Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

On nonlinear diffusion equations with x-dependent convection and absorption

For the 1+1-dimensional nonlinear diffusion equations with x-dependent convection and source terms u_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u), we obtain conditions under which the equations admit the second-order generalized conditional symmetries η(x,u)=u_xx+H(u)u_x²+G(x,u)u_x+F(x,u) and the first-order sign-invariants J(x,u)=u_t−A(u)u_x²−B(x,u)u_x−C(x,u) on the solutions u(x,t). Several different generalized conditional symmetries and first-order sign-invariants for equations in which the diffusion term offers different possibilities (power-law, exponential, Mullin, Fujita) are presented. Exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first-order sign-invariants are constructed.     

Global attractors for nonlinear wave equations with nonlinear dissipative terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x,ut)=a(x)r|ut|ut.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.     

Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1+1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2,R)s_⊕W, where W is the three-dimensional Heisenberg-Weyl algebra. We construct a complete symmetry group of a 1+2 evolution equation ut=(Fy(u)ux) for some functions F using the point symmetries admitted by the equation. The 1+2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377-392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1+2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

On the asymptotic behavior of the solutions of a class of second order nonlinear differential equations

In this paper, we investigate properties of the solutions of a class of second-order nonlinear differential equation such as [p(t)f(x(t))x′(t)]′ + q(t)g(x′(t))e(x(t)) = r(t)c(x(t)). We prove the theorems of monotonicity, nonoscillation and continuation of the solutions of the equation, the sufficient and necessary conditions that the solutions of the equation are bounded, and the asymptotic behavior of the solutions of the equation when t → ∞ on condition that the solutions are bounded. Also we provide the asymptotic relationship between the solutions of this equation and those of the following second-order linear differential equation: [p(t)u′(t)]′ = r(t)u(t)     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

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.     

An abstract doubly nonlinear equation with a measure as initial value

The solvability of the abstract implicit nonlinear nonautonomous differential equation (A(t)u^′(t))+B(t)u(t)+C(t)u(t)∋f(t) will be investigated in the case of a measure as an initial value. It will be shown that this problem has a solution if the inner product of A(t)x and B(t)x+C(t)x is bounded below.     

Similarity stabilizes blow up

The blow-up of solutions to the PDE ψ(x)u_t=[∇·A(x)∇+b(x)]u^m is studied via energy methods. The key step is a similarity transformation of the original unstable equation to a nonlocal stable one.     

Boundary displacement control at one end of a string in the presence of a nonlocal boundary condition of one of four types,and its optimization

In the present paper, we exhaustively solve the problem of boundary control by the displacement u(0, t) = µ(t) at the end x = 0 of the string in the presence of a model nonlocal boundary condition of one of four types relating the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point \(\mathop x\limits^ \circ\).     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

Exponential stability of wave equations with potential and indefinite damping

First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (_∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (_∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω.     

Periodic solutions of the nonlinear telegraph equations with bounded nonlinearities

In this article, using the Leray-Schauder degree theory, we discuss existence, nonexistence and multiplicity for the periodic solutions of the nonlinear telegraph equation

utt−uxx+cut+Φ(u)=f(t,x)+s,     

Scattering length and perturbations of −Δ by positive potentials

This paper extends a result of Fujita [On the blowing up of solutions to the Cauchy problem for u_t = Δu + u^{1 + a}, J. Faculty Science, U. of Tokyo 13 (1966), 109–124] to show that solutions u = u(t, x) for t > 0 and x?R² to the equation u_t = Δu + u² with u(0, x) = a(x) must grow at a rate faster than exp(∥x∥²) at some finite time t, as long as a(x) is nonnegative and not almost everywhere zero.     

A new analytical method for the linearization of dynamic equation on measure chains

In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system xΔ=A(t)x and the nonlinear system xΔ=A(t)x+f(t,x). Some sufficient conditions are obtained to guarantee the existence of a equivalent function H(t,x) sending the (c,d)-quasibounded solutions of nonlinear system xΔ=A(t)x+f(t,x) onto those of linear system xΔ=A(t)x. Our results generalize the Palmer's linearization theorem in [K.J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl. 41 (1973) 753-758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191]. Moreover, we prove that equivalent function H(t,x) is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191] never considered this important property of the equivalent function H(t,x).     

Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation u_tt + c(t)u_t + b(t)u = a(t)u_xx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.     

Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u_t=[D(u)u_x]_x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).