Existence and non-existence of global solutions to a generalized modification of the improved Boussinesq equation

In this paper, the existence, both locally and globally in time, the uniqueness of solutions and the non-existence of global solutions to the initial boundary value problem of a generalized Modification of the Improved Boussinesq equation u_tt−u_xx−u_xxtt=σ(u)_xx are studied and a few examples are discussed. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Über das homogene Dirichlet-Problem bei nichtlinearen partiellen Differentialgleichungen vom Typ der Boussinesq-Gleichung

Semilinear equations of Boussinesq type, e.g. u_tt + u_xx ? u_xxxx + (u²)_xx = 0, u_tt + u_xx ? u_xxxx + u_xu_xx = 0, or certain equations containing the squared wave operator, e.g. u_xxtt ? u^k = 0, k ? N k ≥ 2, are studied. A generalized boundary value problem on bounded domains can be treated using Hilbert space methods. The linear parts of these equations are not elliptic, the latter not even hypoelliptic. A mountain pass lemma is used to prove the existence of nontrivial weak solutions. These solutions are obtained in anisotropic Sobolev spaces.     

On a Universality of the Heat Equation

We prove that there are solutions u(t,x) of the heat equation u_t = u_xx such that every continuous function f : [a, b] → ? can be uniformly approximated by a subsequence of u (n, ·), n? ?.     

Nonexistence for a boundary value problem arising in parabolic theory

The boundary value problemc _t=c _xx−c _yy+q(t,x)c with {fx349-1} was solved by Colton [1] forq analytic int. The solution may be used for mapping solutions of the heat equation into solutions ofu _t=u _xx+q(t,x)u. Solutions (of the boundary value problem) no longer exist ifq is not analytic int. Erica and Ludwig Jesselson Professor of Theoretical Mathematics, The Weizmann Institute of Science. This research was partially supported by the Minerva Foundation.     

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), u_tt−u_xx−a(uⁿ)_xx+b(u^m)_xxxx=0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.     

Properties of solutions to the initial-boundary value problem for a porous media-type equation

The initial-boundary value problem in a semi-infinite strip (0, ∞)×(0, T) for a degenerate parabolic equation of the form u, _t= φ(u)_xx + b(x)φ(u)_x is considered. The properties of solutions in the case where the initial function is compactly supported and for constant initial and boundary data are investigated.     

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.     

Families of Periodic Solutions of Resonant PDEs

Summary. We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation u _tt -u _xx ± u ³ =0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the n ^th family converge to 2π/n when the amplitude tends to zero. Received August 8, 2000; accepted November 21, 2000 Online publication February 26, 2001     

Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ℝ², where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz

A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation u_u - u_xx + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u→∞ as |u|→∞. The proof uses a modified form (PS)_c of the Palais-Smale condition (PS).     

Examples of the absence of a traveling wave for the generalized Korteweg-de Vries-Burgers equation

An example of convex function f(u) for which the generalized Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 has no solutions in the form of a traveling wave with specified limits at infinity is constructed. This example demonstrates the difficulties in analyzing asymptotic behavior of the Cauchy problem for the Korteweg-de Vries-Burgers equation that is not inherent in the type of equation for the conservation law, the Burgers-type equation, and its finite difference analog.     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

Stability of a traveling-wave solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation

The asymptotic behavior of the solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 as t → ∞ is analyzed. Sufficient conditions for the existence and local stability of a traveling-wave solution known in the case of f(u) = u ² are extended to the case of an arbitrary sufficiently smooth convex function f(u).     

On a nonhomogeneous Burgers’ equation

In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u _t +uu _x =μu _xx −kx,x ∈R,t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any L^p(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2)ϕ _t −ϕ _xx = −x ² ϕ. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf’s paper. Especially, we observe that the nonhomogeneous Burgers’ equation (1) is nonlinearly unstable.     

Quenching,nonquenching, and beyond quenching for solution of some parabolic equations

Summary In this paper we examine the first initial boundary value problem for u_t=u_xx + (1 – u)^–, > 0, > 0,on (0, 1) × (0, ) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem u_t=u_xx, 0 <x < 1,t > 0;u(0,t)=0, (1 – u(1, t))^–. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.This research was sponsored by the U.S. Air Force Office of Scientific Research, Air Forse Systems Command Grants 84-0252 and 88-0031. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation therein.     

Oblique derivative problems for second order nonlinear equations of mixed type in multiply connected domains

In this paper, the unique solvability of oblique derivative boundary value problems for second order nonlinear equations of mixed (elliptic-hyperbolic) type in multiply connected domains is proved, which mainly is based on the representation of solutions for the above boundary value problem, and the uniqueness and existence of solutions of the above problem for the equation u_xx + sgn y u_yy = 0.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. I

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u _tt−a ² u _xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998.     

Positive Steady-state Solutions of a Non-linear Reaction–Diffusion System

The steady-state problem of the non-linear reaction-diffusion system u_t−u_xx=u(av−b) , v_t−v_xx=cu is considered. The existence of positive steady-solutions is established by using a fixed point theorem in ordered Banach space. The uniqueness of ordered positive steady-state solutions and an application to the associated reaction-diffusion system are presented. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.