Helmholtz-type solitary wave solutions in nonlinear elastodynamics

The propagation of localised (in space-time) waves is analysed in the context of the dynamic theory of incompressible hyperelastic solids subject to body forces corresponding to a dual power-law substrate potential. A broad class of exact solutions is obtained which, under the assumption of slow modulation, incorporates Helmholtz-type solitary waves. The linear stability of these solutions is studied under the assumption that the speed of propagation of the wave is small enough compared to the speed at which transverse waves travel in the linear regime and in the absence of external actions.     

Some stability theorems for an abstract equation in Hilbert space with applications to linear elastodynamics

Previous results of Knops and Payne concerning continuous data dependence in linear elastodynamics are extended to include the case where the elasticities may be time dependent. Our results, which include stability under perturbations of both the elasticities and the initial geometry, are obtained by applying a logarithmic convexity argument to the Cauchy problem associated with an abstract differential equation in Hilbert space.     

Blow-up of solutions to nonlinear wave equations in two space dimensions

The present paper studies the blow-up of solutions to nonlinear wave equations whose nonlinear terms are proposed by F. John. We shall show that the solutions to the equations in two space dimensions blow up at finite time if the power in nonlinear term is equal to or smaller than three. Our basic idea is to use the fundamental identity for the iterated spherical means.     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

Integral bounds for solutions of nonlinear reaction-diffusion equations

Differential inequality techniques are used to obtain upper bounds for theL ¹ norm of solutions of nonlinear reaction-diffusion equations. Essential use is made of Sobolev type integral inequalities. An extension to third order pseudo-parabolic equations is included.

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Résumé On utilise des techniques d'inégalités differentielles afin d'obtenir des bornes supérieures pour les normes de typeL ¹ des solutions des équations des réaction et diffusion non linéaires. On utilise de façon essentielle des inégalités intégrales du type de Sobolev. On inclut une extension aux équations pseudo-paraboliques du troisième ordre.

     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Decay of solutions of nonlinear wave equations in three space dimensions

Given a solution of the Cauchy problem for nonlinear wave equations of the type $\text{?}{}^2\text{u}\text{?t}{}^2\text{? Δu + f(u) = 0}$ in three space dimensions the asymptotic behaviour in time is considered. It is shown that for nonlinearities which behave like powers $\text{¦u¦}{}^{\mathrm{\sigma \; ?\; 1}}\text{u}$ uniform decay holds with a certain rate depending on σ if $\text{5 > σ >}\text{1}\text{2}\text{+}\text{1}\text{2}\text{√13}$, and moreover scattering states exist if σ is not too small. This improves former results of W. A. Strauss (J. Funct. Anal. 2 (1968), 409–457).     

On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

Stable approximation of solutions to irregular nonlinear operator equations in a Hilbert space under large noise

The class of regularized Gauss-Newton methods for solving inexactly specified irregular nonlinear equations is examined under the condition that additive perturbations of the operator in the problem are close to zero only in the weak topology. By analogy with the well-understood conventional situation where the perturbed and exact operators are close in norm, a stopping criterion is constructed ensuring that the approximate solution is adequate to the errors in the operator.     

Multidimensional nonlinear wave equations with multivalued solutions

We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.