Global existence and exponential stability of small solutions to nonlinear viscoelasticity

DOI10.1007/BF02102372zbMath0779.35066OpenAlexW1990507707MaRDI QIDQ1200508

Shuichi Kawashima, Yoshihiro Shibata

Publication date: 16 January 1993

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf02102372

zbMATH Keywords

exponential stabilitysmooth solutionsDirichlet boundary conditionglobal existencenonlinear wave equationtime periodic solutionsasymptotic exponential decaynonlinear hyperbolic system of second order

Mathematics Subject Classification ID

Related Items (65)

Smoothing effect and large time behavior of solutions to nonlinear elastic wave equations with viscoelastic term ⋮ Some Analytical Issues for Elastic Complex Fluids ⋮ Unnamed Item ⋮ Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping ⋮ MATHEMATICAL ANALYSIS OF A ONE-DIMENSIONAL MODEL FOR AN AGING FLUID ⋮ Energy decay for the quasilinear wave equation with viscosity ⋮ Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity ⋮ Parabolic Approximation of Quasilinear Wave Equations with Applications in Nonlinear Acoustics ⋮ EXISTENCE OF A WEAK SOLUTION IN AN INFINITE VISCOELASTIC STRIP WITH A SEMI-INFINITE CRACK ⋮ Global existence of classical solutions for some Oldroyd-B model via the incompressible limit ⋮ On the control of solutions of a viscoelastic equation ⋮ Arbitrary decays for a viscoelastic equation ⋮ Decay rate estimates for a class of quasilinear hyperbolic equations with damping terms involving p-Laplacian ⋮ The initial boundary value problem for quasi-linear wave equation with viscous damping ⋮ Energy decay rates via convexity for some second-order evolution equation with memory and nonlinear time-dependent dissipation ⋮ Global attractor of the quasi-linear wave equation with strong damping ⋮ Carleman estimate for a strongly damped wave equation and applications to an inverse problem ⋮ Asymptotic behaviour of solutions for \(p\)-Laplacian wave equation with \(m\)-Laplacian dissipation ⋮ Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity ⋮ A high-order discontinuous Galerkin method for nonlinear sound waves ⋮ Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics ⋮ A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation ⋮ Existence and non-existence of global solutions to initial boundary value problems for nonlinear evolution equations with strong dissipation ⋮ On inverse problems for a strongly damped wave equation on compact manifolds ⋮ Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity ⋮ Global existence of solutions for quasi-linear wave equations with viscous damping. ⋮ Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms ⋮ On global existence of solutions of the Neumann problem for spherically symmetric nonlinear viscoelasticity in a ball ⋮ Exponential decay of energy for some nonlinear hyperbolic equations with strong dissipation ⋮ Global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data ⋮ Global non-small data existence of spherically symmetric solutions to nonlinear viscoelasticity in a ball ⋮ Self-adaptive absorbing boundary conditions for quasilinear acoustic wave propagation ⋮ Global solutions for incompressible viscoelastic fluids ⋮ Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain ⋮ A note on ``On the existence of solutions of quasilinear wave equations with viscosity ⋮ The rate at which the energy of solutions for a class of \(p\)-Laplacian wave equation decays ⋮ Well-posedness and attractor on the 2D Kirchhoff-Boussinesq models ⋮ Decay rate of quasilinear wave equation with viscosity ⋮ Global existence of strong solutions to a class of fully nonlinear wave equations with strongly damped terms ⋮ General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms ⋮ Global solutions to quasi-linear hyperbolic systems of viscoelasticity ⋮ Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem ⋮ Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms ⋮ Global nonexistence of solutions for systems of quasilinear hyperbolic equations with damping and source terms ⋮ Cauchy problem for quasi-linear wave equations with viscous damping ⋮ Global existence and asymptotic behavior of solutions for some nonlinear hyperbolic equation ⋮ Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping ⋮ An explicit lower bound for blow up time in a class of nonlinear wave equations with nonlinear damping and source terms ⋮ Time-weighted energy method for quasi-linear hyperbolic systems of viscoelasticity ⋮ Conservative finite difference scheme for the nonlinear fourth-order wave equation ⋮ The long-time stability of solutions for intermittently controlled viscoelastic wave equations with memory terms ⋮ The Work of Yoshihiro Shibata ⋮ Uniform stabilization of solutions of a nonlinear system of viscoelastic equations ⋮ The global attractor for the wave equation with nonlocal strong damping ⋮ On the control of solutions of viscoelastic equations with boundary feedback ⋮ Finite-dimensional attractors for the quasi-linear strongly-damped wave equation ⋮ A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation ⋮ GLOBAL SOLUTIONS AND DECAY PROPERTY WITH REGULARITY-LOSS FOR QUASI-LINEAR HYPERBOLIC SYSTEMS WITH DISSIPATION ⋮ Uniform decay rate estimates for a class of quasi-linear hyperbolic equations with non-linear damping and source terms ⋮ Robust attractors for a Kirchhoff-Boussinesq type equation ⋮ The Global Existence of Small Solutions to the Incompressible Viscoelastic Fluid System in 2 and 3 Space Dimensions ⋮ Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow ⋮ On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity ⋮ Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping ⋮ Asymptotic behavior of an elastic satellite with internal friction

Cites Work

• Unnamed Item
• On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition
• On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficients in Sobolev spaces
• Nonlinear second order equations with applications to partial differential equations
• On global regular solutions of third order partial differential equations
• Some remarks on the equation \(y_{tt}-\sigma (y_x)y_{xx}-y_{x^tx}=f\)
• On the mathematical foundations of elastic stability theory. I
• Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity
• A quasilinear hyperbolic and related third-order equations
• Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation
• Some evolution equations with the quasi-linear strong dissipation
• The global existence of small amplitude solutions to the nonlinear acoustic wave equation
• Systems of nonlinear wave equations with nonlinear viscosity
• On the existence, uniqueness, and stability of solutions of the equation \(p_0 {\mathfrak X}_{tt} = E({\mathfrak X}_ x) {\mathfrak X}_{xx} +\lambda {\mathfrak X}_{xxt}\)
• The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity
• On the Existence and Uniqueness of Solutions of the Equation
• On the Strongly Damped Wave Equation: $u_{tt} - \Delta u - \Delta u_t + f(u) = 0$
• Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations
• On global solutions for mixed problem of a semi-linear differential equation
• On the existence of solutions to the equation \(u_{tt}=u_{xxt}+\sigma (u_x)_x\)

This page was built for publication: Global existence and exponential stability of small solutions to nonlinear viscoelasticity