Series solutions of a semilinear wave equation

We are concerned with the reconstruction of series solutions of a semilinear wave equation with a quadratic nonlinearity. The solution which may blow up in finite time is sought as a sum of exponential functions and is shown to be a classical one. The constructed solutions can be used to benchmark numerical methods used to approximate solutions of nonlinear equations.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

Global solutions and self-similar solutions of semilinear wave equation

We prove existence, uniqueness and regularity results for the global solutions of the semilinear wave equations. In particular, we show existence of regular self-similar solutions. Also, we build some finite-energy asymptotically self-similar solutions. Received: 20 September 1999; in final form: 10 May 2000 / Published online: 25 June 2001     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.     

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Science China Mathematics - We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation...     

Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors Aη, η∈[0,1], corresponding to a singularly perturbed damped wave equation

     

Self-similar solutions of semilinear wave equation with variable speed of propagation

We investigate the issue of existence of the self-similar solutions of the generalized Tricomi equation in the half-space where the equation is hyperbolic. We look for the self-similar solutions via the Cauchy problem. An integral transformation suggested in [K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differential Equations 206 (2004) 227-252] is used to represent solutions of the Cauchy problem for the linear Tricomi-type equation in terms of fundamental solutions of the classical wave equation. This representation allows us to prove decay estimates for the linear Tricomi-type equation with a source term. Obtained in [K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., in press, doi:10.1007/s00033-006-5099-2] estimates for the self-similar solutions of the linear Tricomi-type equation are the key tools to prove existence of the self-similar solutions.     

On positive solutions of a semilinear equation

A semilinear elliptic equation is considered in a domain with smooth boundary. The authors prove the existence and uniqueness of positive solutions of different types, singular at an inner point, subject to the Dirichlet boundary conditions. Bibliography: 13 titles. Translated from Trudy Seminara imeni I., G. Petrovskogo. No. 18, pp. 157–169, 1995.     

Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term

In this paper, we study the existence of positive solutions to the following semilinear elliptic equation with a Sobolev-Hardy term

(0.1)