Positive solutions for second-order nonlinear differential equations

We prove the existence of positive solutions to the scalar equation y^″(x)+F(x,y,y^′)=0$y^{″}(x)+F(x,y,y^{\prime })=0$. Applications to semilinear elliptic equations in exterior domains are considered.     

Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s

Panovsky and Richardson [A family of implicit Chebyshev methods for the numerical integration of second-order differential equations, J. Comput. Appl. Math. 23 (1988) 35–51] presented a method based on Chebyshev approximations for numerically solving the problem y^″=f(x,y)$y^{″}=f(x,y)$, being the steplength constant. Coleman and Booth [Analysis of a Family of Chebyshev Methods for y^″=f(x,y)$y^{″}=f(x,y)$, J. Comput. Appl. Math. 44 (1992) 95–114] made an analysis of the above method and suggested the convenience to design a variable steplength implementation. As far as we know this goal has not been achieved until now. Later on we extended the above method (this journal, 2003), and obtained a scheme for numerically solving the equation y^″-2gy^′+(g²+w²)=f(x,y)$y^{″}-2{\mathit{gy}}^{\prime }+(g^2+w^2)=f(x,y)$. The question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper.     

Regularity results for degenerate elliptic systems

We prove regularity results for certain degenerate quasilinear elliptic systems with coefficients which depend on two different weights. By using Sobolev- and Poincaré inequalities due to Chanillo and Wheeden [S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985) 1191–1226; S. Chanillo, R.L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986) 1111–1134] we derive a new weak Harnack inequality and adapt an idea due to L. Caffarelli [L.A. Caffarelli, Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35 (1982) 833–838] to prove a priori estimates for bounded weak solutions. For example we show that every bounded weak solution of the system −Dα(aαβ(x,u,∇u)Dβui)=0$-D_{\alpha }(a^{\alpha \beta }(x,u,\mathrm{\nabla }u)D_{\beta }{u}^i)=0$ with |x|²|ξ|²?aαβξαξβ?τ|x||ξ|²${|x|}^2{|\xi |}^2?a^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }?{|x|}^{\tau }{|\xi |}^2$, |x|<1$|x|<1$, τ∈(1,2)$\tau \in (1,2)$ is Hölder continuous. Furthermore we derive a Liouville theorem for entire solutions of the above systems.     

The space of 4-ended solutions to the Allen–Cahn equation in the plane

We are interested in entire solutions of the Allen–Cahn equation Δu−F^′(u)=0$\mathrm{\Delta }u-F^{\prime }(u)=0$ which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4-ended solutions  . The main result of our paper states that, for any θ∈(0,π/2)$\theta \in (0,\pi /2)$, there exists a 4-ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ  , π−θ$\pi -\theta$, π+θ$\pi +\theta$ and 2π−θ$2\pi -\theta$ with the x-axis. This paper is part of a program whose aim is to classify all 2k  -ended solutions of the Allen–Cahn equation in dimension 2, for k?2$k?2$.     

Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term

In this paper, the boundedness of all solutions of the oscillator

x^″+f(x,x^′)+ω²x+?(x)=p(t)$x^{″}+f(x,x^{\prime })+{\omega }^{2}x+?(x)=p(t)$

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

Finite time blow-up in nonlinear suspension bridge models

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w⁽⁴⁾+kw^″+f(w)=0$w^{(4)}+k{w}^{″}+f(w)=0$ which arises in models of traveling waves in suspension bridges when k>0$k>0$. Under suitable assumptions on the nonlinearity f   and initial data, we demonstrate blow-up in finite time. The case k≤0$k\le 0$ was first investigated by Gazzola et al., and it is also handled here with a proof that requires less differentiability on f. Our approach is inspired by Gazzola et al. and exhibits the oscillatory mechanism underlying the finite-time blow-up. This blow-up is nonmonotone, with solutions oscillating to higher amplitudes over shrinking time intervals. In the context of bridge dynamics this phenomenon appears to be a consequence of mutually-amplifying interactions between vertical displacements and torsional oscillations.     

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (_t0,_x0)∈R⁺×S¹$(t_0,x_0)\in {\mathbb{R}}^{+}\times {\mathbb{S}}^1$ is the identically zero solution. Such conclusion holds provided the physical parameter γ   of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa–Holm equation, corresponding to γ=1$\gamma =1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.     

Existence and non-existence of positive solutions of BVPs for fractional order elastic beam equations with a non-Caratheodory nonlinearity

In this article, the existence and non-existence results on positive solutions of two classes of boundary value problems for nonlinear singular fractional order elastic beam equations is established. Here f depends on x  x^′$x^{\prime }$ and x^″,f$x^{″}\text{,}f$ may be singular at t=0$t=0$ and t=1$t=1$ and f may be a non-Caratheodory function. The analysis relies on the well known Schauder’s fixed point theorem. By applying iterative techniques, results on the existence of positive solutions are obtained and the iterative scheme which starts off with zero function for approximating the solution is established. The iterative scheme obtained is very useful and feasible for computational purpose. Examples and their numerical simulation are presented to illustrate the main theorems. A conclusion section is also given at the end of this paper.