Remarks on continuous dependence for semilinear wave equation with weak damping / Han, Jiangbo (CC BY)

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Title:: Advances in nonlinear analysis

Publication:: Berlin Boston, Mass.: de Gruyter

Note:: Gesehen am 18.03.2019; Fortsetzung der Druck-Ausgabe; Open Access; Namensnennung 4.0 International

Scope:: Online-Ressource

ISSN:: 2191-950X

ZDB-ID:: 2645915-2

VÖBB-Katalog:: 15630346

Keywords:: Zeitschrift

Classification:: Mathematik

Collection:: Mathematik

Copyright:: Rights reserved

Accessibility:: Free Access

Title:: Advances in nonlinear analysis

Publication:: Berlin Boston, Mass.: de Gruyter

Note:: Gesehen am 18.03.2019; Fortsetzung der Druck-Ausgabe; Open Access; Namensnennung 4.0 International

Scope:: Online-Ressource

ISSN:: 2191-950X

ZDB-ID:: 2645915-2

VÖBB-Katalog:: 15630346

Keywords:: Zeitschrift

Classification:: Mathematik

Collection:: Mathematik

Copyright:: Rights reserved

Accessibility:: Free Access

Article

Author:: Han, Jiangbo; Li, Hong

Title:: Remarks on continuous dependence for semilinear wave equation with weak damping

Publication:: Berlin Boston, Mass.: de Gruyter, 2026

Language:: English

Information:: Abstract: This article provides a rigorous supplement to the work [J. B. Han, K. Y. Wang, R. Z. Xu, and C. Yang, Global quantitative stability of wave equations with strong and weak dampings, J. Differential Equations 390 (2024), 228–344] on the continuous dependence for a semilinear wave equation with weak damping. We consider the general case where the coefficient of the nonlinear source term is real and provide the corresponding analysis and results, which include the special case where this coefficient is unity. In the absence of strong damping, several key estimates and energy identities in the reference were derived through formal calculations, and their validity was uncertain due to the limited regularity of the time derivatives of the solutions. In this article, without imposing additional regularity on the initial data, we supply complete and rigorous justifications for these steps. More precisely, differentiating the energy with respect to time to obtain energy decay estimates and estimating the distance between the solutions (and its time derivative) to improve continuous dependence conclusions via energy decay both involve taking the inner product of both sides of the equations with the time derivatives of the solutions, an operation whose results are merely formal without strong damping. We rigorously justify the results of this operation. For the integral inequality needed to derive the energy decay estimate, we provide a complete construction when the energy vanishes at some time, filling an analytical gap. We further relax the exponent restriction on the nonlinear weak damping and source term, yielding continuous dependence results with wider applicability in the absence of strong damping.

Scope:: Online-Ressource

Note:: Open Access; Archivierung/Langzeitarchivierung gewährleistet

Keywords:: continuous dependence ; stability ; wave equation ; weak damping ; 35L05 ; 35B30

Classification:: Mathematik

URN:: urn:nbn:de:101:1-2609190348568.323678405106

Collection:: Mathematik

Copyright:: CC BY

Accessibility:: Free Access

Author:: Han, Jiangbo; Li, Hong

Title:: Remarks on continuous dependence for semilinear wave equation with weak damping

Publication:: Berlin Boston, Mass.: de Gruyter, 2026

Language:: English

Information:: Abstract: This article provides a rigorous supplement to the work [J. B. Han, K. Y. Wang, R. Z. Xu, and C. Yang, Global quantitative stability of wave equations with strong and weak dampings, J. Differential Equations 390 (2024), 228–344] on the continuous dependence for a semilinear wave equation with weak damping. We consider the general case where the coefficient of the nonlinear source term is real and provide the corresponding analysis and results, which include the special case where this coefficient is unity. In the absence of strong damping, several key estimates and energy identities in the reference were derived through formal calculations, and their validity was uncertain due to the limited regularity of the time derivatives of the solutions. In this article, without imposing additional regularity on the initial data, we supply complete and rigorous justifications for these steps. More precisely, differentiating the energy with respect to time to obtain energy decay estimates and estimating the distance between the solutions (and its time derivative) to improve continuous dependence conclusions via energy decay both involve taking the inner product of both sides of the equations with the time derivatives of the solutions, an operation whose results are merely formal without strong damping. We rigorously justify the results of this operation. For the integral inequality needed to derive the energy decay estimate, we provide a complete construction when the energy vanishes at some time, filling an analytical gap. We further relax the exponent restriction on the nonlinear weak damping and source term, yielding continuous dependence results with wider applicability in the absence of strong damping.

