DECAY PROPERTY OF REGULARITY-LOSS TYPE FOR DISSIPATIVE TIMOSHENKO SYSTEM

2008 ◽  
Vol 18 (05) ◽  
pp. 647-667 ◽  
Author(s):  
KENTARO IDE ◽  
KAZUO HARAMOTO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Decay Property ◽  
General Situation ◽  
Decay Estimates ◽  
Timoshenko System ◽  
Linear Diffusion ◽  
Diffusion Wave ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.

DECAY PROPERTY FOR THE TIMOSHENKO SYSTEM WITH FOURIER'S TYPE HEAT CONDUCTION

2014 ◽  
Vol 11 (01) ◽  
pp. 135-157 ◽  
Author(s):  
NAOFUMI MORI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Heat Conduction ◽  
Energy Method ◽  
Decay Estimates ◽  
Fourier Space ◽  
Pointwise Estimates ◽  
Timoshenko System ◽  
One Dimensional ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide–Haramoto–Kawashima. Moreover, we establish optimal L2decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide–Haramoto–Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.

Decay property of the Timoshenko–Cattaneo system

2016 ◽  
Vol 14 (03) ◽  
pp. 393-413 ◽  
Author(s):  
Naofumi Mori ◽  
Shuichi Kawashima
Keyword(s):  
Heat Conduction ◽  
Decay Estimate ◽  
Low Frequency ◽  
Pointwise Estimate ◽  
Frequency Region ◽  
Decay Property ◽  
General Situation ◽  
High Frequency Region ◽  
Timoshenko System ◽  
Whole Space

We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal [Formula: see text] decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko–Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko–Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135–157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko–Cattaneo system.

scholarly journals On $ L^1 $ estimates of solutions of compressible viscoelastic system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yusuke Ishigaki
Keyword(s):  
Initial Data ◽  
Large Time ◽  
Three Dimensional ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Viscoelastic System ◽  
Diffusion Wave ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
Wave Phenomena

<p style='text-indent:20px;'>We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to <inline-formula><tex-math id="M2">\begin{document}$ W^{2,1} $\end{document}</tex-math></inline-formula>, and is sufficiently small in <inline-formula><tex-math id="M3">\begin{document}$ H^4\cap L^1 $\end{document}</tex-math></inline-formula>, the solutions grow in time at the same rate as <inline-formula><tex-math id="M4">\begin{document}$ t^{\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.</p>

DECAY PROPERTY OF REGULARITY-LOSS TYPE AND NONLINEAR EFFECTS FOR DISSIPATIVE TIMOSHENKO SYSTEM

2008 ◽  
Vol 18 (07) ◽  
pp. 1001-1025 ◽  
Author(s):  
KENTARO IDE ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Nonlinear Effects ◽  
Decay Property ◽  
Regularity Conditions ◽  
Initial Value ◽  
Timoshenko System ◽  
Linear Diffusion ◽  
Diffusion Wave ◽  
Decay Of Solutions ◽  
Nonlinear Version ◽  
Derivatives Of

We consider the initial value problem for a nonlinear version of the dissipative Timoshenko system. This syetem verifies the decay property of regularity-loss type. To overcome this difficulty caused by the regularity-loss property, we employ the time weighed L2energy method which is combined with the optimal L2decay estimates for lower order derivatives of solutions. Then we show the global existence and asymptotic decay of solutions under smallness and enough regularity conditions on the initial data. Moreover, we show that the solution approaches the linear diffusion wave expressed in terms of the superposition of the heat kernels as time tends to infinity.

DECAY ESTIMATES OF SOLUTIONS TO A SEMI-LINEAR DISSIPATIVE PLATE EQUATION

2010 ◽  
Vol 07 (03) ◽  
pp. 471-501 ◽  
Author(s):  
YOUSUKE SUGITANI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Linear Problem ◽  
Dimensional Space ◽  
Weighted Sobolev Spaces ◽  
Decay Estimates ◽  
Plate Equation ◽  
Estimates Of Solutions ◽  
Optimal Decay ◽  
Additional Regularity

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

The asymptotic behavior of the Bresse–Cattaneo system

2016 ◽  
Vol 18 (04) ◽  
pp. 1550045 ◽  
Author(s):  
Belkacem Said-Houari ◽  
Taklit Hamadouche
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Decay Rate ◽  
Stability Number ◽  
Timoshenko System ◽  
Decay Properties ◽  
Bresse System ◽  
Whole Space ◽  
The Stability ◽  
The One

In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the [Formula: see text]-norm of the solution decays with the rate of [Formula: see text]. The behavior of solutions depends on a certain number [Formula: see text] (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations 255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations 253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].

Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Diffusion Equation ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
The Other ◽  
Diffusion Wave ◽  
One Dimensional ◽  
The One ◽  
Propagation Velocities

In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order β,1≤β≤2 are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional diffusion-wave equation is nonrelativistic like the classical diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other.

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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