉+∑_m=1^N λ_m〈u, f^(m)g^(m) 〉, where λ_m≥0 form=1,…,N, anduis a semiclassical, positive definite linear functional. For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated withu.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Global conservative solutions of the generalized hyperelastic-rod wave equation

We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa-Holm equation (f(u)=u²/2 and g(u)=κu+u²) as well as the hyperelastic-rod wave equation (f(u)=γu²/2 and g(u)=(3−γ)u²/2) as special cases. It is shown that the problem is well-posed for initial data in H¹(R) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.     

On singular diffusion equations with applications to self-organized criticality

We consider solutions of the singular diffusion equation _t, = (u^m?1 u_x)_x, m ≦ 0, associated with the flux boundary condition lim_x→?∞ (u^m?1u_x)_x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L¹ for ?1 < m ≦ 0, but have non-integrable tails for m ≦ ?1. We also show that these traveling waves are the same as those for the scalar conservation law u_t = ?[f(u)]_x + u_xx for a particular nonlinear convection term f(u) = f(u;m, λ). © 1993 John Wiley & Sons, Inc.     

Uniqueness of conservative solutions for nonlinear wave equations via characteristics

For some classes of one-dimensional nonlinear wave equations, solutions are Hölder continuous and the ODEs for characteristics admit multiple solutions. Introducing an additional conservation equation and a suitable set of transformed variables, one obtains a new ODE whose right hand side is either Lipschitz continuous or has directionally bounded variation. In this way, a unique characteristic can be singled out through each initial point. This approach yields the uniqueness of conservative solutions to various equations, including the Camassa-Holm and the variational wave equation u_tt ? c(u)(c(u)u_x )_x = 0, for general initial data in H¹(R).     

The ring of an outer von Neumann frame in modular lattices

We prove the following theorem. Let (a ₁, . . . , a _m , c ₁₂, . . . , c _1m ) be a spanning von Neumann m-frame of a modular lattice L, and let (u ₁, . . . , u _n , v ₁₂, . . . , v _1n ) be a spanning von Neumann n-frame of the interval [0, a ₁]. Assume that either m ≥ 4, or L is Arguesian and m ≥ 3. Let R* denote the coordinate ring of (a ₁, . . . , a _m , c ₁₂, . . . , c _1m ). If n ≥ 2, then there is a ring S* such that R* is isomorphic to the ring of all n × n matrices over S*. If n ≥ 4 or L is Arguesian and n ≥ 3, then we can choose S* as the coordinate ring of (u ₁, . . . , u _n , v ₁₂, . . . , v _1n ).     

Critical wave speeds for a family of scalar reaction-diffusion equations

We study the set of traveling waves in a class of reaction-diffusion equations for the family of potentials f_m(U) = 2U^m(1 − U). We use perturbation methods and matched asymptotics to derive expansions for the critical wave speed that separates algebraic and exponential traveling wave front solutions for m → 2 and m → ∞. Also, an integral formulation of the problem shows that nonuniform convergence of the generalized equal area rule occurs at the critical wave speed.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

Cauchy problem for semilinear wave equation with time-dependent metrics

We establish the existence of a weak solution u of the semilinear wave equation where a(t,x) is equal to 1 outside a compact set with respect to x and a non-linear term f_k which satisfies |f_k(u)|≤C|u|^k. For some non-trapping time-periodic perturbations a(t,x), we obtain the long time existence of a solution from little initial data.