Scope:: Online-Ressource

Note:: Open Access; Archivierung/Langzeitarchivierung gewährleistet

Keywords:: continuous dependence ; stability ; wave equation ; weak damping ; 35L05 ; 35B30

Classification:: Mathematik

URN:: urn:nbn:de:101:1-2609190348568.323678405106

Collection:: Mathematik

Copyright:: CC BY

Accessibility:: Free Access

Contents

Table of contents

Advances in nonlinear analysis (Rights reserved)
Strong instability of standing waves for the Hartree equation with a constant magnetic field / Mao, Weifeng (CC BY)
Qualitative properties of solutions to a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films / Duojie, Cairang (CC BY)
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Sharp threshold for global existence and blowup in the focusing nonlinear Schrödinger equation with inverse-square potential / Lin, Qiang (CC BY)
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Critical mass for a no-flux-Dirichlet chemotaxis model with indirect signal production mechanism / Yang, Lan (CC BY)
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Blow-up criteria for compressible viscoelastic flows with general pressure laws / Wang, Yong (CC BY)
The existence and asymptotic behavior of large solutions to the k -Hessian equations with nonlinear gradient terms / Wan, Haitao (CC BY)
Solitary wave solutions of the delayed KP–BBM equation / Xia, Yonghui (CC BY)
Periodic solutions to Mckean–Vlasov SDEs under Lyapunov conditions / Ma, Jun (CC BY)
Solutions of stationary McKean–Vlasov equation on a high-dimensional sphere and other Riemannian manifolds / Shalova, Anna (CC BY)
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Bilinear decompositions for products of local Hardy spaces and inhomogeneous Lipschitz spaces on spaces of homogeneous type / Wang, Fan (CC BY)
Compactness results for prescribed Webster scalar curvature problem on CR sphere / Li, Yan (CC BY)
Infinitely many new solutions for a nonlinear coupled Schrödinger system / Wang, Qingfang (CC BY)
A differential inequality, an invariant set and the blow-up of high-energy solutions of a quasilinear riser equation / Esquivel-Avila, Jorge A. (CC BY)
Blow-up criterion for strong solutions to the two-dimensional compressible two-phase MHD equations with vacuum in a bounded domain / Zhu, Limei (CC BY)
Local existence and finite time blow-up of solutions for a viscoelastic Kirchhoff-like plate equation with logarithmic source term / Li, Junfeng (CC BY)
On various models describing behaviour of thermoviscoelastic rate-type fluids / Bulíček, Miroslav (CC BY)
The ideal flow for planar closed curves with local length constraint / Okabe, Shinya (CC BY)
Propagation pattern for a three-species competition system in shifting patchy environments / Tan, Qian-Qian (CC BY)
On a weakly coupled time-harmonic Maxwell system in R 3 ${\mathbb{R}}^{3}$ / Wen, Yanyun (CC BY)
Wellposedness and dynamics of two types of reaction–nonlocal diffusion systems under the inhomogeneous spectral fractional Laplacian / Yuan, Pu (CC BY)
A priori estimates for free boundary fluid-structure interaction models of magnetohydrodynamic flows with diffusion / Shen, Lin (CC BY)
Global solutions to the Navier–Stokes system with non-decaying forces in Besov spaces / Takeuchi, Taiki (CC BY)
Weak solutions for anisotropic parabolic equations with low-integrability initial data / Li, Zhongqing (CC BY)
Normalized solutions for the Choquard equation with potential and combined nonlinearities / Xing, Wenjun (CC BY)
Chemotaxis-consumption system with Robin boundary conditions coupled to the (Navier–) Stokes equations / Kim, Dongkwang (CC BY)
Connections between coupling and Ishii-Lions methods for tug-of-war with noise stochastic games / Anttila, Riku (CC BY)
Stability and exponential decay for two-dimensional Boussinesq equations with partial viscosity and thermal diffusion / Yu, Tianyuan (CC BY)
Boundedness and finite-time blow-up for the chemotaxis system with indirect signal production and flux limitation / Tu, Xinyu (CC BY)
Bifurcations and vertical distribution in a nonlocal phytoplankton-nutrient-detritus model / Yan, Xiao (CC BY)
Blow-up analysis for parabolic equations and systems via an auxiliary functional approach / Chen, Shaohua (CC BY)
The method of moving planes for nonlocal uniformly fractional parabolic equations in a half space / Tang, Yanjuan (CC BY)
Traveling waves for bistable reaction–diffusion–convection equations with discontinuous density-dependent coefficients / Drábek, Pavel (CC BY)
Global existence and blow-up for a mixed local and nonlocal parabolic equation with logarithmic nonlinearity / Liu, Dengming (CC BY)
Stationary flows for viscous heat-conductive fluid in a perturbed half-space / Li, Mingjie (CC BY)
Locally constrained flows and Michael–Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ / Cui, Jingshi (CC BY)
Capacity and perimeter from logarithmic fractional Sobolev spaces on Carnot groups / Zhao, Nan (CC BY)
Existence and uniqueness for double-phase Poisson equations with variable growth / Khamsi, Mohamed A. (CC BY)
Globally asymptotical stability on delay reaction–diffusion equation involving nonlinear density-dependent mortality / Huang, Chuangxia (CC BY)
Remarks on continuous dependence for semilinear wave equation with weak damping / Han, Jiangbo (CC BY)
Linear asymptotic stability of a free boundary problem modeling tumor growth with two-phase Stokes equations / Wu, Junde (CC BY)
Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system / Wang, Guotao (CC BY)
Normalized solutions for NLS equations with general nonlinearity on compact metric graphs / Zhang, Peng (CC BY)
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Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space / Gupta, Diksha (CC BY)
Generalized quasi-linear fractional Wentzell problems / Mesino-Espinosa, Efren (CC BY)
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Local well-posedness for the two-component Benjamin-Ono equation / Zhao, Min (CC BY)
