Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping

2015 ◽  
Vol 145 (5) ◽  
pp. 959-978 ◽  
Author(s):  
Shifeng Geng ◽  
Lina Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Nonlinear Damping ◽  
Boundary Effects ◽  
Time Behaviour ◽  
Quasilinear Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Quarter Plane

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

CONVERGENCE RATES TO ASYMPTOTIC PROFILE FOR SOLUTIONS OF QUASILINEAR HYPERBOLIC EQUATIONS WITH LINEAR DAMPING

2011 ◽  
Vol 08 (01) ◽  
pp. 115-129 ◽  
Author(s):  
SHIFENG GENG
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Initial Data ◽  
Open Problem ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Asymptotic Profile ◽  
Linear Damping ◽  
Better Than

This paper is concerned with the asymptotic behavior of the solution of quasilinear hyperbolic equations with linear damping. The main novelty lies in the following observation: If we suitably choose the initial data of the corresponding parabolic equation, then the solution Ψ = Ψ(x, t) of the parabolic equation served as the new asymptotic profile satisfies ‖(V-Ψ, (V-Ψ)x, (V-Ψ)t)(t)‖L∞ = O(1)(t-2, t-5/2, t-3). The convergence rates of the new profile Ψ are better than that obtained by H.-J. Zhao (2000, J. Differential Equations167, 467–494), and we need none of the additional technical assumptions (H1) and (H2) therein. Therefore, we answer an open problem posed by Nishihara (1997, J. Differential Equations133, 384–395).

scholarly journals Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hongjun Qiu ◽  
Yinghui Zhang
Keyword(s):  
Low Temperature ◽  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Initial Perturbation ◽  
Strong Solutions ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Quasilinear Hyperbolic Equations ◽  
Global Existence And Uniqueness ◽  
Very Low Temperature

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1≤p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

scholarly journals One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

2021 ◽  
Vol 175 (1) ◽  
Author(s):  
Eugenia Franco ◽  
Mats Gyllenberg ◽  
Odo Diekmann
Keyword(s):  
Mathematical Biology ◽  
Population Dynamics ◽  
Asymptotic Behaviour ◽  
Dimensional Reduction ◽  
Large Time ◽  
Renewal Equation ◽  
Time Behaviour ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Renewal Equations

AbstractDespite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas

1996 ◽  
Vol 126 (6) ◽  
pp. 1277-1296 ◽  
Author(s):  
L. Hsiao ◽  
T. Luo
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Time Behaviour ◽  
Real Gas ◽  
Large Time Behaviour ◽  
One Dimensional ◽  
Viscous Heat

We investigate the large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas. A conclusive answer to the problem of asymptotic behaviour is given in Theorem 1.2.

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