Decay of global solutions of two nonlinear evolution equations

1.IntroductionTherehavebeenconsiderableliteratuxeonthedecayofsolutionstothebestialvalueproblemsforsomenonlinearevolutionequations[3,4,6,7,161.Undercertainassumptions,LZdecayandLoodecayofsolutionstotheseproblemswereestablished.Thereadersinterestedcanfindsuchworksinourreferences.OurillterestisfocusedonthedecayofsolutionsoftheinitialvalueproblemsfornonlinearBenjamin--OnthBurgers(BOB)l"'19--21]andSchlodinger-Burgers(SB)equationwhereHisHilberttransform,definedbyWewallttoshowthattheLZandLoon…     

Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations

In this paper we derive conditions on the equations and data which are sufficient to guarantee the nonexistence of global weak solutions of Cauchy problems for classes of nonlinear partial differential equations.     

Nonexistence of global solutions of a nonlinear hyperbolic system

Consider the initial value problem

with and . We show that there exists a bound such that if all nontrivial solutions with compact support blow up in finite time.

     

Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

We study the global solvability of the Cauchy-Dirichlet problem for two second order in time nonlinear integro-differential equations:

1)

the extensible beam/plate equation

     

Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity

In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.     

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.     

Analyticity of solutions of nonlinear evolution equations

Analyticity in t of solutions u(t) of nonlinear evolution equations of the form $\text{u′ + A(t, u)u = ?(t, u), t > 0, u(0) = u}{}_0$, is established under suitable conditions on $\text{A(t, u), ?(t, u),}\text{and}\text{u}{}_0$. An application is given to quasilinear parabolic equations.     

非线性发展方程新的周期解

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.     

Anti-periodic solutions of some nonlinear evolution equations

Following a recent work of H. Okochi in the case of evolution equations generated by subdifferential operators in a real Hilbert space, we point out that many quasi-autonomous evolution equations of non monotone type associated to odd non linear operators have some anti-periodic solutions provided the forcing term is anti-periodic. This comes from the fact that the space of anti-periodic functions is transversal to the kernel of the linear part and stable under the action of odd non linear operators. The proofs of our results combine strong a priori estimates which depend very little on the non-linearities with an application of Schauder's fixed point theorem to some related dissipative equations.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

Periodic solutions for doubly nonlinear evolution equations

We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u^′(t))+∂?(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂? in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided.     

Smoothness of bounded solutions of nonlinear evolution equations

It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized.     

Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping

This article concerns the Cauchy problem for the damped nonlinear hyperbolic system where ? > 0 is a small parameter, 0 < α ≤ 1,0 < β ≤ 1,p,q ≥ 1 satisfying pq > 1 , and N ≥ 1 is an integer. It is proved that if N ∕ 2α < max((p + 1) ∕ (pq ? 1),(q + 1) ∕ (pq ? 1)), then every weak solution does not exist globally whenever the initial data satisfy or . Copyright © 2012 John Wiley & Sons, Ltd.     

Stability for solutions to nonlinear nonautonomous evolution equations

In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.