Optimal large time behavior of the 3D rate type viscoelastic fluids / Chen, Yangyang (CC BY)
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Existence and multiplicity of solutions for a class of superlinear p -Laplacian equations / Zhao, Tai-Jin (CC BY)
Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system / Hou, Wenhe (CC BY)
Normal forms of piecewise-smooth monodromic systems / Li, Jiahao (CC BY)
Fractional Dirichlet problems with singular and non-locally convective reaction / Gambera, Laura (CC BY)
Sharp forced waves of degenerate diffusion equations in shifting environments / Mei, Ming (CC BY)
On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations / Cao, Yue (CC BY)
Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p -Laplacian / Liu, Mengru (CC BY)
Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities / Yan, Dongze (CC BY)
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Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics / Hu, Yanbo (CC BY)
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One-dimensional boundary blow up problem with a nonlocal term / Inaba, Taketo (CC BY)
General solutions of weakly delayed discrete systems in 3D / Diblík, Josef (CC BY)
Shape of extremal functions for weighted Sobolev-type inequalities / Brock, Friedemann (CC BY)
Homoclinic solutions in periodic partial difference equations / Mei, Peng (CC BY)
Concentrating solutions for double critical fractional Schrödinger-Poisson system with p -Laplacian in ℝ 3 / Liang, Shuaishuai (CC BY)
Existence results for non-coercive problems / Diblík, Josef (CC BY)
Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed / Liu, Mengqian (CC BY)
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Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator / Yang, Zerong (CC BY)
Geometric characterization of generalized Hajłasz-Sobolev embedding domains / Li, Ziwei (CC BY)
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Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space / Gao, Dongmei (CC BY)
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Subharmonic solutions of first-order Hamiltonian systems / Zhou, Yuting (CC BY)
Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces / Li, Haizhong (CC BY)
Properties of minimizers for L 2 -subcritical Kirchhoff energy functionals / Guo, Helin (CC BY)
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Incompressible limit for the compressible viscoelastic fluids in critical space / Han, Bin (CC BY)
Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities / Liu, Yuan-Hao (CC BY)
Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections / Cen, Jinxia (CC BY)
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Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth / Yang, Jie (CC BY)
Bifurcation and multiplicity results for critical problems involving the p -Grushin operator / Malanchini, Paolo (CC BY)
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Sharp viscous shock waves for relaxation model with degeneracy / Xu, Shufang (CC BY)
Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities / Guo, Zhenhua (CC BY)
Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds / Chen, Wenjing (CC BY)
Global dynamics of population-toxicant models with nonlocal dispersals / Ma, Li (CC BY)
Singularity for the macroscopic production model with Chaplygin gas / Liu, Xiaoli (CC BY)
Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting / Li, Yanglei (CC BY)
Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity / Zhang, Nangao (CC BY)
Existence of positive radial solutions of general quasilinear elliptic systems / Devine, Daniel (CC BY)
Remark on the analyticity of the fractional Fokker-Planck equation / Xu, Yan (CC BY)
Continuous dependence on initial data for damped fourth-order wave equation with strain term / Han, Jiangbo (CC BY)
Boundedness of solutions to quasilinear elliptic systems / Mi, Fang (CC BY)
Existence of positive solutions for critical p -Laplacian equation with critical Neumann boundary condition / Deng, Yinbin (CC BY)
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Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals / Bacho, Aras (CC BY)
Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms / Gao, Yanchao (CC BY)
Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians / Mao, Anmin (CC BY)
Unilateral problems for quasilinear operators with fractional Riesz gradients / Campos, Pedro Miguel (CC BY)
High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness / Guan, Wen (CC BY)
α -Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets / Kang, Hyunsuk (CC BY)
On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation / Wang, Yinxia (CC BY)
Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping / Li, Haitong (CC BY)
On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations / Chlebowicz, Agnieszka (CC BY)
Combinatorial p th Calabi flows for total geodesic curvatures in spherical background geometry / Liu, Bin (CC BY)
Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent / Zhang, Jing (CC BY)
Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent / Zheng, Tian-Tian (CC BY)
Modified quasilinear equations with strongly singular and critical exponential nonlinearity / Biswas, Reshmi (CC BY)
Existence and regularity for a p -Laplacian problem in ℝ N with singular, convective, and critical reaction / Baldelli, Laura (CC BY)
Stability on 3D Boussinesq system with mixed partial dissipation / Lin, Hongxia (CC BY)
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Nontrivial solutions for resonance quasilinear elliptic systems / Borgia, Natalino (CC BY)
Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities / Lin, Xiaoyan (CC BY)
Global existence and finite-time blowup for a mixed pseudo-parabolic r (x)-Laplacian equation / Cheng, Jiazhuo (CC BY)
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