scholarly journals Characterization of polynomial decay rate for the solution of linear evolution equation

2005 ◽  
Vol 56 (4) ◽  
pp. 630-644 ◽  
Author(s):  
Zhuangyi Liu ◽  
Bopeng Rao
Keyword(s):  
Evolution Equation ◽  
Decay Rate ◽  
Polynomial Decay ◽  
Linear Evolution ◽  
Linear Evolution Equation

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Modified dissipativity for a non-linear evolution equation arising in turbulence

1979 ◽  
Vol 71 (3) ◽  
pp. 237-256 ◽  
Author(s):  
C. Bardos ◽  
P. Penel ◽  
U. Frisch ◽  
P. L. Sulem
Keyword(s):  
Evolution Equation ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Non Linear

Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation

2018 ◽  
Vol 25 (3) ◽  
pp. 337-348
Author(s):  
Nana Dikhaminjia ◽  
Jemal Rogava ◽  
Mikheil Tsiklauri
Keyword(s):  
Approximate Solution ◽  
Evolution Equation ◽  
Continuous Operator ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Lipschitz Continuous ◽  
Decomposition Scheme ◽  
Linear Evolution ◽  
Numerical Resolution ◽  
Linear Evolution Equation

AbstractIn the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator represents the sum of positive definite self-adjoint operators. The fourth-order accuracy decomposition scheme is constructed for an approximate solution of the problem. The theorem on the error estimate of an approximate solution is proved. Numerical calculations for different model problems are carried out using the constructed scheme. The obtained numerical results confirm the theoretical conclusions.

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