On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity

DOI10.1007/BF01445103zbMath0788.35001OpenAlexW1985938944MaRDI QIDQ1318107

Hartmut Pecher, Takayuki Kobayashi, Yoshihiro Shibata

Publication date: 6 June 1994

Published in: Mathematische Annalen (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/165083

zbMATH Keywords

initial-boundary value problemsmooth solutionsnonlinear wave equationnonlinear viscoelasticitylocal in time solvabilityglobal in time existencesmooth initial datalinear viscosityDirichlet zero conditionmotion of a fixed membrane with viscosity

Mathematics Subject Classification ID

Initial-boundary value problems for second-order hyperbolic equations (35L20) Second-order nonlinear hyperbolic equations (35L70) Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) (74D99)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain
• Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids
• Global existence of small solutions to nonlinear evolution equations
• Nonlinear second order equations with applications to partial differential equations
• On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain
• On global regular solutions of third order partial differential equations
• The initial value problem for the equations of motion of viscous and heat-conductive gases
• Some remarks on the equation \(y_{tt}-\sigma (y_x)y_{xx}-y_{x^tx}=f\)
• Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity
• Long-time behavior of solutions to nonlinear evolution equations
• Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity
• Global existence and exponential stability of small solutions to nonlinear viscoelasticity
• Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation
• Systems of nonlinear wave equations with nonlinear viscosity
• Existence and nonexistence in the large of solutions of quasilinear wave equations
• 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
• Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
• Global existence of small solutions to a class of nonlinear evolution equations
• Global solutions of nonlinear hyperbolic equations for small initial data
• Uniform decay estimates and the lorentz invariance of the classical wave equation
• On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems
• Formation of singularities for wave equations including the nonlinear vibrating string
• On some nonlinear hyperbolic systems with damping boundary conditions
• Formation of singularities in one-dimensional nonlinear wave propagation
• On the Strongly Damped Wave Equation: $u_{tt} - \Delta u - \Delta u_t + f(u) = 0$
• Global, small amplitude solutions to nonlinear evolution equations
• Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations
• On the existence of solutions to the equation \(u_{tt}=u_{xxt}+\sigma (u_x)_x\)