The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative

2006 ◽  
Vol 42 (5) ◽  
pp. 638-649 ◽  
Author(s):  
A. A. Voroshilov ◽  
A. A. Kilbas
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Partial Derivative ◽  
Diffusion Wave ◽  
The Cauchy Problem

Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Rudolf Gorenflo ◽  
Yuri Luchko ◽  
Mirjana Stojanović
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Fundamental Solution ◽  
Probability Density ◽  
Monotone Functions ◽  
Diffusion Wave ◽  
Completely Monotone ◽  
The Cauchy Problem ◽  
Fourier And Laplace Transforms ◽  
Distributed Order

AbstractIn this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation $\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }} {{\partial x^2 }}u(x,t) $ is considered. Here, the time-fractional derivative D tβ is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0, 2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used.

BLOW-UP OF THE SOLUTION FOR SEMILINEAR DAMPED WAVE EQUATION WITH TIME-DEPENDENT DAMPING

2012 ◽  
Vol 14 (05) ◽  
pp. 1250034
Author(s):  
JIAYUN LIN ◽  
JIAN ZHAI
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Upper Bound ◽  
Blow Up ◽  
Time Dependent ◽  
Damped Wave Equation ◽  
Large Initial Data ◽  
Power Type ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped wave equation with time-dependent damping and a power-type nonlinearity |u|ρ. For some large initial data, we will show that the solution to the damped wave equation will blow up within a finite time. Moreover, we can show the upper bound of the life-span of the solution.

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

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