Solutions Of Semilinear Wave Equations Research Articles

Slowly decaying solutions for semilinear wave equations in odd space dimensions

Slowly decaying solutions for semilinear wave equations in odd space dimensions

Local Solutions of Semilinear Wave Equations in Hs1

The Cauchy problem for the wave equation with power type non-linearity ±∣u∣u and data in Hs+1(ℝn)×Hs(ℝn) is considered, where 0<s<(n/2)−1 and n≥3. Under the growth restriction σ*⩽4/(n−2−2s) in many cases the existence of a local solution with u(t)∈Hs+1(ℝn) is shown which is unique in a closely related class.

Local Solutions of Semilinear Wave Equations inHs+1

The Cauchy problem for the wave equation with power type non-linearity ±|u| σ* u and data in H s+1 (R n ) x H s (R n ) is considered, where 0 < s < (n/2)−1 and n ≥ 3. Under the growth restriction σ * ≤ 4/(n−2−2s) in many cases the existence of a local solution with u(t) ∈ H s+1 (R n ) is shown which is unique in a closely related class.

On the existence of periodic solutions for semilinear wave equation on a ball in [formula omitted]n with the space dimension n odd

On the existence of periodic solutions for semilinear wave equation on a ball in [formula omitted]n with the space dimension n odd

Lower bounds for the life span of solutions of semilinear wave equations with data of non compact support

Lower bounds for the life span of solutions of semilinear wave equations with data of non compact support

Weak and Yet Weaker Solutions of Semilinear Wave Equations

Weak and Yet Weaker Solutions of Semilinear Wave Equations

A mountain pass method for the numerical solution of semilinear wave equations

It is well-known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals. In the situation that we studied, there is usually an obvious solution obtained as a solution of linear problem. We formulate a dual variational problem in such a way that the obvious solution is a local minimum. We then find additional non-obvious solutions via a numerical mountain pass algorithm, based on the theorems of Ambrosetti, Rabinowitz and Ekeland. Numerical results are presented.

Lifespan of solutions of semilinear wave equations in two space dimensions

(1993). Lifespan of solutions of semilinear wave equations in two space dimensions. Communications in Partial Differential Equations: Vol. 18, No. 5-6, pp. 895-916.

On the existence of multiple solutions for semilinear equations with monotone nonlinearities crossing a finite number of eigenvalues

LET 0 be a bounded domain in R” (n 2 1) with smooth boundary. In this paper we study in L,(Q) nonlinear equations of the form Lu = N(u), (1.1) where L is a densely defined closed self-adjoint linear operator with closed range and N is a Nemytskii operator generated by a Caratheodory function (w, s) + g(w, s) from Q x R to lR with g( *, 0) = 0. We are interested in the existence of multiple solutions of (1.1) in the case where N interacts with a finite number of eigenvalues of finite multiplicity of L. It is essential in our study that the kernel of L may be infinite dimensional and that L has a pure point spectrum which may be unbounded both from above and below. Hence the results obtained can be applied to the problem of finding periodic solutions for semilinear wave equations of the form

A note on global strong solutions of semilinear wave equations

A note on global strong solutions of semilinear wave equations

Minimal periods for solutions of semilinear wave equations in exterior domains and for solutions of the equations of nonlinear elasticity

Let g ϵ C 2( R 1) and u( r, t) solve u rr + ( (n − 1) r ) u r − u tt − g(u) = 0 classically on D 0 = (0, ∞) × [0, T] and be periodic in time. If lim r → + ∞r 2(n − 1)E(r) = lim r → ∞ ∥u(r,·)∥ ∞ = 0, where E(r) = ∫ 0 T (u r 2 + u t 2) dt, u ϵ L 1(D r1) for some r 1 > 0, and g′(0) > ( 2π T ) 2, then u is independent of T. Related results are obtained for the equations of nonlinear elasticity.

Doubly-periodic solutions of a forced semilinear wave equation

Existence results are obtained for doubly-periodic solutions of a semilinear wave equation when the nonlinearity is bounded in one side.

REGULARITY ESTIMATE OF SOLUTION TO SEMILINEAR WAVE EQUATION IN HIGHER SPACE DIMENSION

In this paper, the propagation and interaction of singularities of solution to semilinearwave equation in higher space dimension are discussed. An estimate of regularity of solutionshas been obtained. Besides, the goneral conclusion is applied to analyzing the constructureof singularities of solutions when several bicharacteristics bearing singularities intersect witheach other, and to discussing the propagation of singularities when three progressive wavesmeet together.

On solutions of semilinear wave equations

under the first initial-boundary condition and prove the existence of functions which exist for every t 3 0 and solve the equations in a sense defined in this paper and for some classes of equations with the first order dissipation we find an unbounded set in a subspace of [L2(Q)12 with a property that if initial value belongs to it, then the equation has a global classical solution. Let (x, t) be a space-time variable in IR” X [0, 03). Let us consider the following semilinear wave equations

On the Asymptotic Behavior of Solutions of Semi-linear Wave Equations

utt -Au + ut +/(«, ut, Vu) - 0 . We show the asymptotic properties of the solutions of the linear equation [(2) with /*=0] in 2. 1 and those of the nonlinear equation (2) in 2. 2 and 2. 3. Notation. We denote by U the space of measurable functions u on Rn whose ^>-th powers (l<r<Joo) are integrable with the norm 1/p U / f = vu(xy J

Global and Unique Weak Solutions of Nonlinear Wave Equations

While the uniqueness of the energy class solutions of semi-linear wave equations with supercritical nonlinearity remains an open problem, there has been a considerable progress recently in understanding of the critical and subcritical cases (see [Brenner & von Wahl], [Pecher], [Ginibre & Velo 1-3], [Ginibre, Soffer & Velo], [Kapitanski 1, 4-6], [Struwe 1, 2], [Grillakis 1-3], [Shatah & Struwe]). In particular, the following result is proved in [Kapitanski 4, 5]. Let L be a second order hyperbolic differential operator of the form (the sum over the repeated indices is assumed)