Exponential stability of classical solutions to a class of semilinear wave equations with boundary dissipation

We consider the Cauchy–Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case.

Download Full-text

Regular generalized solutions to semilinear wave equations

Monatshefte für Mathematik ◽

10.1007/s00605-020-01470-z ◽

2020 ◽

Author(s):

Hideo Deguchi ◽

Michael Oberguggenberger

Keyword(s):

Wave Equations ◽

Uniqueness Result ◽

Generalized Functions ◽

Generalized Solutions ◽

Classical Solutions ◽

Colombeau Algebra ◽

Semilinear Wave Equations ◽

Global Classical Solutions ◽

Colombeau Algebras ◽

Algebras Of Generalized Functions

Abstract The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

Download Full-text

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations

New Trends in the Theory of Hyperbolic Equations - Operator Theory: Advances and Applications ◽

10.1007/3-7643-7386-5_2 ◽

2006 ◽

pp. 113-211 ◽

Cited By ~ 4

Author(s):

Hideo Kubo ◽

Masahito Ohta

Keyword(s):

Wave Equations ◽

Coupled Systems ◽

Classical Solutions ◽

Global Behavior ◽

Semilinear Wave Equations

Download Full-text

Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Download Full-text