Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

Triple Solutions for Second-Order Three-Point Boundary Value Problems

We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u″ + f(t, u) = 0, u(0) = 0, αu(η) = u(1), where η: 0 lt; η < 1, 0 < α < 1/η, and f: [0, 1] × [0, ∞) → [0, ∞) is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.     

A nonlinear parabolic problem from combustion theory: attractors and stability

A parabolic (convection-diffusion) problem in a half-line, arising when modeling the temperature profile of an adiabatic solid in radiation-driven combustion, is considered. Both the coefficient in the “convective” term (the velocity of the burning front) and the Neumann datum (the prescribed heat influx into the burning body) are nonlinearly related to the proper value of the solution at the boundary. In addition, the velocity is allowed to vanish below some threshold value. Under the main assumptions of “intensely irradiated boundary” and initial data that behave suitably as x→-∞, it is proven that there exists a global attractor for the evolution semigroup associated with the problem. Furthermore, the stabilization of all solutions towards the equilibrium solution (a uniformly propagating front) is derived for a class of Neumann data, which are of some interest for applications.     

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A  , in a two-dimensional domain with L²$L^2$ cells. For fixed A  , and L→∞$L\to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when L   is fixed, and A→∞$A\to \infty$. In this case the solution equilibrates along stream lines.     

Grow-up rate of solutions for a supercritical semilinear diffusion equation

We study solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t→∞. We show that the grow-up rate of such solutions depends on the spatial decay rate of initial data.     

Asymptotic behavior for differential equations which cannot be locally linearized

For functions ? which are continuous and locally Lipschitz the authors define a multi-valued differential D_f and prove that if all solutions of the multi-valued differential equation u′ ? D_f(u) approach zero as t → ∞, then all solutions x(·) of x′ = ?(x) with small |x(0)| approach zero exponentially as t → ∞. If ? is continuously differentiable, then D_f coincides with the (single-valued) Frechet differential of ?. Other results on the asymptotic behavior of solutions of perturbed, multi-valued differential equations are presented.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations

With the aid of symbolic computation Maple, an extended Jacobi elliptic function expansion method is presented and successfully applied to variant Boussinesq equations. As a result, abundant periodic wave solutions in terms of the Jacobi elliptic functions are obtained. When the modulus m → 1 or m → 0, exact solitary wave solutions and trigonometric function solutions are also derived. The properties of four new solutions are graphically studied.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator

This paper initiates the investigation of nonlinear integral equations with Erdélyi-Kober fractional operator. Existence and uniqueness results of solutions in a closed ball are obtained by using a directly computational method and Schauder fixed point theorem via a weakly singular integral inequality due to Ma and Pec?ari? [20]. Meanwhile, three certain solutions sets Y_K,σ, Y_1,λ and Y_1,1, which tending to zero at an appropriate rate t^−ν, 0 < ν = σ (or λ or 1) as t → +∞, are constructed and local stability results of solutions are obtained based on these sets respectively under some suitable conditions. Two examples are given to illustrate the results.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Optimal lower bound of the grow-up rate for a supercritical parabolic equation

We study the behavior of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t→∞. It is known that the grow-up rate of such solutions depends on the spatial decay rate of initial data. We give an optimal lower bound on the grow-up rate by using a comparison technique based on a formal asymptotic analysis.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

On a nonlinear elliptic eigenvalue problem

Let N(λ) be the number of the solutions of the equation: , where Ω is a bounded domain in with smooth boundary. Under suitable conditions on f, we proved that N(λ)→+∞ as λ→+∞.     

Boundary blow-up for differential equations with indefinite weight

We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0⁺ and t→1⁻. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1].     

On ground state solutions for singular and semi-linear problems including super-linear terms at infinity

We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈R^N, where f:R^N×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used.